Research ArticleFinite Element Solution of Unsteady Mixed Convection Flow ofMicropolar Fluid over a Porous Shrinking Sheet

Diksha Gupta Lokendra Kumar and Bani Singh

Department of Mathematics Jaypee Institute of Information Technology A-10 Sector 62 Noida Uttar Pradesh 201307 India

Correspondence should be addressed to Lokendra Kumar lokendmagmailcom

Received 14 August 2013 Accepted 24 October 2013 Published 4 February 2014

Academic Editors A Al-Sarkhi C Bao and A Kosar

Copyright copy 2014 Diksha Gupta et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The objective of this investigation is to analyze the effect of unsteadiness on themixed convection boundary layer flow ofmicropolarfluid over a permeable shrinking sheet in the presence of viscous dissipation At the sheet a variable distribution of suction isassumedThe unsteadiness in the flow and temperature fields is caused by the time dependence of the shrinking velocity and surfacetemperature With the aid of similarity transformations the governing partial differential equations are transformed into a set ofnonlinear ordinary differential equations which are solved numerically using variational finite element method The influence ofimportant physical parameters namely suction parameter unsteadiness parameter buoyancy parameter and Eckert number onthe velocity microrotation and temperature functions is investigated and analyzed with the help of their graphical representationsAdditionally skin friction and the rate of heat transfer have also been computed Under special conditions an exact solution forthe flow velocity is compared with the numerical results obtained by finite element method An excellent agreement is observed forthe two sets of solutions Furthermore to verify the convergence of numerical results calculations are conducted with increasingnumber of elements

1 Introduction

In the last few decades the interest for non-Newtonian fluidshas considerably increased due to their connection withapplied sciences The motion of these fluids plays essentialrole not only in theory but also in many industrial pro-cesses Among the various non-Newtonian fluid models themicropolar fluids have acquired the special attention in recentyears due to their applications in polymeric fabricationmate-rials processing and biotechnology Flow and heat transferbehaviour of these fluids cannot be described by the classicaltheory of continuum mechanics Eringen [1] has formulatedthe theory ofmicropolar fluids which describes the physics ofsuch fluids In the micropolar fluid theory two new variablesto the velocity are added which were not presented in theNavier-Stokes model These variables are microrotations thatrepresent spin and microinertia tensors which describe thedistribution of atoms and molecules inside the microscopicfluid particles This class of fluids represents mathematicallymany industrial important fluids such as paints lubricantspolymers human and animal blood colloidal suspensions

and liquid crystalsThe theory and applications ofmicropolarfluids can be found in the books by Eringen [2] and Beg etal [3] Later Eringen [4] extended the theory of micropolarfluids to thermo-microfluids This theory takes into accountthermal effects that is heat conduction convection anddissipation These effects were not included in the classicalfield theories

The boundary layer flow induced by stretching surface isof great practical interest because it occurs in a number ofengineering processes Boundary layer flow of a Newtonianfluid caused by a linearly stretching sheet was first examinedby Crane [5] He gave a similarity solution in closed analyticalform for the steady two-dimensional problem The flow andheat transfer of micropolar fluid past a continuously movingporous plate was analyzed by Takhar and Soundalgekar [6]Hassanien and Gorla [7] conducted numerical computationsto examine the heat transfer characteristics of micropolarfluid past a nonisothermal sheet with suction and blowingThereafter various aspects of micropolar fluid flows froma stretching surface have been reported by El-Arabawy [8]Nazar et al [9] Kumar [10] Ishak [11] and Rawat et al [12]

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 362351 11 pageshttpdxdoiorg1011552014362351

2 The Scientific World Journal

The above-mentioned studies deal with a steady flowonly However in certain cases the flow and heat transfercan be unsteady due to a sudden stretching of the flatsheet or by a change of the temperature or heat flux ofthe sheet Devi et al [13] investigated the unsteady three-dimensional flow caused by a stretching flat surface Usingthe finite difference scheme in combination with the quasi-linearization technique Rajeswari and Nath [14] reportedon the unsteady flow over a stretching surface in a rotatingfluid Exact similarity solution of the heat transfer in a liquidfilm on an unsteady stretching surface was obtained byAndersson et al [15] Abd El-Aziz [16] studied numericallythe effect of radiation on the heat and fluid flow over anunsteady stretching sheet showing that the heat transferrate is an increasing function of radiation and unsteadinessparameters Lok et al [17] obtained the numerical solution forthe unsteady boundary layer flow of a micropolar fluid nearthe stagnation point of a plane surface By taking into accountstrong and weak concentration of microelements Hayat etal [18] examined the effect of magnetic field on the time-dependent flowofmicropolar fluid between stretching sheetsBachok et al [19] studied theoretically the unsteady boundarylayer flow and heat transfer due to a stretching sheet showingthat the surface shear stress and the heat transfer rate are anincreasing function of unsteadiness parameter

Recently the boundary layer flow of incompressible fluidover a shrinking sheet has attracted extensive attention due toits increasing applications in polymeric materials processingThis type of flow was first examined analytically by Wang[20] Later Miklavcic and Wang [21] proved the existenceand uniqueness of the solution for the flow over a shrinkingsheet From the physical point of view steady flow over ashrinking sheet is not possible since the generated vorticityis not confined within the boundary layer To overcomethis difficulty the flow requires a certain amount of exter-nal opposite force at the sheet This has been extensivelydiscussed in the literature Some important references areMiklavcic and Wang [21] Wang [22] and Bachok et al [23]Closed form exact solutions of MHD Newtonian flow over ashrinking sheet were derived by Fang and Zhang [24] Usingshooting method Bhattacharyya and Layek [25] obtainedthe numerical solutions for the flow and heat transfer over aporous shrinking sheet in the presence of thermal radiationThe shrinking sheet problem was extended to micropolarfluids by Ishak et al [26] who considered stagnation pointflow Yacob and Ishak [27] analyzed the flow and heattransfer over a shrinking sheet immersed in a micropolarfluid Recently numerical simulation of mixed convectionflow of micropolar fluid over a shrinking sheet with thermalradiation was conducted by Gupta et al [28]

Many researchers have investigated the unsteady flowover a shrinking sheet Fang et al [29] examined the unsteadyviscous flow over a shrinking surface with mass suction andhighlighted the deviation in flow behaviour for an unsteadyshrinking sheet compared with an unsteady stretching sheetEffect of radiation on the unsteady flow and heat transferinduced by a permeable shrinking sheetwas reported byAli etal [30] Bhattacharyya [31] examined the effects of radiationand heat sourcesink on unsteady flow and heat transfer past

a shrinking sheet with suctioninjection Numerical solutionfor the unsteady stagnation point flow and heat transferover a stretchingshrinking sheet with prescribed heat fluxwas analyzed by Suali et al [32] They found that the skinfriction and local Nusselt number increase with unsteadinessMahapatra and Nandy [33] studied the unsteady stagnation-point flow and heat transfer over a linearly shrinking sheet inthe presence of velocity and thermal slips

In many of the studies described above buoyancy effectplays a significant role The buoyancy force developed fromthe temperature difference induces a longitudinal pressuregradient which in turn modifies the flow field and the rateof heat transfer from the surface A computational studyof unsteady mixed convection flow in stagnation regionadjacent to a vertical surface was conducted by Devi etal [34] Ishak et al [35] presented the numerical solutionfor the unsteady mixed convection flow and heat transferover a stretching vertical sheet Both assisting and opposingflow cases were taken into consideration Unsteady mixedconvection boundary layer flow over a stretching verticalsurface in the presence of velocity slip was presented byMukopadhyay [36] Sharma et al [37] employed element-freeGalerkinmethod to study themixed convection flow andheattransfer of a viscous fluid over an unsteady stretching sheetin a porous medium Using Keller-Box method Vajravelu etal [38] examined the effects of variable thermal conductivitythermal radiation and the thermal buoyancy on the unsteadyfluid flow and heat transfer at a porous stretching sheet

The present paper investigates the unsteady mixed con-vection flow and heat transfer of an incompressible micropo-lar fluid over a vertical shrinking sheet with time-dependentsuction at the sheet Viscous dissipation effects are alsoincluded in the energy equation The velocity and tempera-ture of the sheet are assumed to varywith the horizontal coor-dinate 119909 and time 119905 Similarity transformations are employedfor the conversion of governing time-dependent boundarylayer equations into ordinary differential equations Theseequations are then solved numerically using finite elementmethod The influence of suction parameter unsteadinessparameter buoyancy parameter and Eckert number has beendepicted graphically The skin friction and the rate of heattransfer have also been computed and tabulated for theseparameters The current study has applications in industrialpolymeric materials processing and has not been consideredso far to the knowledge of the authors

2 Mathematical Model

Consider the unsteady mixed convection and boundarylayer flow of an incompressible micropolar fluid past apermeable shrinking sheet A schematic representation ofthe physical model and coordinate system is depicted inFigure 1 It is assumed that for time 119905 lt 0 the fluid andheat flows are steady The unsteady fluid and heat flow startat 119905 = 0 The velocity of shrinking sheet is 119880

119908(119909 119905) =

minus119886119909(1 minus 119890119905) and the temperature of the sheet is 119879119908(119909 119905) =

119879infin

+ 119887119909(1 minus 119890119905) where 119886 119890 are constants (with 119886 gt 0119890 ge 0 wher 119890119905 lt 1) and both have dimension (time)minus1 while

The Scientific World Journal 3

x u

TVw

Tw

Uw

y O

ge

Tinfin

Figure 1 Physical model and coordinate system

119887 is a constant and has dimensions (temperaturelength)The coordinate system is such that 119909-axis is taken along theshrinking sheet in a direction opposite to sheet motion and119910-axis is normal to it Time-dependent suction is considerednormal to the shrinking sheet The physical properties of thefluid are assumed to be constant except density variation dueto temperature difference which is used only to express thebody force term as the buoyancy term The effect of viscousdissipation is also included in the energy equation Underthese assumptions the governing boundary layer equationsfor unsteady flow over a shrinking sheet may be presentedas follows

Continuity equation

120597119906

120597119909

+

120597V120597119910

= 0 (1)

Momentum equation

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

(120583 + 119878)

120588

1205972

119906

1205971199102+

119878

120588

120597119873

120597119910

+ 119892119890120573 (119879 minus 119879

infin)

(2)

Angular momentum equation

(

120597119873

120597119905

+ 119906

120597119873

120597119909

+ V120597119873

120597119910

) =

120574

120588119895

1205972

119873

1205971199102minus

119878

120588119895

(2119873 +

120597119906

120597119910

) (3)

Energy equation

120597119879

120597119905

+ 119906

120597119879

120597119909

+ V120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102+

(120583 + 119878)

120588119888119901

(

120597119906

120597119910

)

2

(4)

The associated boundary conditions are

119910 = 0 119906 = 119880119908(119909 119905) V = 119881

119908

119873 = minus

1

2

120597119906

120597119910

119879 = 119879119908(119909 119905)

119910 997888rarr infin 119906 997888rarr 0 119873 997888rarr 0 119879 997888rarr 119879infin

(5)

It is assumed that 119881119908is a variable distribution of suction

through porous sheet and is given by 119881119908

= (1radic1 minus 119890119905)V0

(Bhattacharyya et al [39])Introducing the similarity variable 120578 and the dimension-

less functions 119891 119892 and 120579 as follows

120578 = (

119886

] (1 minus 119890119905)

)

12

119910 120601 = (

119886](1 minus 119890119905)

)

12

119909119891

119873 = (

119886

(1 minus 119890119905)

)

32

119909

radic]119892 120579 =

119879 minus 119879infin

119879119908minus 119879infin

(6)

where 120601 is a stream function defined as 119906 = 120597120601120597119910 V =

minus120597120601120597119909 which identically satisfies the continuity equation(1) On applying the transformations (2)ndash(4) are reducedto

(1 + 119870)119891101584010158401015840

+ 11989111989110158401015840

minus (1198911015840

)

2

+ 1198701198921015840

+ 120590120579 minus 120591 (1198911015840

+

1

2

12057811989110158401015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus 1198911015840

119892 minus 119870 (2119892 + 11989110158401015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus 1198911015840

120579) + (1 + 119870)PrEc(11989110158401015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(7)

and the corresponding boundary conditions (5) now trans-form to

119891 = minus120582 1198911015840

= minus1 119892 = minus

1

2

11989110158401015840

120579 = 1 at 120578 = 0

1198911015840

= 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(8)

where primedenotes the differentiationwith respect to 120578only119870 = 119878120583 (coupling constant parameter) 120590 = Gr

119909(Re119909)2

(buoyancy parameter) Gr119909

= 119892119890120573 (119879119908minus 119879infin)1199093

]2 (localGrashof number) 120591 = 119886119890 (unsteadiness parameter) Pr =

120583119888119901120581 (Prandtl number) Ec = 119880

2

119908119888119901(119879119908minus 119879infin) (Eckert

number) and 120582 = V0radic119886] (suction parameter) 120582 gt 0

corresponds to suction and 120582 lt 0 corresponds to injection

4 The Scientific World Journal

The quantities of physical interest namely the localskin friction coefficient and the rate of heat transfer arerespectively prescribed by

119862119891=

2120591119908

1205881198802

119908

Nu119909=

119902119909

120581 (119879119908minus 119879infin)

(9)

where the local wall shear stress 120591119908and the heat transfer from

the sheet 119902 are given by

120591119908= minus[(120583 + 119878)

120597119906

120597119910

+ 119878119873]

119910=0

119902 = minus120581(

120597119879

120597119910

)

119910=0

(10)

Using the similarity transformations given in (6) weobtain the following

119862119891(Re119909)12

= minus (2 + 119870)11989110158401015840

(0)

Nu119909

(Re119909)12

= minus1205791015840

(0)

(11)

where Re119909= 119880119908119909] is the local Reynolds number

3 Method of Solution

The set of differential equations given in (7) are non-linearand therefore cannot be solved analytically Thus for thesolution of this problem finite element method has beenimplemented Comprehensive details of this method can befound in Reddy [40] In order to apply finite element methodfirst we assume

1198911015840

= ℎ (12)

Using (12) equation (7) reduces to

(1 + 119870) ℎ10158401015840

+ 119891ℎ1015840

minus ℎ2

+ 1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892 minus 119870 (2119892 + ℎ1015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(13)

and the corresponding boundary conditions now become

119891 = minus120582 ℎ = minus1 119892 = minus

1

2

ℎ1015840

120579 = 1 at 120578 = 0

ℎ = 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(14)

It has been observed that for 120578 gt 8 there is no appre-ciable effect on the results Therefore for the computationalpurposesinfin can be fixed at 8

31 Variational Formulation The variational form associatedwith (12) and (13) over a typical two-noded linear element(120578119890 120578119890+1

) is given by

int

120578119890+1

120578119890

11990811198911015840

minus ℎ 119889120578 = 0

int

120578119890+1

120578119890

1199082 (1 + 119870) ℎ

10158401015840

+ 119891ℎ1015840

minus ℎ2

+1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) 119889120578 = 0

int

120578119890+1

120578119890

1199083(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892

minus119870 (2119892 + ℎ1015840

) minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) 119889120578 = 0

int

120578119890+1

120578119890

119908412057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) 119889120578 = 0

(15)

where119908111990821199083 and119908

4are weight functions which may be

viewed as the variation in 119891 ℎ 119892 and 120579 respectively

32 Finite Element Formulation The finite element modelcan be obtained from (15) by substituting finite elementapproximations of the form

119891 =

2

sum

119895=1

119891119895120595119895 ℎ =

2

sum

119895=1

ℎ119895120595119895

119892 =

2

sum

119895=1

119892119895120595119895 120579 =

2

sum

119895=1

120579119895120595119895

(16)

with 1199081= 1199082= 1199083= 1199084= 120595119894(119894 = 1 2) where 120595

119894are the

shape functions for a typical element (120578119890 120578119890+1

) and are takenas follows

1205951=

120578119890+1

minus 120578

120578119890+1

minus 120578119890

1205952=

120578 minus 120578119890

120578119890+1

minus 120578119890

120578119890le 120578 le 120578

119890+1

(17)

The finite element model of the equations thus formed isgiven by

[

[

[

[

[

[

[

[11987011

] [11987012

] [11987013

] [11987014

]

[11987021

] [11987022

] [11987023

] [11987024

]

[11987031

] [11987032

] [11987033

] [11987034

]

[11987041

] [11987042

] [11987043

] [11987044

]

]

]

]

]

]

]

]

[

[

[

[

[

[

119891

ℎ

119892

120579

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198871

1198872

1198873

1198874

]

]

]

]

]

]

]

(18)

The Scientific World Journal 5

where [119870119898119899

] and [119887119898

] (119898 119899 = 1 2 3 4) are the matrices oforder 2 times 2 and 2 times 1 respectively and are defined as follows

11987011

119894119895= int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987012

119894119895= minusint

120578119890+1

120578119890

120595119894120595119895119889120578

11987013

119894119895= 11987014

119894119895= 0 119870

21

119894119895= 0

11987022

119894119895= minus (1 + 119870)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987023

119894119895= 119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987024

119894119895= 120590int

120578119890+1

120578119890

120595119894120595119895119889120578

11987031

119894119895= 0 119870

32

119894119895= minus119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578

11987033

119894119895= minus(1 +

119870

2

)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 2119870int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

3

2

120591int

120578119890+1

120578119890

120595119894120595119895119889120578 minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987034

119894119895= 0 119870

41

119894119895= 0

11987042

119894119895= (1 + 119870)PrEcint

120578119890+1

120578119890

120595119894

119889ℎ

119889120578

119889120595119895

119889120578

119889120578 11987043

119894119895= 0

11987044

119894119895= minusint

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + Print120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus Print120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus Pr120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus Pr1205912

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

1198871

119894= 0 119887

2

119894= minus (1 + 119870)(120595

119894

119889ℎ

119889120578

)

120578119890+1

120578119890

1198873

119894= minus(1 +

119870

2

)(120595119894

119889119892

119889120578

)

120578119890+1

120578119890

1198874

119894= minus(120595

119894

119889120579

119889120578

)

120578119890+1

120578119890

(19)

where 119891 = sum2

119894=1119891119894120595119894and ℎ = sum

2

119894=1ℎ119894120595119894are assumed to be

known After the assembly of element equations a systemof non-linear equations is obtained therefore an iterativescheme must be utilized to solve it The system is linearizedby incorporating the functions 119891 and ℎ which are assumedto be known at lower iteration level and the computations

Table 1 Convergence of results with the variation of number ofelements 119899 (119870 = 2 Pr = 0733 120582 = 3 120591 = 1 120590 = 3 Ec = 075)

119899 119891 (16) ℎ (16) 119892 (16) 120579(16)

20 27194 00714 minus01087 0185740 26986 00524 minus01198 0177260 26955 00496 minus01223 0175680 26945 00487 minus01231 01751100 26941 00483 minus01236 01748120 26939 00481 minus01238 01747140 26937 00480 minus01239 01746160 26937 00479 minus01240 01745180 26936 00479 minus01241 01745

Table 2 Comparison of the flow velocity 1198911015840

(120578) obtained byanalytical method [24] and FEM (119870 = 0 Pr = 0733 120582 = 3 120591 =

0 120590 = 0 Ec = 0) in the special case

120578

1198911015840

(120578)

[24] FEM0 minus1 minus11 minus007295 minus0072902 minus000532 minus0005313 minus000039 minus0000394 minus000003 minus0000035 6 7 8 000000 000000

for119891 ℎ 119892 and 120579 are then carried out for higher levels Thisprocess is repeated until the desired accuracy of 000005 isattained Convergence of the results with increasing numberof elements is shown in Table 1 It is clear from the tablethat for more than 160 elements no significant variationin the values of 119891 ℎ 119892 and 120579 is observed Thus the finalresults are reported for 160 elementsThis confirms themesh-independence of the present computation

For steady state (120591 = 0) viscous fluid (119870 = 0) and in theabsence of buoyancy force (120590 = 0) the exact solution for119891(120578)as obtained by Fang and Zhang [24] with119872 = 0 is given by119891(120578) = 120582 minus (1 minus 119890

minus120578119911

)119911 where 119911 = 05[120582 + (1205822

minus 4)12

]The comparison of the flow velocity 119891

1015840

(120578) obtained byfinite element method and the exact solution given by Fangand Zhang [24] is shown in Table 2 It is clear from the tablethat numerical results obtained are in complete agreementwith the exact solution and thus confirm the validity andaccuracy of the FEM

4 Results and Discussion

To study the behaviour of velocity microrotation and tem-perature functions comprehensive numerical computationsare carried out for various values of the parameters namelysuction parameter 120582 unsteadiness parameter 120591 buoyancyparameter 120590 and Eckert number Ec The other parameterssuch as coupling constant parameter 119870 and Prandtl numberPr are kept fixed at 20 and 0733 respectively The resultsobtained are presented through the graphs as shown in

6 The Scientific World Journal

Table 3 The skin friction coefficient 11989110158401015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 11989110158401015840

(0) 120591 11989110158401015840

(0) 120590 11989110158401015840

(0) Ec 11989110158401015840

(0)

3 249867 1 249867 0 131709 002 21597135 258229 2 249630 1 174698 025 2260444 270175 3 252259 2 212938 05 23757945 285110 4 257288 3 249867 075 2498675 302241 5 263686 4 287143 1 263105

Table 4 The local Nusselt number minus1205791015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 minus1205791015840

(0) 120591 minus1205791015840

(0) 120590 minus1205791015840

(0) Ec minus1205791015840

(0)

3 minus024924 1 minus024924 0 088895 002 19959635 012914 2 014660 1 054444 025 1394364 045990 3 046597 2 017095 05 06352745 075067 4 071556 3 minus024924 075 minus0249245 101279 5 091656 4 minus073308 1 minus128127

Figures 2ndash13 The skin friction and the rate of heat transferhave also been computed for these parameters and aretabulated in Tables 3 and 4

Figures 2ndash4 display the effect of suction parameter 120582 onthe boundary layer flow induced by an unsteady shrinkingsheet Suction is the most suitable force to sustain the flownear the shrinking sheet by confining the generated vorticityinside the boundary layer It is clear from Figure 2 that theeffect of suction parameter 120582 on the velocity is negligible nearthe sheet whereas away from the sheet velocity decreases withincrease in suction parameter Negative values of velocityclose to the sheet indicate the region of reverse flow Figure 3shows that as suction parameter increases the microrotationdecreases near the sheet After covering a small distancefrom the sheet the opposite behaviour is observed Thenegative values of microrotation show the reverse rotationof microparticles Figure 4 illustrates the influence of suctionparameter 120582 on the temperature distribution It is evidentfrom the figure that temperature of the fluid decreases withincrease in suction parameter As suction is applied thermalboundary layer thickness decreases as a result of whichtemperature of the fluid in the boundary layer decreasesThis observation is in agreement with Bhattacharyya andLayek [25] Thus suction parameter can be used effectivelyfor controlling the flow and heat transfer characteristics

The effect of unsteadiness parameter 120591 on the velocitymicrorotation and temperature functions is shown in Figures5ndash7 Figure 5 shows that increasing the value of unsteadinessparameter 120591 tends to decrease the velocity in the boundarylayer Near the sheet flow velocity is negative whereas awayfrom the sheet it becomes positive and finally satisfies thefree stream boundary condition Thus close to the sheet theflow is strongly reversed Figure 6 demonstrates that near

0 1 2 3 4 5 6 7

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120582 = 3 35 4 45 5120578

Figure 2 Velocity distribution for different 120582 (120591 = 1 120590 = 3Ec =

075)

0 1 2 3

120582 = 3 35 4 45 5

120578

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

Figure 3 Microrotation distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

the sheet the effect of unsteadiness on the microrotation isnegligible but away from the sheet it increases with increasein unsteadiness parameter After covering a certain distancefrom the sheet all profiles converge and finally satisfy thefar field boundary condition Figure 7 reveals that the tem-perature decreases with increase in unsteadiness parameter120591 Temperature at the sheet is invariant for higher values ofunsteadiness parameter These patterns are consistent withthe findings of Ishak et al [35] Physically as the value ofunsteadiness parameter increases the sheet loses more heatas a result of which temperature of the fluid decreases

Figures 8ndash10 display the nature of velocity microrotationand temperature distributions with buoyancy parameter 120590Physically positive values of 120590 are associated with cooling ofthe sheet (assisting flow) negative values of120590 indicate heatingof the sheet (opposing flow) and 120590 = 0 implies vanishing

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

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2 The Scientific World Journal

The above-mentioned studies deal with a steady flowonly However in certain cases the flow and heat transfercan be unsteady due to a sudden stretching of the flatsheet or by a change of the temperature or heat flux ofthe sheet Devi et al [13] investigated the unsteady three-dimensional flow caused by a stretching flat surface Usingthe finite difference scheme in combination with the quasi-linearization technique Rajeswari and Nath [14] reportedon the unsteady flow over a stretching surface in a rotatingfluid Exact similarity solution of the heat transfer in a liquidfilm on an unsteady stretching surface was obtained byAndersson et al [15] Abd El-Aziz [16] studied numericallythe effect of radiation on the heat and fluid flow over anunsteady stretching sheet showing that the heat transferrate is an increasing function of radiation and unsteadinessparameters Lok et al [17] obtained the numerical solution forthe unsteady boundary layer flow of a micropolar fluid nearthe stagnation point of a plane surface By taking into accountstrong and weak concentration of microelements Hayat etal [18] examined the effect of magnetic field on the time-dependent flowofmicropolar fluid between stretching sheetsBachok et al [19] studied theoretically the unsteady boundarylayer flow and heat transfer due to a stretching sheet showingthat the surface shear stress and the heat transfer rate are anincreasing function of unsteadiness parameter

Recently the boundary layer flow of incompressible fluidover a shrinking sheet has attracted extensive attention due toits increasing applications in polymeric materials processingThis type of flow was first examined analytically by Wang[20] Later Miklavcic and Wang [21] proved the existenceand uniqueness of the solution for the flow over a shrinkingsheet From the physical point of view steady flow over ashrinking sheet is not possible since the generated vorticityis not confined within the boundary layer To overcomethis difficulty the flow requires a certain amount of exter-nal opposite force at the sheet This has been extensivelydiscussed in the literature Some important references areMiklavcic and Wang [21] Wang [22] and Bachok et al [23]Closed form exact solutions of MHD Newtonian flow over ashrinking sheet were derived by Fang and Zhang [24] Usingshooting method Bhattacharyya and Layek [25] obtainedthe numerical solutions for the flow and heat transfer over aporous shrinking sheet in the presence of thermal radiationThe shrinking sheet problem was extended to micropolarfluids by Ishak et al [26] who considered stagnation pointflow Yacob and Ishak [27] analyzed the flow and heattransfer over a shrinking sheet immersed in a micropolarfluid Recently numerical simulation of mixed convectionflow of micropolar fluid over a shrinking sheet with thermalradiation was conducted by Gupta et al [28]

Many researchers have investigated the unsteady flowover a shrinking sheet Fang et al [29] examined the unsteadyviscous flow over a shrinking surface with mass suction andhighlighted the deviation in flow behaviour for an unsteadyshrinking sheet compared with an unsteady stretching sheetEffect of radiation on the unsteady flow and heat transferinduced by a permeable shrinking sheetwas reported byAli etal [30] Bhattacharyya [31] examined the effects of radiationand heat sourcesink on unsteady flow and heat transfer past

a shrinking sheet with suctioninjection Numerical solutionfor the unsteady stagnation point flow and heat transferover a stretchingshrinking sheet with prescribed heat fluxwas analyzed by Suali et al [32] They found that the skinfriction and local Nusselt number increase with unsteadinessMahapatra and Nandy [33] studied the unsteady stagnation-point flow and heat transfer over a linearly shrinking sheet inthe presence of velocity and thermal slips

In many of the studies described above buoyancy effectplays a significant role The buoyancy force developed fromthe temperature difference induces a longitudinal pressuregradient which in turn modifies the flow field and the rateof heat transfer from the surface A computational studyof unsteady mixed convection flow in stagnation regionadjacent to a vertical surface was conducted by Devi etal [34] Ishak et al [35] presented the numerical solutionfor the unsteady mixed convection flow and heat transferover a stretching vertical sheet Both assisting and opposingflow cases were taken into consideration Unsteady mixedconvection boundary layer flow over a stretching verticalsurface in the presence of velocity slip was presented byMukopadhyay [36] Sharma et al [37] employed element-freeGalerkinmethod to study themixed convection flow andheattransfer of a viscous fluid over an unsteady stretching sheetin a porous medium Using Keller-Box method Vajravelu etal [38] examined the effects of variable thermal conductivitythermal radiation and the thermal buoyancy on the unsteadyfluid flow and heat transfer at a porous stretching sheet

The present paper investigates the unsteady mixed con-vection flow and heat transfer of an incompressible micropo-lar fluid over a vertical shrinking sheet with time-dependentsuction at the sheet Viscous dissipation effects are alsoincluded in the energy equation The velocity and tempera-ture of the sheet are assumed to varywith the horizontal coor-dinate 119909 and time 119905 Similarity transformations are employedfor the conversion of governing time-dependent boundarylayer equations into ordinary differential equations Theseequations are then solved numerically using finite elementmethod The influence of suction parameter unsteadinessparameter buoyancy parameter and Eckert number has beendepicted graphically The skin friction and the rate of heattransfer have also been computed and tabulated for theseparameters The current study has applications in industrialpolymeric materials processing and has not been consideredso far to the knowledge of the authors

2 Mathematical Model

Consider the unsteady mixed convection and boundarylayer flow of an incompressible micropolar fluid past apermeable shrinking sheet A schematic representation ofthe physical model and coordinate system is depicted inFigure 1 It is assumed that for time 119905 lt 0 the fluid andheat flows are steady The unsteady fluid and heat flow startat 119905 = 0 The velocity of shrinking sheet is 119880

119908(119909 119905) =

minus119886119909(1 minus 119890119905) and the temperature of the sheet is 119879119908(119909 119905) =

119879infin

+ 119887119909(1 minus 119890119905) where 119886 119890 are constants (with 119886 gt 0119890 ge 0 wher 119890119905 lt 1) and both have dimension (time)minus1 while

The Scientific World Journal 3

x u

TVw

Tw

Uw

y O

ge

Tinfin

Figure 1 Physical model and coordinate system

119887 is a constant and has dimensions (temperaturelength)The coordinate system is such that 119909-axis is taken along theshrinking sheet in a direction opposite to sheet motion and119910-axis is normal to it Time-dependent suction is considerednormal to the shrinking sheet The physical properties of thefluid are assumed to be constant except density variation dueto temperature difference which is used only to express thebody force term as the buoyancy term The effect of viscousdissipation is also included in the energy equation Underthese assumptions the governing boundary layer equationsfor unsteady flow over a shrinking sheet may be presentedas follows

Continuity equation

120597119906

120597119909

+

120597V120597119910

= 0 (1)

Momentum equation

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

(120583 + 119878)

120588

1205972

119906

1205971199102+

119878

120588

120597119873

120597119910

+ 119892119890120573 (119879 minus 119879

infin)

(2)

Angular momentum equation

(

120597119873

120597119905

+ 119906

120597119873

120597119909

+ V120597119873

120597119910

) =

120574

120588119895

1205972

119873

1205971199102minus

119878

120588119895

(2119873 +

120597119906

120597119910

) (3)

Energy equation

120597119879

120597119905

+ 119906

120597119879

120597119909

+ V120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102+

(120583 + 119878)

120588119888119901

(

120597119906

120597119910

)

2

(4)

The associated boundary conditions are

119910 = 0 119906 = 119880119908(119909 119905) V = 119881

119908

119873 = minus

1

2

120597119906

120597119910

119879 = 119879119908(119909 119905)

119910 997888rarr infin 119906 997888rarr 0 119873 997888rarr 0 119879 997888rarr 119879infin

(5)

It is assumed that 119881119908is a variable distribution of suction

through porous sheet and is given by 119881119908

= (1radic1 minus 119890119905)V0

(Bhattacharyya et al [39])Introducing the similarity variable 120578 and the dimension-

less functions 119891 119892 and 120579 as follows

120578 = (

119886

] (1 minus 119890119905)

)

12

119910 120601 = (

119886](1 minus 119890119905)

)

12

119909119891

119873 = (

119886

(1 minus 119890119905)

)

32

119909

radic]119892 120579 =

119879 minus 119879infin

119879119908minus 119879infin

(6)

where 120601 is a stream function defined as 119906 = 120597120601120597119910 V =

minus120597120601120597119909 which identically satisfies the continuity equation(1) On applying the transformations (2)ndash(4) are reducedto

(1 + 119870)119891101584010158401015840

+ 11989111989110158401015840

minus (1198911015840

)

2

+ 1198701198921015840

+ 120590120579 minus 120591 (1198911015840

+

1

2

12057811989110158401015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus 1198911015840

119892 minus 119870 (2119892 + 11989110158401015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus 1198911015840

120579) + (1 + 119870)PrEc(11989110158401015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(7)

and the corresponding boundary conditions (5) now trans-form to

119891 = minus120582 1198911015840

= minus1 119892 = minus

1

2

11989110158401015840

120579 = 1 at 120578 = 0

1198911015840

= 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(8)

where primedenotes the differentiationwith respect to 120578only119870 = 119878120583 (coupling constant parameter) 120590 = Gr

119909(Re119909)2

(buoyancy parameter) Gr119909

= 119892119890120573 (119879119908minus 119879infin)1199093

]2 (localGrashof number) 120591 = 119886119890 (unsteadiness parameter) Pr =

120583119888119901120581 (Prandtl number) Ec = 119880

2

119908119888119901(119879119908minus 119879infin) (Eckert

number) and 120582 = V0radic119886] (suction parameter) 120582 gt 0

corresponds to suction and 120582 lt 0 corresponds to injection

4 The Scientific World Journal

The quantities of physical interest namely the localskin friction coefficient and the rate of heat transfer arerespectively prescribed by

119862119891=

2120591119908

1205881198802

119908

Nu119909=

119902119909

120581 (119879119908minus 119879infin)

(9)

where the local wall shear stress 120591119908and the heat transfer from

the sheet 119902 are given by

120591119908= minus[(120583 + 119878)

120597119906

120597119910

+ 119878119873]

119910=0

119902 = minus120581(

120597119879

120597119910

)

119910=0

(10)

Using the similarity transformations given in (6) weobtain the following

119862119891(Re119909)12

= minus (2 + 119870)11989110158401015840

(0)

Nu119909

(Re119909)12

= minus1205791015840

(0)

(11)

where Re119909= 119880119908119909] is the local Reynolds number

3 Method of Solution

The set of differential equations given in (7) are non-linearand therefore cannot be solved analytically Thus for thesolution of this problem finite element method has beenimplemented Comprehensive details of this method can befound in Reddy [40] In order to apply finite element methodfirst we assume

1198911015840

= ℎ (12)

Using (12) equation (7) reduces to

(1 + 119870) ℎ10158401015840

+ 119891ℎ1015840

minus ℎ2

+ 1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892 minus 119870 (2119892 + ℎ1015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(13)

and the corresponding boundary conditions now become

119891 = minus120582 ℎ = minus1 119892 = minus

1

2

ℎ1015840

120579 = 1 at 120578 = 0

ℎ = 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(14)

It has been observed that for 120578 gt 8 there is no appre-ciable effect on the results Therefore for the computationalpurposesinfin can be fixed at 8

31 Variational Formulation The variational form associatedwith (12) and (13) over a typical two-noded linear element(120578119890 120578119890+1

) is given by

int

120578119890+1

120578119890

11990811198911015840

minus ℎ 119889120578 = 0

int

120578119890+1

120578119890

1199082 (1 + 119870) ℎ

10158401015840

+ 119891ℎ1015840

minus ℎ2

+1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) 119889120578 = 0

int

120578119890+1

120578119890

1199083(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892

minus119870 (2119892 + ℎ1015840

) minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) 119889120578 = 0

int

120578119890+1

120578119890

119908412057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) 119889120578 = 0

(15)

where119908111990821199083 and119908

4are weight functions which may be

viewed as the variation in 119891 ℎ 119892 and 120579 respectively

32 Finite Element Formulation The finite element modelcan be obtained from (15) by substituting finite elementapproximations of the form

119891 =

2

sum

119895=1

119891119895120595119895 ℎ =

2

sum

119895=1

ℎ119895120595119895

119892 =

2

sum

119895=1

119892119895120595119895 120579 =

2

sum

119895=1

120579119895120595119895

(16)

with 1199081= 1199082= 1199083= 1199084= 120595119894(119894 = 1 2) where 120595

119894are the

shape functions for a typical element (120578119890 120578119890+1

) and are takenas follows

1205951=

120578119890+1

minus 120578

120578119890+1

minus 120578119890

1205952=

120578 minus 120578119890

120578119890+1

minus 120578119890

120578119890le 120578 le 120578

119890+1

(17)

The finite element model of the equations thus formed isgiven by

[

[

[

[

[

[

[

[11987011

] [11987012

] [11987013

] [11987014

]

[11987021

] [11987022

] [11987023

] [11987024

]

[11987031

] [11987032

] [11987033

] [11987034

]

[11987041

] [11987042

] [11987043

] [11987044

]

]

]

]

]

]

]

]

[

[

[

[

[

[

119891

ℎ

119892

120579

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198871

1198872

1198873

1198874

]

]

]

]

]

]

]

(18)

The Scientific World Journal 5

where [119870119898119899

] and [119887119898

] (119898 119899 = 1 2 3 4) are the matrices oforder 2 times 2 and 2 times 1 respectively and are defined as follows

11987011

119894119895= int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987012

119894119895= minusint

120578119890+1

120578119890

120595119894120595119895119889120578

11987013

119894119895= 11987014

119894119895= 0 119870

21

119894119895= 0

11987022

119894119895= minus (1 + 119870)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987023

119894119895= 119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987024

119894119895= 120590int

120578119890+1

120578119890

120595119894120595119895119889120578

11987031

119894119895= 0 119870

32

119894119895= minus119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578

11987033

119894119895= minus(1 +

119870

2

)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 2119870int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

3

2

120591int

120578119890+1

120578119890

120595119894120595119895119889120578 minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987034

119894119895= 0 119870

41

119894119895= 0

11987042

119894119895= (1 + 119870)PrEcint

120578119890+1

120578119890

120595119894

119889ℎ

119889120578

119889120595119895

119889120578

119889120578 11987043

119894119895= 0

11987044

119894119895= minusint

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + Print120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus Print120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus Pr120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus Pr1205912

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

1198871

119894= 0 119887

2

119894= minus (1 + 119870)(120595

119894

119889ℎ

119889120578

)

120578119890+1

120578119890

1198873

119894= minus(1 +

119870

2

)(120595119894

119889119892

119889120578

)

120578119890+1

120578119890

1198874

119894= minus(120595

119894

119889120579

119889120578

)

120578119890+1

120578119890

(19)

where 119891 = sum2

119894=1119891119894120595119894and ℎ = sum

2

119894=1ℎ119894120595119894are assumed to be

known After the assembly of element equations a systemof non-linear equations is obtained therefore an iterativescheme must be utilized to solve it The system is linearizedby incorporating the functions 119891 and ℎ which are assumedto be known at lower iteration level and the computations

Table 1 Convergence of results with the variation of number ofelements 119899 (119870 = 2 Pr = 0733 120582 = 3 120591 = 1 120590 = 3 Ec = 075)

119899 119891 (16) ℎ (16) 119892 (16) 120579(16)

20 27194 00714 minus01087 0185740 26986 00524 minus01198 0177260 26955 00496 minus01223 0175680 26945 00487 minus01231 01751100 26941 00483 minus01236 01748120 26939 00481 minus01238 01747140 26937 00480 minus01239 01746160 26937 00479 minus01240 01745180 26936 00479 minus01241 01745

Table 2 Comparison of the flow velocity 1198911015840

(120578) obtained byanalytical method [24] and FEM (119870 = 0 Pr = 0733 120582 = 3 120591 =

0 120590 = 0 Ec = 0) in the special case

120578

1198911015840

(120578)

[24] FEM0 minus1 minus11 minus007295 minus0072902 minus000532 minus0005313 minus000039 minus0000394 minus000003 minus0000035 6 7 8 000000 000000

for119891 ℎ 119892 and 120579 are then carried out for higher levels Thisprocess is repeated until the desired accuracy of 000005 isattained Convergence of the results with increasing numberof elements is shown in Table 1 It is clear from the tablethat for more than 160 elements no significant variationin the values of 119891 ℎ 119892 and 120579 is observed Thus the finalresults are reported for 160 elementsThis confirms themesh-independence of the present computation

For steady state (120591 = 0) viscous fluid (119870 = 0) and in theabsence of buoyancy force (120590 = 0) the exact solution for119891(120578)as obtained by Fang and Zhang [24] with119872 = 0 is given by119891(120578) = 120582 minus (1 minus 119890

minus120578119911

)119911 where 119911 = 05[120582 + (1205822

minus 4)12

]The comparison of the flow velocity 119891

1015840

(120578) obtained byfinite element method and the exact solution given by Fangand Zhang [24] is shown in Table 2 It is clear from the tablethat numerical results obtained are in complete agreementwith the exact solution and thus confirm the validity andaccuracy of the FEM

4 Results and Discussion

To study the behaviour of velocity microrotation and tem-perature functions comprehensive numerical computationsare carried out for various values of the parameters namelysuction parameter 120582 unsteadiness parameter 120591 buoyancyparameter 120590 and Eckert number Ec The other parameterssuch as coupling constant parameter 119870 and Prandtl numberPr are kept fixed at 20 and 0733 respectively The resultsobtained are presented through the graphs as shown in

6 The Scientific World Journal

Table 3 The skin friction coefficient 11989110158401015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 11989110158401015840

(0) 120591 11989110158401015840

(0) 120590 11989110158401015840

(0) Ec 11989110158401015840

(0)

3 249867 1 249867 0 131709 002 21597135 258229 2 249630 1 174698 025 2260444 270175 3 252259 2 212938 05 23757945 285110 4 257288 3 249867 075 2498675 302241 5 263686 4 287143 1 263105

Table 4 The local Nusselt number minus1205791015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 minus1205791015840

(0) 120591 minus1205791015840

(0) 120590 minus1205791015840

(0) Ec minus1205791015840

(0)

3 minus024924 1 minus024924 0 088895 002 19959635 012914 2 014660 1 054444 025 1394364 045990 3 046597 2 017095 05 06352745 075067 4 071556 3 minus024924 075 minus0249245 101279 5 091656 4 minus073308 1 minus128127

Figures 2ndash13 The skin friction and the rate of heat transferhave also been computed for these parameters and aretabulated in Tables 3 and 4

Figures 2ndash4 display the effect of suction parameter 120582 onthe boundary layer flow induced by an unsteady shrinkingsheet Suction is the most suitable force to sustain the flownear the shrinking sheet by confining the generated vorticityinside the boundary layer It is clear from Figure 2 that theeffect of suction parameter 120582 on the velocity is negligible nearthe sheet whereas away from the sheet velocity decreases withincrease in suction parameter Negative values of velocityclose to the sheet indicate the region of reverse flow Figure 3shows that as suction parameter increases the microrotationdecreases near the sheet After covering a small distancefrom the sheet the opposite behaviour is observed Thenegative values of microrotation show the reverse rotationof microparticles Figure 4 illustrates the influence of suctionparameter 120582 on the temperature distribution It is evidentfrom the figure that temperature of the fluid decreases withincrease in suction parameter As suction is applied thermalboundary layer thickness decreases as a result of whichtemperature of the fluid in the boundary layer decreasesThis observation is in agreement with Bhattacharyya andLayek [25] Thus suction parameter can be used effectivelyfor controlling the flow and heat transfer characteristics

The effect of unsteadiness parameter 120591 on the velocitymicrorotation and temperature functions is shown in Figures5ndash7 Figure 5 shows that increasing the value of unsteadinessparameter 120591 tends to decrease the velocity in the boundarylayer Near the sheet flow velocity is negative whereas awayfrom the sheet it becomes positive and finally satisfies thefree stream boundary condition Thus close to the sheet theflow is strongly reversed Figure 6 demonstrates that near

0 1 2 3 4 5 6 7

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120582 = 3 35 4 45 5120578

Figure 2 Velocity distribution for different 120582 (120591 = 1 120590 = 3Ec =

075)

0 1 2 3

120582 = 3 35 4 45 5

120578

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

Figure 3 Microrotation distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

the sheet the effect of unsteadiness on the microrotation isnegligible but away from the sheet it increases with increasein unsteadiness parameter After covering a certain distancefrom the sheet all profiles converge and finally satisfy thefar field boundary condition Figure 7 reveals that the tem-perature decreases with increase in unsteadiness parameter120591 Temperature at the sheet is invariant for higher values ofunsteadiness parameter These patterns are consistent withthe findings of Ishak et al [35] Physically as the value ofunsteadiness parameter increases the sheet loses more heatas a result of which temperature of the fluid decreases

Figures 8ndash10 display the nature of velocity microrotationand temperature distributions with buoyancy parameter 120590Physically positive values of 120590 are associated with cooling ofthe sheet (assisting flow) negative values of120590 indicate heatingof the sheet (opposing flow) and 120590 = 0 implies vanishing

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

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DistributedSensor Networks

International Journal of

The Scientific World Journal 3

x u

TVw

Tw

Uw

y O

ge

Tinfin

Figure 1 Physical model and coordinate system

119887 is a constant and has dimensions (temperaturelength)The coordinate system is such that 119909-axis is taken along theshrinking sheet in a direction opposite to sheet motion and119910-axis is normal to it Time-dependent suction is considerednormal to the shrinking sheet The physical properties of thefluid are assumed to be constant except density variation dueto temperature difference which is used only to express thebody force term as the buoyancy term The effect of viscousdissipation is also included in the energy equation Underthese assumptions the governing boundary layer equationsfor unsteady flow over a shrinking sheet may be presentedas follows

Continuity equation

120597119906

120597119909

+

120597V120597119910

= 0 (1)

Momentum equation

120597119906

120597119905

+ 119906

120597119906

120597119909

+ V120597119906

120597119910

=

(120583 + 119878)

120588

1205972

119906

1205971199102+

119878

120588

120597119873

120597119910

+ 119892119890120573 (119879 minus 119879

infin)

(2)

Angular momentum equation

(

120597119873

120597119905

+ 119906

120597119873

120597119909

+ V120597119873

120597119910

) =

120574

120588119895

1205972

119873

1205971199102minus

119878

120588119895

(2119873 +

120597119906

120597119910

) (3)

Energy equation

120597119879

120597119905

+ 119906

120597119879

120597119909

+ V120597119879

120597119910

=

120581

120588119888119901

1205972

119879

1205971199102+

(120583 + 119878)

120588119888119901

(

120597119906

120597119910

)

2

(4)

The associated boundary conditions are

119910 = 0 119906 = 119880119908(119909 119905) V = 119881

119908

119873 = minus

1

2

120597119906

120597119910

119879 = 119879119908(119909 119905)

119910 997888rarr infin 119906 997888rarr 0 119873 997888rarr 0 119879 997888rarr 119879infin

(5)

It is assumed that 119881119908is a variable distribution of suction

through porous sheet and is given by 119881119908

= (1radic1 minus 119890119905)V0

(Bhattacharyya et al [39])Introducing the similarity variable 120578 and the dimension-

less functions 119891 119892 and 120579 as follows

120578 = (

119886

] (1 minus 119890119905)

)

12

119910 120601 = (

119886](1 minus 119890119905)

)

12

119909119891

119873 = (

119886

(1 minus 119890119905)

)

32

119909

radic]119892 120579 =

119879 minus 119879infin

119879119908minus 119879infin

(6)

where 120601 is a stream function defined as 119906 = 120597120601120597119910 V =

minus120597120601120597119909 which identically satisfies the continuity equation(1) On applying the transformations (2)ndash(4) are reducedto

(1 + 119870)119891101584010158401015840

+ 11989111989110158401015840

minus (1198911015840

)

2

+ 1198701198921015840

+ 120590120579 minus 120591 (1198911015840

+

1

2

12057811989110158401015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus 1198911015840

119892 minus 119870 (2119892 + 11989110158401015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus 1198911015840

120579) + (1 + 119870)PrEc(11989110158401015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(7)

and the corresponding boundary conditions (5) now trans-form to

119891 = minus120582 1198911015840

= minus1 119892 = minus

1

2

11989110158401015840

120579 = 1 at 120578 = 0

1198911015840

= 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(8)

where primedenotes the differentiationwith respect to 120578only119870 = 119878120583 (coupling constant parameter) 120590 = Gr

119909(Re119909)2

(buoyancy parameter) Gr119909

= 119892119890120573 (119879119908minus 119879infin)1199093

]2 (localGrashof number) 120591 = 119886119890 (unsteadiness parameter) Pr =

120583119888119901120581 (Prandtl number) Ec = 119880

2

119908119888119901(119879119908minus 119879infin) (Eckert

number) and 120582 = V0radic119886] (suction parameter) 120582 gt 0

corresponds to suction and 120582 lt 0 corresponds to injection

4 The Scientific World Journal

The quantities of physical interest namely the localskin friction coefficient and the rate of heat transfer arerespectively prescribed by

119862119891=

2120591119908

1205881198802

119908

Nu119909=

119902119909

120581 (119879119908minus 119879infin)

(9)

where the local wall shear stress 120591119908and the heat transfer from

the sheet 119902 are given by

120591119908= minus[(120583 + 119878)

120597119906

120597119910

+ 119878119873]

119910=0

119902 = minus120581(

120597119879

120597119910

)

119910=0

(10)

Using the similarity transformations given in (6) weobtain the following

119862119891(Re119909)12

= minus (2 + 119870)11989110158401015840

(0)

Nu119909

(Re119909)12

= minus1205791015840

(0)

(11)

where Re119909= 119880119908119909] is the local Reynolds number

3 Method of Solution

The set of differential equations given in (7) are non-linearand therefore cannot be solved analytically Thus for thesolution of this problem finite element method has beenimplemented Comprehensive details of this method can befound in Reddy [40] In order to apply finite element methodfirst we assume

1198911015840

= ℎ (12)

Using (12) equation (7) reduces to

(1 + 119870) ℎ10158401015840

+ 119891ℎ1015840

minus ℎ2

+ 1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892 minus 119870 (2119892 + ℎ1015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(13)

and the corresponding boundary conditions now become

119891 = minus120582 ℎ = minus1 119892 = minus

1

2

ℎ1015840

120579 = 1 at 120578 = 0

ℎ = 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(14)

It has been observed that for 120578 gt 8 there is no appre-ciable effect on the results Therefore for the computationalpurposesinfin can be fixed at 8

31 Variational Formulation The variational form associatedwith (12) and (13) over a typical two-noded linear element(120578119890 120578119890+1

) is given by

int

120578119890+1

120578119890

11990811198911015840

minus ℎ 119889120578 = 0

int

120578119890+1

120578119890

1199082 (1 + 119870) ℎ

10158401015840

+ 119891ℎ1015840

minus ℎ2

+1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) 119889120578 = 0

int

120578119890+1

120578119890

1199083(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892

minus119870 (2119892 + ℎ1015840

) minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) 119889120578 = 0

int

120578119890+1

120578119890

119908412057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) 119889120578 = 0

(15)

where119908111990821199083 and119908

4are weight functions which may be

viewed as the variation in 119891 ℎ 119892 and 120579 respectively

32 Finite Element Formulation The finite element modelcan be obtained from (15) by substituting finite elementapproximations of the form

119891 =

2

sum

119895=1

119891119895120595119895 ℎ =

2

sum

119895=1

ℎ119895120595119895

119892 =

2

sum

119895=1

119892119895120595119895 120579 =

2

sum

119895=1

120579119895120595119895

(16)

with 1199081= 1199082= 1199083= 1199084= 120595119894(119894 = 1 2) where 120595

119894are the

shape functions for a typical element (120578119890 120578119890+1

) and are takenas follows

1205951=

120578119890+1

minus 120578

120578119890+1

minus 120578119890

1205952=

120578 minus 120578119890

120578119890+1

minus 120578119890

120578119890le 120578 le 120578

119890+1

(17)

The finite element model of the equations thus formed isgiven by

[

[

[

[

[

[

[

[11987011

] [11987012

] [11987013

] [11987014

]

[11987021

] [11987022

] [11987023

] [11987024

]

[11987031

] [11987032

] [11987033

] [11987034

]

[11987041

] [11987042

] [11987043

] [11987044

]

]

]

]

]

]

]

]

[

[

[

[

[

[

119891

ℎ

119892

120579

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198871

1198872

1198873

1198874

]

]

]

]

]

]

]

(18)

The Scientific World Journal 5

where [119870119898119899

] and [119887119898

] (119898 119899 = 1 2 3 4) are the matrices oforder 2 times 2 and 2 times 1 respectively and are defined as follows

11987011

119894119895= int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987012

119894119895= minusint

120578119890+1

120578119890

120595119894120595119895119889120578

11987013

119894119895= 11987014

119894119895= 0 119870

21

119894119895= 0

11987022

119894119895= minus (1 + 119870)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987023

119894119895= 119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987024

119894119895= 120590int

120578119890+1

120578119890

120595119894120595119895119889120578

11987031

119894119895= 0 119870

32

119894119895= minus119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578

11987033

119894119895= minus(1 +

119870

2

)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 2119870int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

3

2

120591int

120578119890+1

120578119890

120595119894120595119895119889120578 minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987034

119894119895= 0 119870

41

119894119895= 0

11987042

119894119895= (1 + 119870)PrEcint

120578119890+1

120578119890

120595119894

119889ℎ

119889120578

119889120595119895

119889120578

119889120578 11987043

119894119895= 0

11987044

119894119895= minusint

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + Print120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus Print120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus Pr120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus Pr1205912

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

1198871

119894= 0 119887

2

119894= minus (1 + 119870)(120595

119894

119889ℎ

119889120578

)

120578119890+1

120578119890

1198873

119894= minus(1 +

119870

2

)(120595119894

119889119892

119889120578

)

120578119890+1

120578119890

1198874

119894= minus(120595

119894

119889120579

119889120578

)

120578119890+1

120578119890

(19)

where 119891 = sum2

119894=1119891119894120595119894and ℎ = sum

2

119894=1ℎ119894120595119894are assumed to be

known After the assembly of element equations a systemof non-linear equations is obtained therefore an iterativescheme must be utilized to solve it The system is linearizedby incorporating the functions 119891 and ℎ which are assumedto be known at lower iteration level and the computations

Table 1 Convergence of results with the variation of number ofelements 119899 (119870 = 2 Pr = 0733 120582 = 3 120591 = 1 120590 = 3 Ec = 075)

119899 119891 (16) ℎ (16) 119892 (16) 120579(16)

20 27194 00714 minus01087 0185740 26986 00524 minus01198 0177260 26955 00496 minus01223 0175680 26945 00487 minus01231 01751100 26941 00483 minus01236 01748120 26939 00481 minus01238 01747140 26937 00480 minus01239 01746160 26937 00479 minus01240 01745180 26936 00479 minus01241 01745

Table 2 Comparison of the flow velocity 1198911015840

(120578) obtained byanalytical method [24] and FEM (119870 = 0 Pr = 0733 120582 = 3 120591 =

0 120590 = 0 Ec = 0) in the special case

120578

1198911015840

(120578)

[24] FEM0 minus1 minus11 minus007295 minus0072902 minus000532 minus0005313 minus000039 minus0000394 minus000003 minus0000035 6 7 8 000000 000000

for119891 ℎ 119892 and 120579 are then carried out for higher levels Thisprocess is repeated until the desired accuracy of 000005 isattained Convergence of the results with increasing numberof elements is shown in Table 1 It is clear from the tablethat for more than 160 elements no significant variationin the values of 119891 ℎ 119892 and 120579 is observed Thus the finalresults are reported for 160 elementsThis confirms themesh-independence of the present computation

For steady state (120591 = 0) viscous fluid (119870 = 0) and in theabsence of buoyancy force (120590 = 0) the exact solution for119891(120578)as obtained by Fang and Zhang [24] with119872 = 0 is given by119891(120578) = 120582 minus (1 minus 119890

minus120578119911

)119911 where 119911 = 05[120582 + (1205822

minus 4)12

]The comparison of the flow velocity 119891

1015840

(120578) obtained byfinite element method and the exact solution given by Fangand Zhang [24] is shown in Table 2 It is clear from the tablethat numerical results obtained are in complete agreementwith the exact solution and thus confirm the validity andaccuracy of the FEM

4 Results and Discussion

To study the behaviour of velocity microrotation and tem-perature functions comprehensive numerical computationsare carried out for various values of the parameters namelysuction parameter 120582 unsteadiness parameter 120591 buoyancyparameter 120590 and Eckert number Ec The other parameterssuch as coupling constant parameter 119870 and Prandtl numberPr are kept fixed at 20 and 0733 respectively The resultsobtained are presented through the graphs as shown in

6 The Scientific World Journal

Table 3 The skin friction coefficient 11989110158401015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 11989110158401015840

(0) 120591 11989110158401015840

(0) 120590 11989110158401015840

(0) Ec 11989110158401015840

(0)

3 249867 1 249867 0 131709 002 21597135 258229 2 249630 1 174698 025 2260444 270175 3 252259 2 212938 05 23757945 285110 4 257288 3 249867 075 2498675 302241 5 263686 4 287143 1 263105

Table 4 The local Nusselt number minus1205791015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 minus1205791015840

(0) 120591 minus1205791015840

(0) 120590 minus1205791015840

(0) Ec minus1205791015840

(0)

3 minus024924 1 minus024924 0 088895 002 19959635 012914 2 014660 1 054444 025 1394364 045990 3 046597 2 017095 05 06352745 075067 4 071556 3 minus024924 075 minus0249245 101279 5 091656 4 minus073308 1 minus128127

Figures 2ndash13 The skin friction and the rate of heat transferhave also been computed for these parameters and aretabulated in Tables 3 and 4

Figures 2ndash4 display the effect of suction parameter 120582 onthe boundary layer flow induced by an unsteady shrinkingsheet Suction is the most suitable force to sustain the flownear the shrinking sheet by confining the generated vorticityinside the boundary layer It is clear from Figure 2 that theeffect of suction parameter 120582 on the velocity is negligible nearthe sheet whereas away from the sheet velocity decreases withincrease in suction parameter Negative values of velocityclose to the sheet indicate the region of reverse flow Figure 3shows that as suction parameter increases the microrotationdecreases near the sheet After covering a small distancefrom the sheet the opposite behaviour is observed Thenegative values of microrotation show the reverse rotationof microparticles Figure 4 illustrates the influence of suctionparameter 120582 on the temperature distribution It is evidentfrom the figure that temperature of the fluid decreases withincrease in suction parameter As suction is applied thermalboundary layer thickness decreases as a result of whichtemperature of the fluid in the boundary layer decreasesThis observation is in agreement with Bhattacharyya andLayek [25] Thus suction parameter can be used effectivelyfor controlling the flow and heat transfer characteristics

The effect of unsteadiness parameter 120591 on the velocitymicrorotation and temperature functions is shown in Figures5ndash7 Figure 5 shows that increasing the value of unsteadinessparameter 120591 tends to decrease the velocity in the boundarylayer Near the sheet flow velocity is negative whereas awayfrom the sheet it becomes positive and finally satisfies thefree stream boundary condition Thus close to the sheet theflow is strongly reversed Figure 6 demonstrates that near

0 1 2 3 4 5 6 7

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120582 = 3 35 4 45 5120578

Figure 2 Velocity distribution for different 120582 (120591 = 1 120590 = 3Ec =

075)

0 1 2 3

120582 = 3 35 4 45 5

120578

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

Figure 3 Microrotation distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

the sheet the effect of unsteadiness on the microrotation isnegligible but away from the sheet it increases with increasein unsteadiness parameter After covering a certain distancefrom the sheet all profiles converge and finally satisfy thefar field boundary condition Figure 7 reveals that the tem-perature decreases with increase in unsteadiness parameter120591 Temperature at the sheet is invariant for higher values ofunsteadiness parameter These patterns are consistent withthe findings of Ishak et al [35] Physically as the value ofunsteadiness parameter increases the sheet loses more heatas a result of which temperature of the fluid decreases

Figures 8ndash10 display the nature of velocity microrotationand temperature distributions with buoyancy parameter 120590Physically positive values of 120590 are associated with cooling ofthe sheet (assisting flow) negative values of120590 indicate heatingof the sheet (opposing flow) and 120590 = 0 implies vanishing

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

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4 The Scientific World Journal

The quantities of physical interest namely the localskin friction coefficient and the rate of heat transfer arerespectively prescribed by

119862119891=

2120591119908

1205881198802

119908

Nu119909=

119902119909

120581 (119879119908minus 119879infin)

(9)

where the local wall shear stress 120591119908and the heat transfer from

the sheet 119902 are given by

120591119908= minus[(120583 + 119878)

120597119906

120597119910

+ 119878119873]

119910=0

119902 = minus120581(

120597119879

120597119910

)

119910=0

(10)

Using the similarity transformations given in (6) weobtain the following

119862119891(Re119909)12

= minus (2 + 119870)11989110158401015840

(0)

Nu119909

(Re119909)12

= minus1205791015840

(0)

(11)

where Re119909= 119880119908119909] is the local Reynolds number

3 Method of Solution

The set of differential equations given in (7) are non-linearand therefore cannot be solved analytically Thus for thesolution of this problem finite element method has beenimplemented Comprehensive details of this method can befound in Reddy [40] In order to apply finite element methodfirst we assume

1198911015840

= ℎ (12)

Using (12) equation (7) reduces to

(1 + 119870) ℎ10158401015840

+ 119891ℎ1015840

minus ℎ2

+ 1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) = 0

(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892 minus 119870 (2119892 + ℎ1015840

)

minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) = 0

12057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) = 0

(13)

and the corresponding boundary conditions now become

119891 = minus120582 ℎ = minus1 119892 = minus

1

2

ℎ1015840

120579 = 1 at 120578 = 0

ℎ = 0 119892 = 0 120579 = 0 as 120578 997888rarr infin

(14)

It has been observed that for 120578 gt 8 there is no appre-ciable effect on the results Therefore for the computationalpurposesinfin can be fixed at 8

31 Variational Formulation The variational form associatedwith (12) and (13) over a typical two-noded linear element(120578119890 120578119890+1

) is given by

int

120578119890+1

120578119890

11990811198911015840

minus ℎ 119889120578 = 0

int

120578119890+1

120578119890

1199082 (1 + 119870) ℎ

10158401015840

+ 119891ℎ1015840

minus ℎ2

+1198701198921015840

+ 120590120579 minus 120591 (ℎ +

1

2

120578ℎ1015840

) 119889120578 = 0

int

120578119890+1

120578119890

1199083(1 +

119870

2

)11989210158401015840

+ 1198911198921015840

minus ℎ119892

minus119870 (2119892 + ℎ1015840

) minus 120591 (

3

2

119892 +

1

2

1205781198921015840

) 119889120578 = 0

int

120578119890+1

120578119890

119908412057910158401015840

+ Pr (1198911205791015840 minus ℎ120579) + (1 + 119870)PrEc(ℎ1015840)2

minus Pr120591 (120579 + 1

2

1205781205791015840

) 119889120578 = 0

(15)

where119908111990821199083 and119908

4are weight functions which may be

viewed as the variation in 119891 ℎ 119892 and 120579 respectively

32 Finite Element Formulation The finite element modelcan be obtained from (15) by substituting finite elementapproximations of the form

119891 =

2

sum

119895=1

119891119895120595119895 ℎ =

2

sum

119895=1

ℎ119895120595119895

119892 =

2

sum

119895=1

119892119895120595119895 120579 =

2

sum

119895=1

120579119895120595119895

(16)

with 1199081= 1199082= 1199083= 1199084= 120595119894(119894 = 1 2) where 120595

119894are the

shape functions for a typical element (120578119890 120578119890+1

) and are takenas follows

1205951=

120578119890+1

minus 120578

120578119890+1

minus 120578119890

1205952=

120578 minus 120578119890

120578119890+1

minus 120578119890

120578119890le 120578 le 120578

119890+1

(17)

The finite element model of the equations thus formed isgiven by

[

[

[

[

[

[

[

[11987011

] [11987012

] [11987013

] [11987014

]

[11987021

] [11987022

] [11987023

] [11987024

]

[11987031

] [11987032

] [11987033

] [11987034

]

[11987041

] [11987042

] [11987043

] [11987044

]

]

]

]

]

]

]

]

[

[

[

[

[

[

119891

ℎ

119892

120579

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198871

1198872

1198873

1198874

]

]

]

]

]

]

]

(18)

The Scientific World Journal 5

where [119870119898119899

] and [119887119898

] (119898 119899 = 1 2 3 4) are the matrices oforder 2 times 2 and 2 times 1 respectively and are defined as follows

11987011

119894119895= int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987012

119894119895= minusint

120578119890+1

120578119890

120595119894120595119895119889120578

11987013

119894119895= 11987014

119894119895= 0 119870

21

119894119895= 0

11987022

119894119895= minus (1 + 119870)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987023

119894119895= 119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987024

119894119895= 120590int

120578119890+1

120578119890

120595119894120595119895119889120578

11987031

119894119895= 0 119870

32

119894119895= minus119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578

11987033

119894119895= minus(1 +

119870

2

)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 2119870int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

3

2

120591int

120578119890+1

120578119890

120595119894120595119895119889120578 minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987034

119894119895= 0 119870

41

119894119895= 0

11987042

119894119895= (1 + 119870)PrEcint

120578119890+1

120578119890

120595119894

119889ℎ

119889120578

119889120595119895

119889120578

119889120578 11987043

119894119895= 0

11987044

119894119895= minusint

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + Print120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus Print120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus Pr120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus Pr1205912

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

1198871

119894= 0 119887

2

119894= minus (1 + 119870)(120595

119894

119889ℎ

119889120578

)

120578119890+1

120578119890

1198873

119894= minus(1 +

119870

2

)(120595119894

119889119892

119889120578

)

120578119890+1

120578119890

1198874

119894= minus(120595

119894

119889120579

119889120578

)

120578119890+1

120578119890

(19)

where 119891 = sum2

119894=1119891119894120595119894and ℎ = sum

2

119894=1ℎ119894120595119894are assumed to be

known After the assembly of element equations a systemof non-linear equations is obtained therefore an iterativescheme must be utilized to solve it The system is linearizedby incorporating the functions 119891 and ℎ which are assumedto be known at lower iteration level and the computations

Table 1 Convergence of results with the variation of number ofelements 119899 (119870 = 2 Pr = 0733 120582 = 3 120591 = 1 120590 = 3 Ec = 075)

119899 119891 (16) ℎ (16) 119892 (16) 120579(16)

20 27194 00714 minus01087 0185740 26986 00524 minus01198 0177260 26955 00496 minus01223 0175680 26945 00487 minus01231 01751100 26941 00483 minus01236 01748120 26939 00481 minus01238 01747140 26937 00480 minus01239 01746160 26937 00479 minus01240 01745180 26936 00479 minus01241 01745

Table 2 Comparison of the flow velocity 1198911015840

(120578) obtained byanalytical method [24] and FEM (119870 = 0 Pr = 0733 120582 = 3 120591 =

0 120590 = 0 Ec = 0) in the special case

120578

1198911015840

(120578)

[24] FEM0 minus1 minus11 minus007295 minus0072902 minus000532 minus0005313 minus000039 minus0000394 minus000003 minus0000035 6 7 8 000000 000000

for119891 ℎ 119892 and 120579 are then carried out for higher levels Thisprocess is repeated until the desired accuracy of 000005 isattained Convergence of the results with increasing numberof elements is shown in Table 1 It is clear from the tablethat for more than 160 elements no significant variationin the values of 119891 ℎ 119892 and 120579 is observed Thus the finalresults are reported for 160 elementsThis confirms themesh-independence of the present computation

For steady state (120591 = 0) viscous fluid (119870 = 0) and in theabsence of buoyancy force (120590 = 0) the exact solution for119891(120578)as obtained by Fang and Zhang [24] with119872 = 0 is given by119891(120578) = 120582 minus (1 minus 119890

minus120578119911

)119911 where 119911 = 05[120582 + (1205822

minus 4)12

]The comparison of the flow velocity 119891

1015840

(120578) obtained byfinite element method and the exact solution given by Fangand Zhang [24] is shown in Table 2 It is clear from the tablethat numerical results obtained are in complete agreementwith the exact solution and thus confirm the validity andaccuracy of the FEM

4 Results and Discussion

To study the behaviour of velocity microrotation and tem-perature functions comprehensive numerical computationsare carried out for various values of the parameters namelysuction parameter 120582 unsteadiness parameter 120591 buoyancyparameter 120590 and Eckert number Ec The other parameterssuch as coupling constant parameter 119870 and Prandtl numberPr are kept fixed at 20 and 0733 respectively The resultsobtained are presented through the graphs as shown in

6 The Scientific World Journal

Table 3 The skin friction coefficient 11989110158401015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 11989110158401015840

(0) 120591 11989110158401015840

(0) 120590 11989110158401015840

(0) Ec 11989110158401015840

(0)

3 249867 1 249867 0 131709 002 21597135 258229 2 249630 1 174698 025 2260444 270175 3 252259 2 212938 05 23757945 285110 4 257288 3 249867 075 2498675 302241 5 263686 4 287143 1 263105

Table 4 The local Nusselt number minus1205791015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 minus1205791015840

(0) 120591 minus1205791015840

(0) 120590 minus1205791015840

(0) Ec minus1205791015840

(0)

3 minus024924 1 minus024924 0 088895 002 19959635 012914 2 014660 1 054444 025 1394364 045990 3 046597 2 017095 05 06352745 075067 4 071556 3 minus024924 075 minus0249245 101279 5 091656 4 minus073308 1 minus128127

Figures 2ndash13 The skin friction and the rate of heat transferhave also been computed for these parameters and aretabulated in Tables 3 and 4

Figures 2ndash4 display the effect of suction parameter 120582 onthe boundary layer flow induced by an unsteady shrinkingsheet Suction is the most suitable force to sustain the flownear the shrinking sheet by confining the generated vorticityinside the boundary layer It is clear from Figure 2 that theeffect of suction parameter 120582 on the velocity is negligible nearthe sheet whereas away from the sheet velocity decreases withincrease in suction parameter Negative values of velocityclose to the sheet indicate the region of reverse flow Figure 3shows that as suction parameter increases the microrotationdecreases near the sheet After covering a small distancefrom the sheet the opposite behaviour is observed Thenegative values of microrotation show the reverse rotationof microparticles Figure 4 illustrates the influence of suctionparameter 120582 on the temperature distribution It is evidentfrom the figure that temperature of the fluid decreases withincrease in suction parameter As suction is applied thermalboundary layer thickness decreases as a result of whichtemperature of the fluid in the boundary layer decreasesThis observation is in agreement with Bhattacharyya andLayek [25] Thus suction parameter can be used effectivelyfor controlling the flow and heat transfer characteristics

The effect of unsteadiness parameter 120591 on the velocitymicrorotation and temperature functions is shown in Figures5ndash7 Figure 5 shows that increasing the value of unsteadinessparameter 120591 tends to decrease the velocity in the boundarylayer Near the sheet flow velocity is negative whereas awayfrom the sheet it becomes positive and finally satisfies thefree stream boundary condition Thus close to the sheet theflow is strongly reversed Figure 6 demonstrates that near

0 1 2 3 4 5 6 7

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120582 = 3 35 4 45 5120578

Figure 2 Velocity distribution for different 120582 (120591 = 1 120590 = 3Ec =

075)

0 1 2 3

120582 = 3 35 4 45 5

120578

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

Figure 3 Microrotation distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

the sheet the effect of unsteadiness on the microrotation isnegligible but away from the sheet it increases with increasein unsteadiness parameter After covering a certain distancefrom the sheet all profiles converge and finally satisfy thefar field boundary condition Figure 7 reveals that the tem-perature decreases with increase in unsteadiness parameter120591 Temperature at the sheet is invariant for higher values ofunsteadiness parameter These patterns are consistent withthe findings of Ishak et al [35] Physically as the value ofunsteadiness parameter increases the sheet loses more heatas a result of which temperature of the fluid decreases

Figures 8ndash10 display the nature of velocity microrotationand temperature distributions with buoyancy parameter 120590Physically positive values of 120590 are associated with cooling ofthe sheet (assisting flow) negative values of120590 indicate heatingof the sheet (opposing flow) and 120590 = 0 implies vanishing

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

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The Scientific World Journal 5

where [119870119898119899

] and [119887119898

] (119898 119899 = 1 2 3 4) are the matrices oforder 2 times 2 and 2 times 1 respectively and are defined as follows

11987011

119894119895= int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987012

119894119895= minusint

120578119890+1

120578119890

120595119894120595119895119889120578

11987013

119894119895= 11987014

119894119895= 0 119870

21

119894119895= 0

11987022

119894119895= minus (1 + 119870)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987023

119894119895= 119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578 11987024

119894119895= 120590int

120578119890+1

120578119890

120595119894120595119895119889120578

11987031

119894119895= 0 119870

32

119894119895= minus119870int

120578119890+1

120578119890

120595119894

119889120595119895

119889120578

119889120578

11987033

119894119895= minus(1 +

119870

2

)int

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + int

120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus int

120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus 2119870int

120578119890+1

120578119890

120595119894120595119895119889120578

minus

3

2

120591int

120578119890+1

120578119890

120595119894120595119895119889120578 minus

120591

2

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

11987034

119894119895= 0 119870

41

119894119895= 0

11987042

119894119895= (1 + 119870)PrEcint

120578119890+1

120578119890

120595119894

119889ℎ

119889120578

119889120595119895

119889120578

119889120578 11987043

119894119895= 0

11987044

119894119895= minusint

120578119890+1

120578119890

119889120595119894

119889120578

119889120595119895

119889120578

119889120578 + Print120578119890+1

120578119890

119891120595119894

119889120595119895

119889120578

119889120578

minus Print120578119890+1

120578119890

ℎ120595119894120595119895119889120578 minus Pr120591int

120578119890+1

120578119890

120595119894120595119895119889120578

minus Pr1205912

int

120578119890+1

120578119890

120578120595119894

119889120595119895

119889120578

119889120578

1198871

119894= 0 119887

2

119894= minus (1 + 119870)(120595

119894

119889ℎ

119889120578

)

120578119890+1

120578119890

1198873

119894= minus(1 +

119870

2

)(120595119894

119889119892

119889120578

)

120578119890+1

120578119890

1198874

119894= minus(120595

119894

119889120579

119889120578

)

120578119890+1

120578119890

(19)

where 119891 = sum2

119894=1119891119894120595119894and ℎ = sum

2

119894=1ℎ119894120595119894are assumed to be

known After the assembly of element equations a systemof non-linear equations is obtained therefore an iterativescheme must be utilized to solve it The system is linearizedby incorporating the functions 119891 and ℎ which are assumedto be known at lower iteration level and the computations

Table 1 Convergence of results with the variation of number ofelements 119899 (119870 = 2 Pr = 0733 120582 = 3 120591 = 1 120590 = 3 Ec = 075)

119899 119891 (16) ℎ (16) 119892 (16) 120579(16)

20 27194 00714 minus01087 0185740 26986 00524 minus01198 0177260 26955 00496 minus01223 0175680 26945 00487 minus01231 01751100 26941 00483 minus01236 01748120 26939 00481 minus01238 01747140 26937 00480 minus01239 01746160 26937 00479 minus01240 01745180 26936 00479 minus01241 01745

Table 2 Comparison of the flow velocity 1198911015840

(120578) obtained byanalytical method [24] and FEM (119870 = 0 Pr = 0733 120582 = 3 120591 =

0 120590 = 0 Ec = 0) in the special case

120578

1198911015840

(120578)

[24] FEM0 minus1 minus11 minus007295 minus0072902 minus000532 minus0005313 minus000039 minus0000394 minus000003 minus0000035 6 7 8 000000 000000

for119891 ℎ 119892 and 120579 are then carried out for higher levels Thisprocess is repeated until the desired accuracy of 000005 isattained Convergence of the results with increasing numberof elements is shown in Table 1 It is clear from the tablethat for more than 160 elements no significant variationin the values of 119891 ℎ 119892 and 120579 is observed Thus the finalresults are reported for 160 elementsThis confirms themesh-independence of the present computation

For steady state (120591 = 0) viscous fluid (119870 = 0) and in theabsence of buoyancy force (120590 = 0) the exact solution for119891(120578)as obtained by Fang and Zhang [24] with119872 = 0 is given by119891(120578) = 120582 minus (1 minus 119890

minus120578119911

)119911 where 119911 = 05[120582 + (1205822

minus 4)12

]The comparison of the flow velocity 119891

1015840

(120578) obtained byfinite element method and the exact solution given by Fangand Zhang [24] is shown in Table 2 It is clear from the tablethat numerical results obtained are in complete agreementwith the exact solution and thus confirm the validity andaccuracy of the FEM

4 Results and Discussion

To study the behaviour of velocity microrotation and tem-perature functions comprehensive numerical computationsare carried out for various values of the parameters namelysuction parameter 120582 unsteadiness parameter 120591 buoyancyparameter 120590 and Eckert number Ec The other parameterssuch as coupling constant parameter 119870 and Prandtl numberPr are kept fixed at 20 and 0733 respectively The resultsobtained are presented through the graphs as shown in

6 The Scientific World Journal

Table 3 The skin friction coefficient 11989110158401015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 11989110158401015840

(0) 120591 11989110158401015840

(0) 120590 11989110158401015840

(0) Ec 11989110158401015840

(0)

3 249867 1 249867 0 131709 002 21597135 258229 2 249630 1 174698 025 2260444 270175 3 252259 2 212938 05 23757945 285110 4 257288 3 249867 075 2498675 302241 5 263686 4 287143 1 263105

Table 4 The local Nusselt number minus1205791015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 minus1205791015840

(0) 120591 minus1205791015840

(0) 120590 minus1205791015840

(0) Ec minus1205791015840

(0)

3 minus024924 1 minus024924 0 088895 002 19959635 012914 2 014660 1 054444 025 1394364 045990 3 046597 2 017095 05 06352745 075067 4 071556 3 minus024924 075 minus0249245 101279 5 091656 4 minus073308 1 minus128127

Figures 2ndash13 The skin friction and the rate of heat transferhave also been computed for these parameters and aretabulated in Tables 3 and 4

Figures 2ndash4 display the effect of suction parameter 120582 onthe boundary layer flow induced by an unsteady shrinkingsheet Suction is the most suitable force to sustain the flownear the shrinking sheet by confining the generated vorticityinside the boundary layer It is clear from Figure 2 that theeffect of suction parameter 120582 on the velocity is negligible nearthe sheet whereas away from the sheet velocity decreases withincrease in suction parameter Negative values of velocityclose to the sheet indicate the region of reverse flow Figure 3shows that as suction parameter increases the microrotationdecreases near the sheet After covering a small distancefrom the sheet the opposite behaviour is observed Thenegative values of microrotation show the reverse rotationof microparticles Figure 4 illustrates the influence of suctionparameter 120582 on the temperature distribution It is evidentfrom the figure that temperature of the fluid decreases withincrease in suction parameter As suction is applied thermalboundary layer thickness decreases as a result of whichtemperature of the fluid in the boundary layer decreasesThis observation is in agreement with Bhattacharyya andLayek [25] Thus suction parameter can be used effectivelyfor controlling the flow and heat transfer characteristics

The effect of unsteadiness parameter 120591 on the velocitymicrorotation and temperature functions is shown in Figures5ndash7 Figure 5 shows that increasing the value of unsteadinessparameter 120591 tends to decrease the velocity in the boundarylayer Near the sheet flow velocity is negative whereas awayfrom the sheet it becomes positive and finally satisfies thefree stream boundary condition Thus close to the sheet theflow is strongly reversed Figure 6 demonstrates that near

0 1 2 3 4 5 6 7

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120582 = 3 35 4 45 5120578

Figure 2 Velocity distribution for different 120582 (120591 = 1 120590 = 3Ec =

075)

0 1 2 3

120582 = 3 35 4 45 5

120578

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

Figure 3 Microrotation distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

the sheet the effect of unsteadiness on the microrotation isnegligible but away from the sheet it increases with increasein unsteadiness parameter After covering a certain distancefrom the sheet all profiles converge and finally satisfy thefar field boundary condition Figure 7 reveals that the tem-perature decreases with increase in unsteadiness parameter120591 Temperature at the sheet is invariant for higher values ofunsteadiness parameter These patterns are consistent withthe findings of Ishak et al [35] Physically as the value ofunsteadiness parameter increases the sheet loses more heatas a result of which temperature of the fluid decreases

Figures 8ndash10 display the nature of velocity microrotationand temperature distributions with buoyancy parameter 120590Physically positive values of 120590 are associated with cooling ofthe sheet (assisting flow) negative values of120590 indicate heatingof the sheet (opposing flow) and 120590 = 0 implies vanishing

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 The Scientific World Journal

Table 3 The skin friction coefficient 11989110158401015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 11989110158401015840

(0) 120591 11989110158401015840

(0) 120590 11989110158401015840

(0) Ec 11989110158401015840

(0)

3 249867 1 249867 0 131709 002 21597135 258229 2 249630 1 174698 025 2260444 270175 3 252259 2 212938 05 23757945 285110 4 257288 3 249867 075 2498675 302241 5 263686 4 287143 1 263105

Table 4 The local Nusselt number minus1205791015840(0) for different values of120582 120591 120590 and Ec (119870 = 2Pr = 0733)

120591 = 1 120590 = 3

Ec = 075

120582 = 3 120590 = 3

Ec = 075

120582 = 3 120591 = 1

Ec = 075

120582 = 3 120591 = 1

120590 = 3

120582 minus1205791015840

(0) 120591 minus1205791015840

(0) 120590 minus1205791015840

(0) Ec minus1205791015840

(0)

3 minus024924 1 minus024924 0 088895 002 19959635 012914 2 014660 1 054444 025 1394364 045990 3 046597 2 017095 05 06352745 075067 4 071556 3 minus024924 075 minus0249245 101279 5 091656 4 minus073308 1 minus128127

Figures 2ndash13 The skin friction and the rate of heat transferhave also been computed for these parameters and aretabulated in Tables 3 and 4

Figures 2ndash4 display the effect of suction parameter 120582 onthe boundary layer flow induced by an unsteady shrinkingsheet Suction is the most suitable force to sustain the flownear the shrinking sheet by confining the generated vorticityinside the boundary layer It is clear from Figure 2 that theeffect of suction parameter 120582 on the velocity is negligible nearthe sheet whereas away from the sheet velocity decreases withincrease in suction parameter Negative values of velocityclose to the sheet indicate the region of reverse flow Figure 3shows that as suction parameter increases the microrotationdecreases near the sheet After covering a small distancefrom the sheet the opposite behaviour is observed Thenegative values of microrotation show the reverse rotationof microparticles Figure 4 illustrates the influence of suctionparameter 120582 on the temperature distribution It is evidentfrom the figure that temperature of the fluid decreases withincrease in suction parameter As suction is applied thermalboundary layer thickness decreases as a result of whichtemperature of the fluid in the boundary layer decreasesThis observation is in agreement with Bhattacharyya andLayek [25] Thus suction parameter can be used effectivelyfor controlling the flow and heat transfer characteristics

The effect of unsteadiness parameter 120591 on the velocitymicrorotation and temperature functions is shown in Figures5ndash7 Figure 5 shows that increasing the value of unsteadinessparameter 120591 tends to decrease the velocity in the boundarylayer Near the sheet flow velocity is negative whereas awayfrom the sheet it becomes positive and finally satisfies thefree stream boundary condition Thus close to the sheet theflow is strongly reversed Figure 6 demonstrates that near

0 1 2 3 4 5 6 7

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120582 = 3 35 4 45 5120578

Figure 2 Velocity distribution for different 120582 (120591 = 1 120590 = 3Ec =

075)

0 1 2 3

120582 = 3 35 4 45 5

120578

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

Figure 3 Microrotation distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

the sheet the effect of unsteadiness on the microrotation isnegligible but away from the sheet it increases with increasein unsteadiness parameter After covering a certain distancefrom the sheet all profiles converge and finally satisfy thefar field boundary condition Figure 7 reveals that the tem-perature decreases with increase in unsteadiness parameter120591 Temperature at the sheet is invariant for higher values ofunsteadiness parameter These patterns are consistent withthe findings of Ishak et al [35] Physically as the value ofunsteadiness parameter increases the sheet loses more heatas a result of which temperature of the fluid decreases

Figures 8ndash10 display the nature of velocity microrotationand temperature distributions with buoyancy parameter 120590Physically positive values of 120590 are associated with cooling ofthe sheet (assisting flow) negative values of120590 indicate heatingof the sheet (opposing flow) and 120590 = 0 implies vanishing

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 7

0

02

04

06

08

1

12

0 1 2 3 4

120582 = 3 35 4 45 5

120578

120579

Figure 4 Temperature distribution for different 120582 (120591 = 1 120590 =

3Ec = 075)

0 1 2 3 4 5 6 7

f998400

120578

minus1

minus08

minus06

minus02

minus04

0

02

120591 = 1 2 3 4 5

Figure 5 Velocity distribution for different 120591 (120582 = 3 120590 = 3Ec =

075)

buoyancy effects FromFigure 8 it can be seen that velocity ofthe fluid increases with increase in buoyancy parameter Thisis due to the reason that an increase in the value of buoyancyparameter leads to an increase in the temperature difference(119879119908

minus 119879infin) This leads to an increase in the convection

currents as a result of which fluid velocity increases andthus boundary layer thickness decreases This trend has alsobeen identified by Shit and Haldar [41] Figure 9 indicatesthat the microrotation decreases initially with increase inbuoyancy parameter For 120578 asymp 115 all profiles intersectand then increases with increases in buoyancy parameter Inall cases microrotation is negative which shows the reverserotation of microelements A crossing over point appears inthe temperature profile as shown in Figure 10 This point isspecial where all temperature curves cross each other that isthe temperature profile shows different behaviour before and

0 1 2 3 4

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

g

120578

120591 = 1 2 3 4 5

Figure 6 Microrotation distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

0

02

04

06

08

1

12

0 1 2 3 4 5

120579

120578

120591 = 1 2 3 4 5

Figure 7 Temperature distribution for different 120591 (120582 = 3 120590 =

3Ec = 075)

after this point It is observed that as the buoyancy parameterincreases fluid temperature increases up to this point anddecreases after this pointMaximum temperature occurs nearthe sheet corresponding to 120590 = 4 Temperature at the sheet isinvariant for lower values of buoyancy parameter

Figures 11ndash13 present the distributions of velocity micro-rotation and temperature functions with Eckert numberEc This parameter is called the fluid motion controllingparameter From Figure 11 it is observed that the velocityincreases with increase in Eckert number Ec Boundary layerthickness will therefore decrease as Eckert number increasesFlow reversal arises near the sheet as testified by the negativevalues of velocity Figure 12 demonstrates that in the vicinityof sheet microrotation decreases with increase in Eckertnumber Minimum value of microrotation corresponds toEc = 1 and this value is minus132 approximately After a

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

0 2 4 6 8

minus1

minus08

minus06

minus04

minus02

0

02

f998400

120578

120590 = 0 1 2 3 4

Figure 8 Velocity distribution for different 120590 (120582 = 3 120591 = 1Ec =

075)

0 1 2 3 4 5 6120578

120590 = 0 1 2 3 4

g

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Figure 9 Microrotation distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

small distance from the sheet all profiles converge and finallysatisfy the far field boundary condition Figure 13 reveals thatthe temperature as well as thermal boundary layer thicknessincreases with increase in Eckert number Ec A temperatureovershoot near the sheet has been observed for higher valuesof Eckert numberThis is due to the fact that for higher valuesof Eckert number there is a significant generation of heat dueto viscous dissipation near the sheet so that the temperaturein the region close to the sheet exceeds the temperature of thewall 119879

119908

Table 3 gives the skin friction for different values of 120582120591 120590 and Ec It is evident that the skin friction decreasesslightly with a small increase in unsteadiness parameter whileit increases with increase in suction parameter buoyancyparameter Eckert number and large values of unsteadinessparameters Physically positive values of the skin friction

0

02

04

06

08

1

12

0 1 2 3 4 5120578

120579

120590 = 0 1 2 3 4

Figure 10 Temperature distribution for different 120590 (120582 = 3 120591 =

1Ec = 075)

0 2 4 6 8

f998400

120578

minus1

minus08

minus06

minus04

minus02

0

02

002 025 05 075 1Ec =

Figure 11 Velocity distribution for different Ec (120582 = 3 120591 = 1 120590 =

3)

show that the fluid exerts a drag force on the sheet It has alsobeen observed that the skin friction is lower in the absenceof buoyancy parameter Thus skin friction can be reducedeffectively by assigning lower values to unsteadiness parame-ter and in the absence of buoyancy parameter FromTable 4 itmay be noted that the rate of heat transfer decreases initiallywith increase in buoyancy parameter Eckert number smallvalues of suction and unsteadiness parameters It is alsoobserved that heat transfer rate increases numerically withincrease in suction unsteadiness parameters large values ofbuoyancy parameter and Eckert number Positive values ofthe heat transfer rate indicate that the heat is transferredfrom the surface of the sheet to the fluid and negativevalues mean the opposite Thus effective cooling of thesheet can be achieved by the judicious selection of theseparameters

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

0 1 2 3 4

g

120578

002 025 05 075 1

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Ec =

Figure 12 Microrotation distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

0

01

02

03

04

05

06

07

08

09

1

11

0 1 2 3 4 5120578

120579

002 025 05 075 1Ec =

Figure 13 Temperature distribution for different Ec (120582 = 3 120591 =

1 120590 = 3)

5 Conclusions

The present study has addressed theoretically and numeri-cally the unsteady mixed convection flow and heat transfer ofan incompressible micropolar fluid over a porous shrinkingsheet in the presence of viscous dissipation Using similaritytransformations the governing time-dependent boundarylayer equations are reduced to a set of nonlinear ordinarydifferential equations A variational finite element methodhas been employed to solve the nondimensional momentumangular momentum and thermal boundary layer equationssubject to physically realistic boundary conditions Underlimiting cases the numerical results obtained for the flowvelocity are compared very well with the exact solutionavailable in the literature Numerical computations haveclearly demonstrated that the drag can be reduced effectively

with the judicious selection of unsteadiness parameter andbuoyancy parameter It has also been found that a fast rate ofcooling can be achieved by implementing suction parameterunsteadiness parameter higher values of buoyancy parameterand Eckert number The present study has neglected velocityand thermal slip effects at the sheetThese may be consideredin the future It is hoped that the results obtained from thepresentworkmay be useful for differentmodel investigations

Nomenclature119888119901 Specific heat at constant pressure

119862119891 Skin friction coefficient

119891 Dimensionless velocity119892 Dimensionless microrotation119892119890 Gravitational acceleration

Gr119909 Local Grashof number

119895 Microinertia density119870 Coupling constant parameter119873 Microrotation componentNu119909 Local Nusselt number

Pr Prandtl number119902 Heat fluxRe119909 Local Reynolds number

119878 Constant characteristic of the fluid119879 Temperature of the fluid119879infin Temperature of the ambient fluid

119905 Time119906 Velocity in the 119909-direction119880119908 Velocity of the sheet

V Velocity in the 119910-direction119881119908 Velocity at the wall

119908119894 Weight functions

119909 Distance along the surface119910 Distance normal to the surface

Greek Symbols

120574 Spin gradient viscosity120578 Similarity variables120583 The dynamic viscosity120588 Density of the fluid120581 Thermal conductivity] Kinematic viscosity120579 Dimensionless temperature120582 Suction parameter120601 Stream function120590 Buoyancy parameter120591 Unsteadiness parameter120591119908 Wall shear stress

Subscripts

119908 Surface conditioninfin Conditions far away from the surface

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

References

[1] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matical Fluid Mechanics vol 16 pp 1ndash18 1966

[2] A C Eringen Microcontinuum Field Theories II-Fluent MediaSpringer New York NY USA 1st edition 2001

[3] O A Beg R Bhargava and M M Rashidi NumericalSimulation in Micropolar Fluid Dynamics Lambert AcademicPublishing 2011

[4] A C Eringen ldquoTheory of thermomicrofluidsrdquo Journal ofMathematical Analysis and Applications vol 38 no 2 pp 480ndash496 1972

[5] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4 pp645ndash647 1970

[6] H S Takhar and V M Soundalgekar ldquoFlow and heat transferof a micropolar fluid past a continuously moving porous platerdquoInternational Journal of Engineering Science vol 23 no 2 pp201ndash205 1985

[7] I AHassanien andR S RGorla ldquoHeat transfer to amicropolarfluid from a non-isothermal stretching sheet with suction andblowingrdquo Acta Mechanica vol 84 no 1ndash4 pp 191ndash199 1990

[8] H A M El-Arabawy ldquoEffect of suctioninjection on the flowof a micropolar fluid past a continuously moving plate in thepresence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[9] R Nazar N Amin D Filip and I Pop ldquoStagnation point flowof a micropolar fluid towards a stretching sheetrdquo InternationalJournal of Non-Linear Mechanics vol 39 no 7 pp 1227ndash12352004

[10] L Kumar ldquoFinite element analysis of combined heat and masstransfer in hydromagnetic micropolar flow along a stretchingsheetrdquo Computational Materials Science vol 46 no 4 pp 841ndash848 2009

[11] A Ishak ldquoThermal boundary layer flow over a stretching sheetin a micropolar fluid with radiation effectrdquo Meccanica vol 45no 3 pp 367ndash373 2010

[12] S Rawat R Bhargava S Kapoor and O A Beg ldquoHeatand mass transfer of a chemically reacting micropolar fluidover a linear stretching sheet in darcy forchheimer porousmediumrdquo International Journal of Computational and AppliedMathematics vol 44 no 6 pp 40ndash51 2012

[13] C D S Devi H S Takhar and G Nath ldquoUnsteady three-dimensional boundary-layer flow due to a stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 29 no 12pp 1996ndash1999 1986

[14] V Rajeswari and G Nath ldquoUnsteady flow over a stretchingsurface in a rotating fluidrdquo International Journal of EngineeringScience vol 30 no 6 pp 747ndash756 1992

[15] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[16] M Abd El-Aziz ldquoRadiation effect on the flow and heat transferover an unsteady stretching sheetrdquo International Communica-tions in Heat andMass Transfer vol 36 no 5 pp 521ndash524 2009

[17] Y Y Lok N Amin and I Pop ldquoUnsteady boundary layer flowof a micropolar fluid near the rear stagnation point of a planesurfacerdquo International Journal of Thermal Sciences vol 42 no11 pp 995ndash1001 2003

[18] T Hayat M Nawaz and S Obaidat ldquoAxisymmetric magne-tohydrodynamic flow of micropolar fluid between unsteady

stretching surfacesrdquo Applied Mathematics and Mechanics vol32 no 3 pp 361ndash374 2011

[19] N Bachok A Ishak and R Nazar ldquoFlow and heat transfer overan unsteady stretching sheet in a micropolar fluidrdquo Meccanicavol 46 no 5 pp 935ndash942 2011

[20] C Y Wang ldquoLiquid film on an unsteady stretching sheetrdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[21] M Miklavcic and C Y Wang ldquoViscous flow due to a shrinkingsheetrdquoQuarterly of AppliedMathematics vol 64 no 2 pp 283ndash290 2006

[22] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[23] N Bachok A Ishak and I Pop ldquoUnsteady three-dimensionalboundary layer flow due to a permeable shrinking sheetrdquoApplied Mathematics and Mechanics vol 31 no 11 pp 1421ndash1428 2010

[24] T Fang and J Zhang ldquoClosed-form exact solutions ofMHDvis-cous flow over a shrinking sheetrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 7 pp 2853ndash28572009

[25] K Bhattacharyya and G C Layek ldquoEffects of suctionblowingon steady boundary layer stagnation-point flow and heattransfer towards a shrinking sheet with thermal radiationrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1pp 302ndash307 2011

[26] A Ishak Y Y Lok and I Pop ldquoStagnation-point flow over ashrinking sheet in a micropolar fluidrdquo Chemical EngineeringCommunications vol 197 no 11 pp 1417ndash1427 2010

[27] N A Yacob and A Ishak ldquoMicropolar fluid flow over ashrinking sheetrdquoMeccanica vol 47 no 2 pp 293ndash299 2012

[28] D Gupta L Kumar O A Beg and B Singh ldquoFinite elementsimulation of mixed convection flow of micropolar fluid overa shrinking sheet with thermal radiationrdquo Journal of ProcessMechanical Engineering 2013

[29] T G Fang J Zhang and S S Yao ldquoViscous flow over anunsteady shrinking sheet with mass transferrdquo Chinese PhysicsLetters vol 26 no 1 Article ID 014703 2009

[30] F M Ali R Nazar N M Arifin and I Pop ldquoUnsteady flowand heat transfer past an axisymmetric permeable shrinkingsheet with radiation effectrdquo International Journal for NumericalMethods in Fluids vol 67 no 10 pp 1310ndash1320 2011

[31] K Bhattacharyya ldquoEffects of radiation and heat sourcesink onunsteady MHD boundary layer flow and heat transfer over ashrinking sheet with suctioninjectionrdquo Frontiers of ChemicalEngineering in China vol 5 no 3 pp 376ndash384 2011

[32] M Suali N M A N Long and A Ishak ldquoUnsteady stagnationpoint flow and heat transfer over a stretchingshrinking sheetwith prescribed surface heat fluxrdquo Applied Mathematics andComputational Intelligence vol 1 no 1 pp 1ndash11 2012

[33] T R Mahapatra and S K Nandy ldquoSlip effects on unsteadystagnation-point flow and heat transfer over a shrinking sheetrdquoMecannica vol 48 no 7 pp 1599ndash1606 2013

[34] C D S Devi H S Takhar and G Nath ldquoUnsteady mixedconvection flow in stagnation region adjacent to a verticalsurfacerdquo Heat Mass Transfer vol 26 no 2 pp 71ndash79 1991

[35] A Ishak R Nazar and I Pop ldquoBoundary layer flow andheat transfer over an unsteady stretching vertical surfacerdquoMeccanica vol 44 no 4 pp 369ndash375 2009

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 11

[36] SMukhopadhyay ldquoEffects of slip on unsteadymixed convectiveflow and heat transfer past a stretching surfacerdquo Chinese PhysicsLetters vol 27 no 12 Article ID 124401 2010

[37] R Sharma R Bhargava and I V Singh ldquoA numerical solutionof MHD convection heat transfer over an unsteady stretchingsurface embedded in a porous medium using element freeGalerkinmethodrdquo International Journal of AppliedMathematicsand Mechanics vol 8 no 10 pp 83ndash103 2012

[38] K Vajravelu K V Prasad and C O Ng ldquoUnsteady convectiveboundary layer flow of a viscous fluid at a vertical surfacewith variable fluid propertiesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 455ndash464 2013

[39] K Bhattacharyya S Mukhopadhyay and G C LayekldquoUnsteady MHD boundary layer flow with diffusion and firstorder chemical reaction over a permeable stretching sheet withsuction or blowingrdquo Chemical Engineering Communicationsvol 200 no 3 pp 379ndash397 2013

[40] J N Reddy Introduction to the Finite ElementMethodMcGraw-Hill 2005

[41] G C Shit and R Haldar ldquoEffects of thermal radiation onMHDviscous fluid flow and heat transfer over nonlinear shrinkingporous sheetrdquo Applied Mathematics and Mechanics vol 32 no6 pp 677ndash688 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of