The Influence of Slip Boundary Condition on Casson Nanofluid ...downloads.hindawi.com/journals/mpe/2017/3804751.pdfMathematicalProblemsinEngineering 3 x y Slit Force Nanofluid boundary

Research ArticleThe Influence of Slip Boundary Condition onCasson Nanofluid Flow over a Stretching Sheet in the Presenceof Viscous Dissipation and Chemical Reaction

Ahmed A Afify12

1Department of Mathematics Deanship of Educational Services Qassim University PO Box 6595 Buraidah 51452 Saudi Arabia2Department of Mathematics Faculty of Science Helwan University PO Box 11795 Ain Helwan Cairo Egypt

Correspondence should be addressed to Ahmed A Afify afify65yahoocom

Received 2 March 2017 Revised 19 May 2017 Accepted 30 May 2017 Published 26 July 2017

Academic Editor Efstratios Tzirtzilakis

Copyright copy 2017 Ahmed A Afify 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 impacts of multiple slips with viscous dissipation on the boundary layer flow and heat transfer of a non-Newtonian nanofluidover a stretching surface have been investigated numerically The Casson fluid model is applied to characterize the non-Newtonianfluid behavior Physical mechanisms responsible for Brownian motion and thermophoresis with chemical reaction are accountedfor in the model The governing nonlinear boundary layer equations through appropriate transformations are reduced into a set ofnonlinear ordinary differential equations which are solved numerically using a shooting method with fourth-order Runge-Kuttaintegration scheme Comparisons of the numerical method with the existing results in the literature are made and an excellentagreement is obtained The heat transfer rate is enhanced with generative chemical reaction and concentration slip parameterwhereas the reverse trend is observed with destructive chemical reaction and thermal slip parameter It is also noticed that themass transfer rate is boosted with destructive chemical reaction and thermal slip parameter Further the opposite influence isfound with generative chemical reaction and concentration slip parameter

1 Introduction

Nowadays nanofluids have been utilized as the workingfluids instead of the base fluids due to their high thermalconductivity Choi [1] is the first researcher who establishedfluids containing a suspension of nanosize particles whichare termed the nanofluids Lee et al [2] confirmed that thenanofluids possess outstanding heat transfer characteristicscompared to those of base fluids Later onmany investigators[3ndash7] have indicated that the nanofluids enhanced thermo-physical characteristics and heat transfer behavior comparedto the base fluids The thermal conductivity enhanced by40 when copper nanoparticles with the volume fractionless than 1 are added to the ethylene glycol or oil wasstudied by Eastman et al [3] The enhancement of thermalconductivity of various nanofluids was reviewed by Aybaret al [4] They confirmed that the addition of nanoparticlesin the fluids increases the thermal conductivity Wang et al[5] discussed the viscosity of Al2O3 and CuO nanoparticles

dispersed in water vacuum pump fluid engine oil and ethy-lene glycol Their results indicated that 30 enhancementwith Al2O3water nanofluid at 3 volume concentration isobtained Afify and Bazid [6] investigated the impacts ofvariable viscosity and viscous dissipation on the boundarylayer flow and heat transfer along amoving permeable surfaceimmersed in nanofluids The influences of thermal radiationand particle shape on the Marangoni boundary layer flowand heat transfer of nanofluid driven by an exponentialtemperature were examined by Lin et al [7]

Buongiorno [8] modified the reasons behind theenhancement of heat transfer of nanofluids Recently manyresearchers [9ndash15] have effectively applied Buongiornorsquosmodel [8] Nield andKuznetsov [9] presented the influence ofBrownianmotion and thermophoresis on natural convectionboundary layer flow past a vertical plate suspended in aporousmediumThe steady two-dimensional boundary layerflow of a nanofluid past a stretching surface was examinedby Khan and Pop [10] Nield and Kuznetsov [11] discussed

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 3804751 12 pageshttpsdoiorg10115520173804751

2 Mathematical Problems in Engineering

the impact of Brownian motion and thermophoresis on thedouble-diffusive nanofluid convection past a vertical plateembedded in a porous medium Makinde and Aziz [12]discussed the influence of a convective boundary Brownianmotion and thermophoresis on the steady two-dimensionalboundary layer flow of a nanofluid past a stretching sheetRana et al [13] numerically examined the Brownian motionand thermophoresis effects on the steady mixed convectionboundary layer flow of nanofluid past an inclined plateembedded in a porous medium The influence of variablefluid properties with Brownian motion and thermophoresison the natural convective boundary layer flow in a nanofluidpast a vertical plate was numerically studied by Afifyand Bazid [14] Recently Makinde et al [15] numericallyexamined the effects of variable viscosity thermal radiationand magnetic field on convective heat transfer of nanofluidover a stretching surface

Non-Newtonian nanofluids are widely encountered inmany industrial and technological applications such as thedissolved polymers biological solutions paints asphalts andglues The power-law non-Newtonian nanofluid along a ver-tical plate and a truncated cone saturated in a porousmediumwere analyzed byHady et al [16] andCheng [17] respectivelyRashad et al [18] investigated the natural convection of non-Newtonian nanofluid around a vertical permeable cone Theinfluence of Soret and Dufour on the mixed convective flowof Maxwell nanofluid over a permeable stretched surfacewas examined by Ramzan et al [19] Several authors [20ndash23] have analytically solved the problem of non-Newtoniannanofluids in various aspects using the homotopy analysismethod (HAM) Abou-Zeid et al [24] examined the impactof viscous dispersion on the mixed convection of glidingmotion of bacteria on power-law nanofluids through a non-Darcy porous medium The influence of nonuniform heatsourcesink with the Brownian motion and thermophoresison non-Newtonian nanofluids over a cone was illustrated byRaju et al [25]

The Casson fluid model is classified as a subclass of non-Newtonian fluid which has several applications in food pro-cessing metallurgy drilling operations and bioengineeringoperationsTheCasson fluidmodelwas discovered byCassonin 1959 for the prediction of the flow behavior of pigment-oil suspensions [26] Boundary layer flow of Casson fluid andCasson nanofluid flow over different geometries was consid-ered by many authors [27ndash30] Mustafa et al [27] analyticallydiscussed the unsteady boundary layer flow of a Casson fluidover a moving flat plate using the homotopy analysis method(HAM) Nadeem et al [28] examined the analytical solutionof convective boundary conditions for the steady stagnation-point flow of a Casson nanofluid Makanda et al [29] studiedthe numerical solution of MHD Casson fluid flow over anunsteady stretching surface saturated in a porous mediumwith a chemical reaction effect Recently Hayat et al [30]investigated the combined effects of the variable thermalconductivity and viscous dissipation on the boundary layerflow of a Casson fluid due to a stretching cylinder

The chemical reaction influence is a significant factorin the study of heat and mass transfer for many branchesof science and engineering A chemical reaction between

the base liquid and nanoparticles may regularly occur eitherthroughout a given phase (homogeneous reaction) or inan enclosed region (boundary) of the phase (heterogeneousreaction) Das et al [31] investigated numerically the effectsof chemical reaction and thermal radiation on the heat andmass transfer of an electrically conducting incompressiblenanofluid over a heated stretching sheet The effect of time-dependent chemical reaction on stagnation-point flow andheat transfer of nanofluid over a stretching sheet was pre-sented by Abd El-Aziz [32] El-Dabe et al [33] studied theimpact of chemical reaction heat generation and radiationon the MHD flow of non-Newtonian nanofluid over astretching sheet embedded in a porous medium Eid [34]proposed the numerical analysis of MHD mixed convectiveboundary layer flow of a nanofluid through a porousmediumalong an exponentially stretching sheet in the presence ofchemical reaction and heat generation or absorption effectsRecently Afify and Elgazery [35] discussed the influences ofthe convective boundary condition and chemical reaction onthe MHD boundary layer flow of a Maxwell nanofluid over astretching surface

All these previous studies restricted their discussions onconventional no-slip boundary conditions However bothvelocity slip and temperature jump at the wall have numerousbenefits in many practical applications such as micro- andnanoscale devices In 1823 Navier [36] became the firstperson to create the slip boundary condition and suggestedthat the slip velocity is linearly proportional to the shearstress at the wall Following him many researchers [37ndash42] have widely studied the velocity slip and temperaturejump at the wall over various geometries placed in viscousfluids and nanofluids The impacts of slip and convectiveboundary condition on the unsteady three-dimensional flowof nanofluid over an inclined stretching surface embeddedin a porous medium were numerically discussed by Rashad[37] Afify et al [38] used Lie symmetry analysis to investigatethe effects of slip flow Newtonian heating and thermalradiation on MHD flow and heat transfer along the per-meable stretching sheet EL-Kabeir et al [39] examined thenumerical solution for mixed convection boundary layerflow of Casson non-Newtonian fluid about a solid spherein the presence of thermal and solutal slip conditions Afify[40] examined the impact of slips and generationabsorptionon an unsteady boundary layer flow and heat transfer overa stretching surface immersed in nanofluids Abolbashariet al [41] presented an analytical solution to analyze theheat and mass transfer characteristics of Casson nanofluidflow induced by a stretching sheet with velocity slip andconvective surface boundary conditions Recently Uddin etal [42] discussed the effects of Navier slip and variablefluid properties on the forced convection of nanofluid andheat transfer over a wedge The objective of the presentpaper is to investigate the influences of chemical reactionviscous dissipation velocity and thermal and concentrationslip boundary conditions in the presence of nanoparticlesattributable to Brownian diffusion and thermophoresis onflow and heat transfer of Casson fluid over a stretching sur-face To the best of the authorrsquos knowledge this work has notbeen previously studied in the scientific research Numerical

Mathematical Problems in Engineering 3

x

y

ForceSlit

Nanofluid boundary layer

Stretching sheet uw(x)

Figure 1 Physical model and coordinate system

results for the velocity temperature and nanoparticle con-centration fields are plotted The friction factor and the heatand mass transfer rates are also tabulated and discussed Thepresent paper confirmed that the nanoparticles embeddedin Casson fluid have many practical applications such asnuclear reactors microelectronics chemical production andbiomedical fields

2 Mathematical Formulation

Consider the steady boundary layer flow of an incompressibleCasson nanofluid past a stretching surface The sheet isstretched with a linear velocity 119906119908(119909) = 119887119909 where 119887 is thepositive constantThe 119909-axis is directed along the continuousstretching sheet and the 119910-axis is measured normal to the 119909-axis It is assumed that the flow takes place for119910 gt 0The tem-perature and the nanoparticle concentration are maintainedat prescribed constant values 119879119908 119862119908 at the surface and 119879infinand119862infin are the fixed values far away from the surface It is alsoassumed that there is a first-order homogeneous chemicalreaction of species with reaction rate constant 1198700 The flowconfiguration is shown in Figure 1 The rheological equationof state for an isotropic and incompressible flow of Cassonfluid is given by Eldabe and Salwa [43]

120591119894119895 =

2(120583119861 + 119901119910radic2120587) 119890119894119895 120587 gt 120587119888

2 (120583119861 + 119901119910radic2120587119888) 119890119894119895 120587 lt 120587119888

(1)

where 120583119861 is the plastic dynamic viscosity of the non-Newtonian fluid119901119910 is the yield stress of fluid120587 is the productof the component of deformation rate and itself namely 120587 =119890119894119895119890119894119895 119890119894119895 is the (119894 119895) component of the deformation rate and120587119888 is a critical value of 120587 based on non-Newtonian modelUnder the boundary layer approximations the governingequations of Casson nanofluid can be expressed as follows(Buongiorno [8] and Haq et al [44])

120597119906120597119909 +

120597V120597119910 = 0 (2)

119906120597119906120597119909 + V120597119906120597119910 = 120592(1 + 1120573)

12059721199061205971199102 (3)

119906120597119879120597119909 + V120597119879120597119910 = 12057212059721198791205971199102

+ 120591119863119861 (120597119862120597119910120597119879120597119910 ) +

119863119879119879infin (120597119879120597119910 )2

+ (1 + 1120573)120583120588119888119901 (

120597119906120597119910)2

(4)

119906120597119862120597119909 + V120597119862120597119910 = 119863119861 120597

21198621205971199102 +

119863119879119879infin12059721198791205971199102 minus 1198700 (119862 minus 119862infin) (5)

subject to the boundary conditions

119906 = 119906119908 + (1 + 1120573)119873120588]120597119906120597119910

V = 0119879 = 119879119908 + 1198701 120597119879120597119910

119862 = 119862119908 + 1198702 120597119862120597119910at 119910 = 0

119906 = 0119879 = 119879infin119862 = 119862infin

as 119910 997888rarr infin

(6)

where 119906 and V are the velocity components along the 119909-and 119910-axes respectively 120588119891 is the density of base fluid ] isthe kinematic viscosity of the base fluid 120572 = 119896120588119888119901 is thethermal diffusivity of the base fluid 120591 = (120588119888)119901(120588119888)119891 is theratio of nanoparticle heat capacity and the base fluid heatcapacity 119863119861 is the Brownian diffusion coefficient and 119863119879is the thermophoretic diffusion coefficient Furthermore 1198731198701 and1198702 are velocity thermal and concentration slip factorThe following nondimensional variables are defined as

120578 = (119887])12 119910

120595 (119909 119910) = (119887])12 119909119891 (120578) 120579 (120578) = 119879 minus 119879infin119879119908 minus 119879infin

120601 (120578) = 119862 minus 119862infin119862119908 minus 119862infin

(7)

The continuity equation (2) is satisfied by introducing thestream function 120595(119909 119910) such that

119906 = 120597120595120597119910

V = minus120597120595120597119909 (8)


In view of the above-mentioned transformations (3)ndash(6) arereduced to

(1 + 1120573)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 = 0 (9)

1Pr12057910158401015840 + 1198911205791015840 + 11987311988712060110158401205791015840 + 11987311990512057910158402 + (1 + 1120573)Ec119891101584010158402= 0

(10)

12060110158401015840 + Le1198911206011015840 + 11987311990511987311988712057910158401015840 minus Le120594120601 = 0 (11)

119891 (0) = 01198911015840 (0) = 1 + 120582(1 + 1120573)11989110158401015840 (0) 120579 (0) = 1 + 1205741205791015840 (0) 120601 (0) = 1 + 1205751206011015840 (0) 1198911015840 (infin) = 0120579 (infin) = 0120601 (infin) = 0

(12)

Here prime denotes differentiation with respect to 120578 119891 issimilarity function 120579 is the dimensionless temperature 120601is the dimensionless nanoparticle volume fraction Pr =]120572 is Prandtl number Le = ]119863119861 is Lewis number 120574 =1198701(119887])12 is the thermal slip parameter 120573 = 120583119861radic2120587119888119901119910 isthe Casson parameter 120582 = 119873120588(]119887)12 is the slip parameter120575 = 1198702(119887])12 is the concentration slip parameter Ec =1199062119908119888119901(119879119908 minus 119879infin) is the Eckert number 120594 = 1198700119887 is the chem-ical reaction parameter 119873119887 = (120588119888)119901119863119861(119862119908 minus 119862infin)(120588119888)119891] isthe Brownian motion parameter and 119873119905 = (120588119888)119901119863119879(119879119908 minus119879infin)(120588119888)119891]119879infin is the thermophoresis parameter respectivelyIt should be mentioned here that (120594 ≻ 0) indicates adestructive chemical reaction while (120594 ≺ 0) correspondsto a generative chemical reaction The quantities of physicalinterest in this problem are the local skin friction coefficient119862119891119909 the local Nusselt number Nu119909 and local Sherwoodnumber Sh119909 which are defined as

119862119891119909 = 1205911199081205881199062119908 Nu119909 = 119909119902119908119896 (119879119908 minus 119879infin) Sh119909 = 119909119902119898119863119861 (119862119908 minus 119862infin)

(13)

where 120591119908 is the shear stress and 119902119908 and 119902119898 are respectivelythe surface heat and mass flux which are given by thefollowing expressions

120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0

119902119898 = minus119863119861 (120597119862120597119910 )119910=0

(14)

The dimensionless forms of skin friction the local Nusseltnumber and the local Sherwood number become

Re12119909 119862119891 = (1 + 1120573)11989110158401015840 (0) Nu119909Re12119909

= minus1205791015840 (0) Sh119909Re12119909

= minus1206011015840 (0)

(15)

where Re119909 = 119909119906119908] is the local Reynolds number

3 Numerical Procedure

The nonlinear differential equations (9)ndash(11) along with theboundary conditions (12) constitute a two-point boundaryvalue problem (BVP) and they are solved numerically usingshooting method by converting them into an initial valueproblem (IVP) In this method the system of (9)ndash(11) isreduced to the following system of first-order ordinarydifferential equations

1198911015840 = 1199011199011015840 = 1199021199021015840 = ( 120573

1 + 120573) [1199012 minus 119891119902] 1205791015840 = 1199111206011015840 = 1199041199111015840 = minusPr(119891119911 + 119873119887119904119911 + 1198731199051199112 + (1 + 1120573)Ec1199022)

1199041015840 = Le (120594120601 minus 119891119904) minus 1198731199051198731198871199111015840

(16)

with the initial conditions

119891 (0) = 0119901 (0) = 1 + 120582(1 + 1120573) 119902 (0) 120579 (0) = 1 + 120574119911 (0) 120601 (0) = 1 + 120575119904 (0)

(17)

The initial guess values of 119902(0) that is 11989110158401015840(0) 119911(0) thatis 1205791015840(0) and 119904(0) that is 1206011015840(0) are chosen and the initialvalue problem (16)-(17) is solved repeatedly by using fourth-order Runge-Kutta method Then the calculated values of1198911015840(120578) 120579(120578) and 120601(120578) at 120578infin(=8) are compared with the givenboundary conditions 1198911015840(120578infin) = 0 120579(120578infin) = 0 and 120601(120578infin) =0 and the values of 11989110158401015840(0) 1205791015840(0) and 1206011015840(0) are adjustedby ldquosecant methodrdquo to give a better approximation for thesolution The step size is taken as Δ120578 = 0001 A conver-gence criterion based on the relative difference between the


Table 1 Comparison of results for minus1205791015840(0) with different values of Pr when 120573 rarr infin and Ec = 120575 = 120594 = 120574 = 120582 = 119873 = 119873119905 = Le = 0Pr

minus1205791015840 (0) minus1205791015840 (0) minus1205791015840 (0) minus1205791015840 (0) minus1205791015840 (0)Khan and Pop [10] Makinde and Aziz [12] Alsaedi et al [45] Abolbashari et al [41] Present results

007 00663 00656 00663 mdash 00662020 01691 01691 01691 01691 01691070 04539 04539 04539 04539 0453820 09113 09114 09113 09114 0911370 18954 18954 mdash 18954 18954

Table 2 Comparison of results for minus1205791015840(0) and minus1206011015840(0) with119873119887 and119873119905 for 120573 rarr infin Ec = 120575 = 120594 = 120574 = 120582 = 0 and Le = Pr = 10119873119887 119873119905 minus1205791015840 (0) minus1205791015840 (0) minus1206011015840(0) minus1206011015840(0)

Noghrehabadi et al [46] Present results Noghrehabadi et al [46] Present results

0101 0952377 0952376 2129394 212938903 0520079 0520079 2528638 252863505 0321054 0321054 3035142 3035144

0201 0505581 0505581 2381871 238187603 0273096 0273096 2655459 265545705 0168077 0168077 2888339 2888336

0301 0252156 0252156 2410019 241001503 0135514 0135514 2608819 260881705 0083298 0083298 2751875 2751873

Table 3 Comparison of results for minus1205791015840(0) and minus1205931015840(0) with119873119887 and119873119905 for 120573 rarr infin Ec = 120575 = 120594 = 120574 = 0 120582 = 05 and Le = Pr = 10119873119887 119873119905 minus1205791015840 (0) minus1205791015840 (0) minus1206011015840(0) minus1206011015840(0)


0101 0799317 0799310 1787171 178714203 0436495 0436492 2122251 212224305 0269457 0269454 2547353 2547347

0201 0424328 0424324 199907 19990403 0229206 0229204 2228691 222868805 0141064 0141061 2424144 2424139

0301 0211631 0211630 2022696 202267203 0113735 0113731 2189547 218953905 0069911 0069921 2309612 2309609

current and previous iteration values is employed When thedifference reaches less than 10minus6 the solution is assumedto converge and the iterative process is terminated Tables1ndash3 ensure the validation of the present numerical solutionswith the previous literature It is observed that the obtainedresults are in excellent agreement with the published work[10 12 41 45 46] Numerical calculations were performed inthe ranges 03 le 120573 le infin 0 le 120582 le 3 0 le 120574 lt 4 0 le 120575 le 301 le 119873119905 le 06 01 le 119873119887 le 06 1 le Pr le 10 minus03 le 120594 le 05and 1 le Le le 10 with 01 le Ec le 14 Results and Discussions

Thenonlinear ordinary differential equations (9)ndash(11) subjectto the boundary conditions (12) are solved numerically byusing a shooting method with the fourth-order Runge-Kuttaintegration scheme For various physical parameters the

numerical values of the friction factor (in terms of thewall velocity gradient 11989110158401015840(0)) Nusselt number (in terms ofheat transfer rate minus1205791015840(0)) and local Sherwood number (interms of mass transfer rate minus1206011015840(0)) are shown in Tables 4and 5 for both cases of Newtonian (120573 rarr infin) and non-Newtonian flows From Table 4 it is seen that the magnitudeof the friction factor and the heat and mass transfer ratesreduce with an increase in the slip parameter 120582 Physicallythe presence of a slip parameter generates a resistive forceadjacent to a stretching surface which diminishes the frictionfactor and heat and mass transfer rates One can observethat the heat transfer rate reduces with an increase in thethermal slip parameter 120574 whereas the opposite results effectis seen in the mass transfer rate Additionally it is noticedthat the heat transfer rate is enhanced with an increase inconcentration slip parameter 120575 whereas the opposite resultseffect is seen in the mass transfer rate It is clear that the


Table 4 Numerical values of (1 + 1120573)11989110158401015840(0) minus1205791015840(0) and minus1206011015840(0) with 120582 120574 120575 and 120573 for119873119905 = 119873119887 = 01 Ec = 120594 = 02 Pr = 4 and Le = 5120582 120574 120575 120573 (1 + 1120573) 11989110158401015840 (0) minus1205791015840 (0) minus1206011015840(0)0 02 02 05 minus1733110 0628232 132441 02 02 05 minus0541057 0587859 104733 02 02 05 minus0243961 0473596 097252802 0 02 05 minus1164996 076304 11727302 1 02 05 minus1164996 0411102 12419702 3 02 05 minus1164996 0208327 12884702 02 0 05 minus1164996 0620139 16393602 02 1 05 minus1164996 0707216 056935202 02 3 05 minus1164996 0734532 024676102 02 02 03 minus1319520 0651801 11884202 02 02 4 minus0846526 0655087 11852902 02 02 infin minus0776388 0652042 117971

Table 5 Numerical values of minus1205791015840(0) and minus1206011015840(0)with119873119905119873119887 120594 andEc for 120582 = 120574 = 120575 = 02 120573 = 05 Pr = 4 and Le = 5119873119905 119873119887 120594 Ec minus1205791015840 (0) minus1206011015840(0)01 01 02 02 0655854 11921303 01 02 02 0510010 12259805 01 02 02 0394448 14570301 02 02 02 0550359 12766501 04 02 02 0371446 13124501 06 02 02 0235575 13189101 01 minus02 02 0664530 063746701 01 00 02 0659002 096186101 01 04 02 0654009 136783001 01 02 00 0795783 111352101 01 02 10 0082093 151573701 01 02 13 minus0139103 1641030

magnitude of the friction factor and the mass transfer ratediminish with an increase in Casson parameter 120573 For thenon-Newtonian nanofluid case (03 le 120573 le 4) the heattransfer rate is enhanced whereas the opposite results holdwith the Newtonian nanofluid case (120573 rarr infin) From Table 5it is also noticed that the heat transfer rate diminishes withan increase in Brownian motion 119873119887 and thermophoresisparameter119873119905 Physically the presence of solid nanoparticleswithin the conventional working fluid generates a forcenormal to the imposed temperature gradient which is definedas thermophoretic forceThis force has the tendency to movethe nanoparticles of high thermal conductivity towards thecold fluid at the ambient This causes an increase in thetemperature of the fluid but on the contrary the heat transferrate reduces within the thermal boundary layer as shown inFigure 9 On the other hand the mass transfer rate augmentswith an increase in Brownianmotion119873119887 and thermophore-sis parameter119873119905These results are identical to those declaredby Khan and Pop [10] In addition the heat transfer rateis boosted with generative chemical reaction case (120594 ≺ 0)whereas the opposite results hold with destructive chemicalreaction case (120594 ≻ 0) With destructive chemical reaction

= 05 (Casson fluid) = infin (Newtonian fluid)

= 0 1 3

= 0 1 3

f()

f()

2 4 6 80

minus10

minus05

00

05

10

f ()f

(

)

Figure 2 Velocity profile and skin friction coefficient for variousvalues of 120573 and 120582

case (120594 ≻ 0) the mass transfer rate is enhanced whereasthe opposite results hold with generative chemical reactioncase (120594 ≺ 0) It is noticed that the heat transfer reduces withincreasing Eckert number Ec whereas the reverse trend isseen in the mass transfer Physically the presence of Eckertnumber produces viscous heating which decreases the heattransfer rate at the moving plate surface Figures 2ndash18 aredrawn in order to see the influence of slip parameter 120582thermal slip parameter 120574 concentration slip parameter 120575Brownianmotion parameter119873119887 thermophoresis parameter119873119905 chemical reaction parameter 120594 and Eckert numberEc on the velocity the temperature and the nanoparticleconcentration distributions for both cases of Newtonian andnon-Newtonian fluids respectivelyThe influences of the slipparameter 120582 on the velocity 1198911015840(120578) the magnitude of theskin friction coefficient 11989110158401015840(0) the temperature 120579(120578) andthe nanoparticle concentration profiles 120601(120578) are shown inFigures 2ndash4 It is noted from Figures 2ndash4 that the velocity


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)

Figure 3 Temperature profile for various values of 120573 and 120582

= 0 1 3


1 2 3 40

00

02

04

06

08

(

)

Figure 4 Nanoparticle concentration profile for various values of 120573and 120582

profile and the magnitude of the skin friction coefficientreduce with an increase in slip parameter whereas the reversetrend is seen for temperature and nanoparticle concentrationprofiles This is due to the fact that a rise in slip parameter120582 causes a decrease in the surface skin friction between thestretching sheet and the fluid On the other hand an increasein the slip factor generates the friction force which allowsmore fluid to slip past the sheet and the flow deceleratingand the temperature and the nanoparticle concentrationfields are enhanced owing to the occurrence of the forceFor non-Newtonian fluids the impact of slip factor is morepronounced in comparison with the Newtonian flow Theseresults are similar to those reported by Noghrehabadi et al[46] Figures 5 and 6 depict the impacts of the thermal slip

= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07



parameter 120574 on the temperature and the nanoparticle con-centration distributions It is found from Figures 5 and 6 thatthe temperature and the nanoparticle concentration distribu-tions reduce with an increase in the thermal slip parameterPhysically an increase in the thermal slip parameter leadsto diminishing the heat transfer inside the boundary layerregime The temperature distribution for non-Newtonianfluid is more pronounced for all values of 120574 than Newtonianfluid The concentration distribution of Newtonian fluid ismore pronounced for all values of 120574 than non-Newtonianfluid Figures 7 and 8 illustrate the effects of the concentrationslip parameter 120575 on the temperature and the nanoparticleconcentration profiles It is observed from Figures 7 and 8that the temperature and nanoparticle concentration profilesreduce with an increase in concentration slip parameterThe temperature profile for non-Newtonian fluid is more



1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3


pronounced with a rise in 120575 than Newtonian fluid Thenanoparticle concentration in the case of Newtonian flowis higher than that of non-Newtonian fluid for all values of120575 The influences of the thermophoresis parameter 119873119905 onthe temperature and the concentration fields are displayedin Figures 9 and 10 It is observed that the temperature andthe nanoparticle concentration fields are enhanced with anincrease in thermophoresis parameter Physically an increasein the thermophoresis parameter leads to an increase in thethermophoretic force inside a fluid regime which causes anenhancement of the temperature andnanoparticle concentra-tion fields It is found that the increment in the temperatureand the nanoparticle concentration fields for Newtonian fluiddue to thermophoresis parameter is higher than that of non-Newtonian fluid Figures 11 and 12 present the impacts of theBrownianmotion parameter119873119887 on the temperature and the


2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05

Figure 9 Temperature profile for various values of 120573 and119873119905


Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12

Figure 10 Nanoparticle concentration profile for various values of120573 and119873119905

concentration distributions It is noticed that the temperaturedistribution augments with an increase in the Brownianmotion parameter whereas the reverse trend is observedfor the concentration profile Physically the increase inBrownian motion helps to warm the boundary layer whichtends to move nanoparticles from the stretching sheet tothe quiescent fluid Hence the concentration nanoparticlereduces It is easily noticeable that the temperature andconcentration distributions of Newtonian fluid are morepronounced for all values of 119873119887 than non-Newtonian fluidThe effects of the thermophoresis parameter 119873119905 Brownianmotion parameter 119873119887 and chemical reaction parameter 120594on the concentration profile for a non-Newtonian fluid arepresented in Figures 13 and 14 It is noticed from Figures13 and 14 that the concentration profile diminishes with an



Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06


increase in the chemical reaction parameter It is also noticedthat nanoparticle concentration and concentration boundarylayer thickness increase with the generative chemical reactioncase (120594 lt 0) whereas the reverse trend is observed in thedestructive chemical reaction case (120594 gt 0) Physically thepresence of a destructive case (120594 gt 0) causes the transfor-mation of the species as a reason of chemical reaction whichdecreases the concentration distribution in the concentrationboundary layer thickness For generative chemical reactioncase (120594 lt 0) a generative chemical reaction is illustrated thatis the species which diffuses from the stretching sheet in thefree stream and thereby augments the concentration in theconcentration boundary layer thickness It is observed thatthe concentration field reduces with an increase in the Brow-nian motion parameter 119873119887 whereas the opposite trend isnoticedwith thermophoresis parameter119873119905 Figures 15 and 16

2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25


elucidate the influences of the thermophoresis parameter119873119905 Brownian motion parameter 119873119887 and Lewis numberLe on the nanoparticle concentration profile for a non-Newtonian fluid It is observed from Figures 15 and 16 thatthe concentration distribution augments with an increasein thermophoresis parameter whereas the reverse trend isobserved with an increase in the Brownianmotion parameterfor all values of Lewis number On the other hand theconcentration profile diminishes with an increase in Lewisnumber Physically Lewis number is the ratio of momentumdiffusivity to Brownian diffusion coefficient Increasing Lewisnumber leads to a decrease in the Brownian diffusion coeffi-cient which causes a reduction in the concentration fieldTheimpact of the Eckert number on the temperature and the con-centration fields for both cases is plotted in Figures 17 and 18


00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05

Figure 15 Nanoparticle concentration profile for various values ofLe and119873119905

Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05


It is found from Figure 17 that the temperature field isenhanced with an increase in Eckert number PhysicallyEckert number is the ratio of kinetic energy to enthalpyTherefore an increase in Eckert number causes an increasein thermal energy which in turn enhances the temperatureand thermal boundary layer thickness of nanofluid Thisagrees with the fact that the heat transfer rate at the surfacedecreases with an increase in Eckert number as shown inTable 5 It is also noticed fromFigure 18 that the concentrationdistribution reduces with an increase in Eckert number nearthe surface whereas the reverse trend is observed far awayfrom the surface in both cases

00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1

Figure 17 Temperature profile for various values of Ec and 120573

00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1


Figure 18 Nanoparticle concentration profile for various values ofEc and 120573

5 Conclusions

The steady boundary layer flow of Casson nanofluid overa stretching surface with chemical reaction slip boundaryconditions and viscous dissipation has been numericallystudied The effects of various parameters on velocity tem-perature and nanoparticle concentration fields as well asthe friction factor the heat transfer and the mass transferrates are discussed through graphs and tables The validity ofthe present analysis is established by comparing the existingresults with previously published data The main conclusionsemerging from this study are as follows

(1) The heat transfer rate is augmented whereas themass transfer rate is clearly decreased with generativechemical reaction case


(2) The mass transfer rate is elevated whereas the heattransfer rate ismarkedly deceleratedwith the destruc-tive chemical reaction case

(3) The magnitudes of the friction factor the heat trans-fer and the mass transfer rates are reduced with anincrease in slip parameter

(4) The heat transfer rate is diminished whereas theopposite effect is found in the mass transfer rate withan increase in the thermal slip parameter

(5) The heat transfer rate is augmented whereas the masstransfer rate is considerably reduced with a rise inconcentration slip parameter

(6) The heat transfer rate is decreased whereas the masstransfer rate is distinctly boosted with an increase inBrownian motion and thermophoresis parameter

(7) The heat transfer rate is reduced whereas the masstransfer rate is enhanced with an increase in Eckertnumber

(8) The magnitudes of the friction factor and the masstransfer rate are enhanced with an increase in theCasson parameter in both cases

(9) The heat transfer rate is enhanced in the non-Newtonian nanofluid case whereas the oppositeinfluence is noticed in the Newtonian nanofluid case

(10) The nanoparticle concentration distribution isenhanced with the generative chemical reactionwhereas the opposite influence is noticed with thedestructive chemical reaction

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the in Proceedings of the1995 ASME International Mechanical Engineering Congress andExposition vol 66 pp 99ndash105 San Francisco Calif USA 1995

[2] S Lee S U Choi S Li and J A Eastman ldquoMeasuring thermalconductivity of fluids containing oxide nanoparticlesrdquo Journalof Heat Transfer vol 121 no 2 pp 280ndash289 1999

[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001

[4] H S Aybar M Sharifpur M R Azizian M Mehrabi andJ P Meyer ldquoA review of thermal conductivity models fornanofluidsrdquoHeat Transfer Engineering vol 36 no 13 pp 1085ndash1110 2015

[5] X Wang X Xu and S U S Choi ldquoThermal conductivity ofnanoparticle-fluid mixturerdquo Journal of Thermophysics and HeatTransfer vol 13 no 4 pp 474ndash480 1999

[6] A A Afify and M A A Bazid ldquoFlow and heat transferanalysis of nanofluids over a moving surface with temperature-dependent viscosity and viscous dissipationrdquo Journal of Com-putational andTheoretical Nanoscience vol 11 no 12 pp 2440ndash2448 2014

[7] Y Lin B Li L Zheng and G Chen ldquoParticle shape andradiation effects on Marangoni boundary layer flow and heattransfer of copper-water nanofluid driven by an exponentialtemperaturerdquo Powder Technology vol 301 pp 379ndash386 2016

[8] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006

[9] D A Nield and A V Kuznetsov ldquoThermal instability in aporous medium layer saturated by a nanofluidrdquo InternationalJournal of Heat and Mass Transfer vol 52 no 25-26 pp 5796ndash5801 2009

[10] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[11] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[12] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[13] P Rana R Bhargava and O A Beg ldquoNumerical solution formixed convection boundary layer flow of a nanofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012

[14] A A Afify and M A A Bazid ldquoEffects of variable fluidproperties on the natural convective boundary layer flow ofa nanofluid past a vertical plate numerical studyrdquo Journal ofComputational and Theoretical Nanoscience vol 11 no 1 pp210ndash218 2014

[15] O D Makinde F Mabood W A Khan and M S TshehlaldquoMHD flow of a variable viscosity nanofluid over a radiallystretching convective surface with radiative heatrdquo Journal ofMolecular Liquids vol 219 pp 624ndash630 2016

[16] F M Hady F S Ibrahim S M Abdel-Gaied and M R EidldquoBoundary-layer non-Newtonian flow over vertical plate inporousmedium saturatedwith nanofluidrdquoAppliedMathematicsand Mechanics vol 32 no 12 pp 1577ndash1586 2011

[17] C-Y Cheng ldquoFree convection of non-Newtonian nanofluidsabout a vertical truncated cone in a porous mediumrdquo Interna-tional Communications in Heat andMass Transfer vol 39 no 9pp 1348ndash1353 2012

[18] A M Rashad M A El-Hakiem and M M Abdou ldquoNaturalconvection boundary layer of a non-Newtonian fluid abouta permeable vertical cone embedded in a porous mediumsaturated with a nanofluidrdquo Computers and Mathematics withApplications vol 62 pp 3140ndash3151 2011

[19] M Ramzan M Bilal J D Chung and U Farooq ldquoMixedconvective flow of Maxwell nanofluid past a porous verticalstretched surfacemdashAn optimal solutionrdquo Results in Physics vol6 pp 1072ndash1079 2016

[20] T Hayat T Muhammad S A Shehzad and A AlsaedildquoThree-dimensional boundary layer flow ofMaxwell nanofluidmathematical modelrdquo Applied Mathematics and Mechanics vol36 no 6 pp 747ndash762 2015


[21] M Y Abou-Zeid ldquoHomotopy perturbation method to glidingmotion of bacteria on a layer of power-law nanoslime with heattransferrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 10 pp 3605ndash3614 2015

[22] M Y Abou-Zeid ldquoEffects of thermal-diffusion and viscousdissipation on peristaltic flow of micropolar non-Newtoniannanofluid application of homotopy perturbation methodrdquoResults in Physics vol 6 pp 481ndash495 2016

[23] R Ellahi ldquoThe effects of MHD and temperature dependentviscosity on the flow of non-Newtonian nanofluid in a pipeanalytical solutionsrdquo Applied Mathematical Modelling vol 37no 3 pp 1451ndash1467 2013

[24] M Y Abou-Zeid A A Shaaban andM Y Alnour ldquoNumericaltreatment and global error estimation of natural convectiveeffects on gliding motion of bacteria on a power-lawnanoslimethrough a non-darcy porousmediumrdquo Journal of PorousMediavol 18 no 11 pp 1091ndash1106 2015

[25] C S K Raju N Sandeep and A Malvandi ldquoFree convectiveheat and mass transfer of MHD non-Newtonian nanofluidsover a cone in the presence of non-uniform heat sourcesinkrdquoJournal of Molecular Liquids vol 221 pp 108ndash115 2016

[26] N Casson In Reheology of Dipersed System Peragamon PressOxford UK 1959

[27] M Mustafa T Hayat I Pop and A Aziz ldquoUnsteady boundarylayer flowof aCasson fluid due to an impulsively startedmovingflat platerdquoHeat TransfermdashAsianResearch vol 40 no 6 pp 563ndash576 2011

[28] S Nadeem R Mehmood and N S Akbar ldquoOptimizedanalytical solution for oblique flow of a Casson-nano fluidwith convective boundary conditionsrdquo International Journal ofThermal Sciences vol 78 pp 90ndash100 2014

[29] G Makanda S Shaw and P Sibanda ldquoDiffusion of chemicallyreactive species in Casson fluid flow over an unsteady stretchingsurface in porous medium in the presence of a magnetic fieldrdquoMathematical Problems in Engineering vol 2015 Article ID724596 10 pages 2015

[30] T Hayat S Asad and A Alsaedi ldquoFlow of Casson fluid withnanoparticlesrdquo Applied Mathematics and Mechanics vol 37 no4 pp 459ndash470 2016

[31] K Das P R Duari and P K Kundu ldquoNumerical simulation ofnanofluid flow with convective boundary conditionrdquo Journal ofthe Egyptian Mathematical Society vol 23 no 2 pp 435ndash4392015

[32] MAbdEl-Aziz ldquoEffect of time-dependent chemical reaction onstagnation point flow and heat transfer over a stretching sheetin a nanofluidrdquo Physica Scripta vol 89 no 8 Article ID 0852052014

[33] N T M El-Dabe A A Shaaban M Y Abou-Zeid and HA Ali ldquoMagnetohydrodynamic non-Newtonian nanofluid flowover a stretching sheet through a non-Darcy porous mediumwith radiation and chemical reactionrdquo Journal of ComputationalandTheoretical Nanoscience vol 12 no 12 pp 5363ndash5371 2015

[34] M R Eid ldquoChemical reaction effect on MHD boundary-layer flow of two-phase nanofluid model over an exponentiallystretching sheet with a heat generationrdquo Journal of MolecularLiquids vol 220 pp 718ndash725 2016

[35] A A Afify and N S Elgazery ldquoEffect of a chemical reaction onmagnetohydrodynamic boundary layer flow of a Maxwell fluidover a stretching sheet with nanoparticlesrdquo Particuology vol 29pp 154ndash161 2016

[36] C L M H Navier ldquoMemoire sur les lois du mouvement desfluidesrdquo Memoires de Academie Royale des Sciences de Institutde France vol 6 pp 389ndash440 1823

[37] A Rashad ldquoUnsteady nanofluid flowover an inclined stretchingsurface with convective boundary condition and anisotropicslip impactrdquo International Journal of Heat and Technology vol35 no 1 pp 82ndash90 2017

[38] A A Afify M J Uddin and M Ferdows ldquoScaling grouptransformation for MHD boundary layer flow over permeablestretching sheet in presence of slip flowwithNewtonian heatingeffectsrdquo Applied Mathematics and Mechanics vol 35 no 11 pp1375ndash1386 2014

[39] S M M El-Kabeir E R El-Zahar and A M Rashad ldquoEffectof chemical reaction on heat and mass transfer by mixedconvection flow of casson fluid about a sphere with partial sliprdquoJournal of Computational and Theoretical Nanoscience vol 13no 8 pp 5218ndash5226 2016

[40] A A Afify ldquoSlip effects on the flow and heat transfer ofnanofluids over an unsteady stretching surface with heat gen-erationabsorptionrdquo Journal of Computational and TheoreticalNanoscience vol 12 no 3 pp 484ndash491 2015

[41] M H Abolbashari N Freidoonimehr F Nazari and M MRashidi ldquoAnalytical modeling of entropy generation for Cassonnano-fluid flow induced by a stretching surfacerdquo AdvancedPowder Technology vol 26 no 2 pp 542ndash552 2015

[42] M J Uddin W A Khan and A I M Ismail ldquoEffect of variableproperties Navier slip and convective heating on hydromag-netic transport phenomenardquo Indian Journal of Physics vol 90no 6 pp 627ndash637 2016

[43] N TM Eldabe andMG E Salwa ldquoHeat transfer ofMHDnon-Newtonian Casson fluid flow between two rotating cylindersrdquoJournal of the Physical vol 64 pp 41ndash64 1995

[44] R U Haq S Nadeem Z H Khan and T G Okedayo ldquoCon-vective heat transfer andMHDeffects onCasson nanofluid flowover a shrinking sheetrdquoCentral European Journal of Physics vol12 no 12 pp 862ndash871 2014

[45] A Alsaedi M Awais and T Hayat ldquoEffects of heat genera-tionabsorption on stagnation point flow of nanofluid over asurface with convective boundary conditionsrdquoCommunicationsinNonlinear Science andNumerical Simulation vol 17 no 11 pp4210ndash4223 2012

[46] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the impact of Brownian motion and thermophoresis on thedouble-diffusive nanofluid convection past a vertical plateembedded in a porous medium Makinde and Aziz [12]discussed the influence of a convective boundary Brownianmotion and thermophoresis on the steady two-dimensionalboundary layer flow of a nanofluid past a stretching sheetRana et al [13] numerically examined the Brownian motionand thermophoresis effects on the steady mixed convectionboundary layer flow of nanofluid past an inclined plateembedded in a porous medium The influence of variablefluid properties with Brownian motion and thermophoresison the natural convective boundary layer flow in a nanofluidpast a vertical plate was numerically studied by Afifyand Bazid [14] Recently Makinde et al [15] numericallyexamined the effects of variable viscosity thermal radiationand magnetic field on convective heat transfer of nanofluidover a stretching surface

Non-Newtonian nanofluids are widely encountered inmany industrial and technological applications such as thedissolved polymers biological solutions paints asphalts andglues The power-law non-Newtonian nanofluid along a ver-tical plate and a truncated cone saturated in a porousmediumwere analyzed byHady et al [16] andCheng [17] respectivelyRashad et al [18] investigated the natural convection of non-Newtonian nanofluid around a vertical permeable cone Theinfluence of Soret and Dufour on the mixed convective flowof Maxwell nanofluid over a permeable stretched surfacewas examined by Ramzan et al [19] Several authors [20ndash23] have analytically solved the problem of non-Newtoniannanofluids in various aspects using the homotopy analysismethod (HAM) Abou-Zeid et al [24] examined the impactof viscous dispersion on the mixed convection of glidingmotion of bacteria on power-law nanofluids through a non-Darcy porous medium The influence of nonuniform heatsourcesink with the Brownian motion and thermophoresison non-Newtonian nanofluids over a cone was illustrated byRaju et al [25]

The Casson fluid model is classified as a subclass of non-Newtonian fluid which has several applications in food pro-cessing metallurgy drilling operations and bioengineeringoperationsTheCasson fluidmodelwas discovered byCassonin 1959 for the prediction of the flow behavior of pigment-oil suspensions [26] Boundary layer flow of Casson fluid andCasson nanofluid flow over different geometries was consid-ered by many authors [27ndash30] Mustafa et al [27] analyticallydiscussed the unsteady boundary layer flow of a Casson fluidover a moving flat plate using the homotopy analysis method(HAM) Nadeem et al [28] examined the analytical solutionof convective boundary conditions for the steady stagnation-point flow of a Casson nanofluid Makanda et al [29] studiedthe numerical solution of MHD Casson fluid flow over anunsteady stretching surface saturated in a porous mediumwith a chemical reaction effect Recently Hayat et al [30]investigated the combined effects of the variable thermalconductivity and viscous dissipation on the boundary layerflow of a Casson fluid due to a stretching cylinder

The chemical reaction influence is a significant factorin the study of heat and mass transfer for many branchesof science and engineering A chemical reaction between

the base liquid and nanoparticles may regularly occur eitherthroughout a given phase (homogeneous reaction) or inan enclosed region (boundary) of the phase (heterogeneousreaction) Das et al [31] investigated numerically the effectsof chemical reaction and thermal radiation on the heat andmass transfer of an electrically conducting incompressiblenanofluid over a heated stretching sheet The effect of time-dependent chemical reaction on stagnation-point flow andheat transfer of nanofluid over a stretching sheet was pre-sented by Abd El-Aziz [32] El-Dabe et al [33] studied theimpact of chemical reaction heat generation and radiationon the MHD flow of non-Newtonian nanofluid over astretching sheet embedded in a porous medium Eid [34]proposed the numerical analysis of MHD mixed convectiveboundary layer flow of a nanofluid through a porousmediumalong an exponentially stretching sheet in the presence ofchemical reaction and heat generation or absorption effectsRecently Afify and Elgazery [35] discussed the influences ofthe convective boundary condition and chemical reaction onthe MHD boundary layer flow of a Maxwell nanofluid over astretching surface

All these previous studies restricted their discussions onconventional no-slip boundary conditions However bothvelocity slip and temperature jump at the wall have numerousbenefits in many practical applications such as micro- andnanoscale devices In 1823 Navier [36] became the firstperson to create the slip boundary condition and suggestedthat the slip velocity is linearly proportional to the shearstress at the wall Following him many researchers [37ndash42] have widely studied the velocity slip and temperaturejump at the wall over various geometries placed in viscousfluids and nanofluids The impacts of slip and convectiveboundary condition on the unsteady three-dimensional flowof nanofluid over an inclined stretching surface embeddedin a porous medium were numerically discussed by Rashad[37] Afify et al [38] used Lie symmetry analysis to investigatethe effects of slip flow Newtonian heating and thermalradiation on MHD flow and heat transfer along the per-meable stretching sheet EL-Kabeir et al [39] examined thenumerical solution for mixed convection boundary layerflow of Casson non-Newtonian fluid about a solid spherein the presence of thermal and solutal slip conditions Afify[40] examined the impact of slips and generationabsorptionon an unsteady boundary layer flow and heat transfer overa stretching surface immersed in nanofluids Abolbashariet al [41] presented an analytical solution to analyze theheat and mass transfer characteristics of Casson nanofluidflow induced by a stretching sheet with velocity slip andconvective surface boundary conditions Recently Uddin etal [42] discussed the effects of Navier slip and variablefluid properties on the forced convection of nanofluid andheat transfer over a wedge The objective of the presentpaper is to investigate the influences of chemical reactionviscous dissipation velocity and thermal and concentrationslip boundary conditions in the presence of nanoparticlesattributable to Brownian diffusion and thermophoresis onflow and heat transfer of Casson fluid over a stretching sur-face To the best of the authorrsquos knowledge this work has notbeen previously studied in the scientific research Numerical


x

y

ForceSlit







120591119894119895 =

2(120583119861 + 119901119910radic2120587) 119890119894119895 120587 gt 120587119888

2 (120583119861 + 119901119910radic2120587119888) 119890119894119895 120587 lt 120587119888

(1)


120597119906120597119909 +

120597V120597119910 = 0 (2)

119906120597119906120597119909 + V120597119906120597119910 = 120592(1 + 1120573)

12059721199061205971199102 (3)

119906120597119879120597119909 + V120597119879120597119910 = 12057212059721198791205971199102

+ 120591119863119861 (120597119862120597119910120597119879120597119910 ) +

119863119879119879infin (120597119879120597119910 )2

+ (1 + 1120573)120583120588119888119901 (

120597119906120597119910)2

(4)

119906120597119862120597119909 + V120597119862120597119910 = 119863119861 120597

21198621205971199102 +



119906 = 119906119908 + (1 + 1120573)119873120588]120597119906120597119910

V = 0119879 = 119879119908 + 1198701 120597119879120597119910

119862 = 119862119908 + 1198702 120597119862120597119910at 119910 = 0

119906 = 0119879 = 119879infin119862 = 119862infin


(6)


120578 = (119887])12 119910



(7)


119906 = 120597120595120597119910

V = minus120597120595120597119909 (8)



(1 + 1120573)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 = 0 (9)

1Pr12057910158401015840 + 1198911205791015840 + 11987311988712060110158401205791015840 + 11987311990512057910158402 + (1 + 1120573)Ec119891101584010158402= 0

(10)

12060110158401015840 + Le1198911206011015840 + 11987311990511987311988712057910158401015840 minus Le120594120601 = 0 (11)


(12)



(13)


120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0

119902119898 = minus119863119861 (120597119862120597119910 )119910=0

(14)


Re12119909 119862119891 = (1 + 1120573)11989110158401015840 (0) Nu119909Re12119909

= minus1205791015840 (0) Sh119909Re12119909

= minus1206011015840 (0)

(15)




1198911015840 = 1199011199011015840 = 1199021199021015840 = ( 120573

1 + 120573) [1199012 minus 119891119902] 1205791015840 = 1199111206011015840 = 1199041199111015840 = minusPr(119891119911 + 119873119887119904119911 + 1198731199051199112 + (1 + 1120573)Ec1199022)

1199041015840 = Le (120594120601 minus 119891119904) minus 1198731199051198731198871199111015840

(16)


119891 (0) = 0119901 (0) = 1 + 120582(1 + 1120573) 119902 (0) 120579 (0) = 1 + 120574119911 (0) 120601 (0) = 1 + 120575119904 (0)

(17)





007 00663 00656 00663 mdash 00662020 01691 01691 01691 01691 01691070 04539 04539 04539 04539 0453820 09113 09114 09113 09114 0911370 18954 18954 mdash 18954 18954



0101 0952377 0952376 2129394 212938903 0520079 0520079 2528638 252863505 0321054 0321054 3035142 3035144

0201 0505581 0505581 2381871 238187603 0273096 0273096 2655459 265545705 0168077 0168077 2888339 2888336

0301 0252156 0252156 2410019 241001503 0135514 0135514 2608819 260881705 0083298 0083298 2751875 2751873



0101 0799317 0799310 1787171 178714203 0436495 0436492 2122251 212224305 0269457 0269454 2547353 2547347

0201 0424328 0424324 199907 19990403 0229206 0229204 2228691 222868805 0141064 0141061 2424144 2424139

0301 0211631 0211630 2022696 202267203 0113735 0113731 2189547 218953905 0069911 0069921 2309612 2309609









= 0 1 3

= 0 1 3

f()

f()

2 4 6 80

minus10

minus05

00

05

10

f ()f

(

)




= 0 1 3


1 2 3 40

00

02

04

06

08

(

)


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)



= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





x

y

ForceSlit







120591119894119895 =

2(120583119861 + 119901119910radic2120587) 119890119894119895 120587 gt 120587119888

2 (120583119861 + 119901119910radic2120587119888) 119890119894119895 120587 lt 120587119888

(1)


120597119906120597119909 +

120597V120597119910 = 0 (2)

119906120597119906120597119909 + V120597119906120597119910 = 120592(1 + 1120573)

12059721199061205971199102 (3)

119906120597119879120597119909 + V120597119879120597119910 = 12057212059721198791205971199102

+ 120591119863119861 (120597119862120597119910120597119879120597119910 ) +

119863119879119879infin (120597119879120597119910 )2

+ (1 + 1120573)120583120588119888119901 (

120597119906120597119910)2

(4)

119906120597119862120597119909 + V120597119862120597119910 = 119863119861 120597

21198621205971199102 +



119906 = 119906119908 + (1 + 1120573)119873120588]120597119906120597119910

V = 0119879 = 119879119908 + 1198701 120597119879120597119910

119862 = 119862119908 + 1198702 120597119862120597119910at 119910 = 0

119906 = 0119879 = 119879infin119862 = 119862infin


(6)


120578 = (119887])12 119910



(7)


119906 = 120597120595120597119910

V = minus120597120595120597119909 (8)



(1 + 1120573)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 = 0 (9)

1Pr12057910158401015840 + 1198911205791015840 + 11987311988712060110158401205791015840 + 11987311990512057910158402 + (1 + 1120573)Ec119891101584010158402= 0

(10)

12060110158401015840 + Le1198911206011015840 + 11987311990511987311988712057910158401015840 minus Le120594120601 = 0 (11)


(12)



(13)


120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0

119902119898 = minus119863119861 (120597119862120597119910 )119910=0

(14)


Re12119909 119862119891 = (1 + 1120573)11989110158401015840 (0) Nu119909Re12119909

= minus1205791015840 (0) Sh119909Re12119909

= minus1206011015840 (0)

(15)




1198911015840 = 1199011199011015840 = 1199021199021015840 = ( 120573

1 + 120573) [1199012 minus 119891119902] 1205791015840 = 1199111206011015840 = 1199041199111015840 = minusPr(119891119911 + 119873119887119904119911 + 1198731199051199112 + (1 + 1120573)Ec1199022)

1199041015840 = Le (120594120601 minus 119891119904) minus 1198731199051198731198871199111015840

(16)


119891 (0) = 0119901 (0) = 1 + 120582(1 + 1120573) 119902 (0) 120579 (0) = 1 + 120574119911 (0) 120601 (0) = 1 + 120575119904 (0)

(17)





007 00663 00656 00663 mdash 00662020 01691 01691 01691 01691 01691070 04539 04539 04539 04539 0453820 09113 09114 09113 09114 0911370 18954 18954 mdash 18954 18954



0101 0952377 0952376 2129394 212938903 0520079 0520079 2528638 252863505 0321054 0321054 3035142 3035144

0201 0505581 0505581 2381871 238187603 0273096 0273096 2655459 265545705 0168077 0168077 2888339 2888336

0301 0252156 0252156 2410019 241001503 0135514 0135514 2608819 260881705 0083298 0083298 2751875 2751873



0101 0799317 0799310 1787171 178714203 0436495 0436492 2122251 212224305 0269457 0269454 2547353 2547347

0201 0424328 0424324 199907 19990403 0229206 0229204 2228691 222868805 0141064 0141061 2424144 2424139

0301 0211631 0211630 2022696 202267203 0113735 0113731 2189547 218953905 0069911 0069921 2309612 2309609









= 0 1 3

= 0 1 3

f()

f()

2 4 6 80

minus10

minus05

00

05

10

f ()f

(

)




= 0 1 3


1 2 3 40

00

02

04

06

08

(

)


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)



= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(1 + 1120573)119891101584010158401015840 + 11989111989110158401015840 minus 11989110158402 = 0 (9)

1Pr12057910158401015840 + 1198911205791015840 + 11987311988712060110158401205791015840 + 11987311990512057910158402 + (1 + 1120573)Ec119891101584010158402= 0

(10)

12060110158401015840 + Le1198911206011015840 + 11987311990511987311988712057910158401015840 minus Le120594120601 = 0 (11)


(12)



(13)


120591119908 = (120583119861 + 119901119910radic2120587119888)(

120597119906120597119910)119910=0

119902119908 = minus119896(120597119879120597119910 )119910=0

119902119898 = minus119863119861 (120597119862120597119910 )119910=0

(14)


Re12119909 119862119891 = (1 + 1120573)11989110158401015840 (0) Nu119909Re12119909

= minus1205791015840 (0) Sh119909Re12119909

= minus1206011015840 (0)

(15)




1198911015840 = 1199011199011015840 = 1199021199021015840 = ( 120573

1 + 120573) [1199012 minus 119891119902] 1205791015840 = 1199111206011015840 = 1199041199111015840 = minusPr(119891119911 + 119873119887119904119911 + 1198731199051199112 + (1 + 1120573)Ec1199022)

1199041015840 = Le (120594120601 minus 119891119904) minus 1198731199051198731198871199111015840

(16)


119891 (0) = 0119901 (0) = 1 + 120582(1 + 1120573) 119902 (0) 120579 (0) = 1 + 120574119911 (0) 120601 (0) = 1 + 120575119904 (0)

(17)





007 00663 00656 00663 mdash 00662020 01691 01691 01691 01691 01691070 04539 04539 04539 04539 0453820 09113 09114 09113 09114 0911370 18954 18954 mdash 18954 18954



0101 0952377 0952376 2129394 212938903 0520079 0520079 2528638 252863505 0321054 0321054 3035142 3035144

0201 0505581 0505581 2381871 238187603 0273096 0273096 2655459 265545705 0168077 0168077 2888339 2888336

0301 0252156 0252156 2410019 241001503 0135514 0135514 2608819 260881705 0083298 0083298 2751875 2751873



0101 0799317 0799310 1787171 178714203 0436495 0436492 2122251 212224305 0269457 0269454 2547353 2547347

0201 0424328 0424324 199907 19990403 0229206 0229204 2228691 222868805 0141064 0141061 2424144 2424139

0301 0211631 0211630 2022696 202267203 0113735 0113731 2189547 218953905 0069911 0069921 2309612 2309609









= 0 1 3

= 0 1 3

f()

f()

2 4 6 80

minus10

minus05

00

05

10

f ()f

(

)




= 0 1 3


1 2 3 40

00

02

04

06

08

(

)


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)



= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







007 00663 00656 00663 mdash 00662020 01691 01691 01691 01691 01691070 04539 04539 04539 04539 0453820 09113 09114 09113 09114 0911370 18954 18954 mdash 18954 18954



0101 0952377 0952376 2129394 212938903 0520079 0520079 2528638 252863505 0321054 0321054 3035142 3035144

0201 0505581 0505581 2381871 238187603 0273096 0273096 2655459 265545705 0168077 0168077 2888339 2888336

0301 0252156 0252156 2410019 241001503 0135514 0135514 2608819 260881705 0083298 0083298 2751875 2751873



0101 0799317 0799310 1787171 178714203 0436495 0436492 2122251 212224305 0269457 0269454 2547353 2547347

0201 0424328 0424324 199907 19990403 0229206 0229204 2228691 222868805 0141064 0141061 2424144 2424139

0301 0211631 0211630 2022696 202267203 0113735 0113731 2189547 218953905 0069911 0069921 2309612 2309609









= 0 1 3

= 0 1 3

f()

f()

2 4 6 80

minus10

minus05

00

05

10

f ()f

(

)




= 0 1 3


1 2 3 40

00

02

04

06

08

(

)


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)



= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









= 0 1 3

= 0 1 3

f()

f()

2 4 6 80

minus10

minus05

00

05

10

f ()f

(

)




= 0 1 3


1 2 3 40

00

02

04

06

08

(

)


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)



= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





= 0 1 3


1 2 3 40

00

02

04

06

08

(

)


= 0 1 3


1 2 3 40

00

02

04

06

08

(

)



= 0 08 4

05 10 15 20 2500 30

00

02

04

06

08

10


(

)


= 0 08 4

05 10 15 20 2500 30

00

01

02

(

)

03

04

05

06

07






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






1 2 3 40

00

02

04

06

08

(

)

= 0 05 3



05 10 15 20 2500

00

02

04

(

)

06

08

10

= 0 05 3




2 4 6 80

00

02

04

06

08

(

)

Nt = 01 03 05



Nt = 01 03 05

2 4 6 80

00

02

04

(

)

06

08

10

12





Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Nb = 01 025 06

2 4 6 80

00

02

04

06

08

(

)



00

02

04

(

)

06

08

2 4 6 80

Nb = 01 025 06



2 4 6 80

00

02

04(

)

06

08

10

= minus03 00 05

Nb = 01Nb = 05


Nt = 01Nt = 05

= minus03 00 05

2 4 6 80

00

05

10

(

) 15

20

25




00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





00

02

04

(

)

06

08

10

12

2 4 6 80

Le = 1Le = 5

Nt = 01 03 05


Le = 1Le = 5

2 4 6 80

00

02

(

)

04

06

08

Nb = 01 03 05



00

02

04

06

08

10

(

)

1 2 3 4 5 60


Ec = 01 05 1


00

02

04

(

)

06

08

1 2 40 3

Ec = 01 05 1

Ec = 01 05 1



5 Conclusions















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of
















References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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