Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation

Ain Shams Engineering Journal (2012) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal

www.elsevier.com/locate/asejwww.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Slip effects on MHD boundary layer flow over

an exponentially stretching sheet with suction/blowing

and thermal radiation

Swati Mukhopadhyay *

Department of Mathematics, The University of Burdwan, Burdwan 713 104, WB, India

Received 12 April 2012; revised 5 October 2012; accepted 15 October 2012

*

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KEYWORDS

Exponential stretching;

MHD;

Suction/blowing;

Velocity slip;

Thermal slip;

Radiation;

Similarity solutions

Tel.: +91 342 2657741; fax:

mail address: swati_bumath@

er review under responsibilit

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lease cite this article in preet with suction/blowing

+91 342

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Universit

.2012.10.0

ess as: Mand the

Abstract The boundary layer flow and heat transfer towards a porous exponential stretching sheet

in presence of a magnetic field is presented in this analysis. Velocity slip and thermal slip are con-

sidered instead of no-slip conditions at the boundary. Thermal radiation term is incorporated in the

temperature equation. Similarity transformations are used to convert the partial differential equa-

tions corresponding to the momentum and energy equations into non-linear ordinary differential

equations. Numerical solutions of these equations are obtained by shooting method. It is found that

the horizontal velocity decreases with increasing slip parameter as well as with the increasing mag-

netic parameter. Temperature increases with the increasing values of magnetic parameter. Temper-

ature is found to decrease with an increase of thermal slip parameter. Thermal radiation enhances

the effective thermal diffusivity and the temperature rises.� 2012 Ain Shams University. Production and hosting by Elsevier B.V.

All rights reserved.

1. Introduction

The study of laminar flow and heat transfer over a stretching

sheet in a viscous fluid is of considerable interest because ofits ever increasing industrial applications and important bear-ings on several technological processes. Examples are numer-

ous and they include the cooling of an infinite metallic plate

2530452.

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Shams University.

g by Elsevier

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ukhopadhyay S, Slip effects ormal radiation, Ain Shams En

in a cooling bath, the boundary layer along material handlingconveyers, the aerodynamic extrusion of plastic sheets, the

boundary layer along a liquid film in condensation processes,paper production, glass blowing, metal spinning, drawing plas-tic films and polymer extrusion, to name just a few [1]. Crane

[2] investigated the flow caused by the stretching of a sheet.Under different physical situations, many researchers extendedthe work of Crane [2]. Most of the available literature dealswith the study of boundary layer flow over a stretching surface

where the velocity of the stretching surface is assumed linearlyproportional to the distance from the fixed origin.

However, realistically stretching of plastic sheet may not

necessarily be linear [3]. Flow and heat transfer characteristicspast an exponentially stretching sheet has a wider applicationsin technology. For example, in case of annealing and thinning

ier B.V. All rights reserved.

n MHD boundary layer flow over an exponentially stretchingg J (2012), http://dx.doi.org/10.1016/j.asej.2012.10.007

mailto:[email protected]

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x

y

O

Too

u

v

V(x) UTw

Boundary layer

B(x)

Figure 1 Sketch of physical problem.

2 S. Mukhopadhyay

of copper wires the final product depends on the rate of heattransfer at the stretching continuous surface with exponentialvariations of stretching velocity and temperature distribution.

During such processes, both the kinematics of stretching andthe simultaneous heating or cooling have a decisive influenceon the quality of the final products [4]. Magyari and Keller

[5] focused on heat and mass transfer on boundary layer flowdue to an exponentially continuous stretching sheet. Elbashbe-shy [6] studied the flow past an exponentially stretching

surface. Khan [7] and Sanjayanand and Khan [8] discussedthe viscous–elastic boundary layer flow and heat transfer dueto an exponentially stretching sheet. Later, Sajid and Hayat[9] find the analytic solution using homotopy analysis method

and discussed the influence of thermal radiation on the bound-ary layer flow due to an exponentially stretching sheet.Recently, the effect of thermal radiation on the steady laminar

two-dimensional boundary layer flow and heat transfer overan exponentially stretching sheet was reported by Bidin andNazar [10]. Of late, El-Aziz [11] and Ishak [12] described the

flow and heat transfer past an exponentially stretching sheet.All the above mentioned studies continued their discussions

by assuming the no slip boundary conditions. Theno-slip bound-

ary condition (the assumption that a liquid adheres to a solidboundary) is one of the central tenets of the Navier–Stokes the-ory. However, there are situations wherein this condition doesnot hold. Partial velocity slipmay occur on the stretching bound-

ary when the fluid is particulate such as emulsions, suspensions,foams andpolymer solutions. Thenon-adherence of the fluid to asolid boundary, also known as velocity slip, is a phenomenon

that has been observed under certain circumstances [13]. Re-cently, many researchers [14–18], etc. investigated the flow prob-lems taking slip flow condition at the boundary. The fluids that

exhibit boundary slip have important technological applicationssuch as in the polishing of artificial heart valves and internal cav-ities. For some coated surfaces, such as Teflon, resist adhesion,

the no slip condition is replaced by Navier’s partial slip condi-tion, where the slip velocity is proportional to the local shearstress. However, experiments suggest that the slip velocity alsodepends on the normal stress. A number ofmodels have been ad-

vanced for describing the slip that occurs at solid boundaries. Anew dimension is added to the abovementioned study by consid-ering the effects of partial slip at the stretching wall. The study of

magnetohydrodynamics boundary layer flow of a conductingfluid is also important as it finds applications in a variety ofstretching sheet problems. Representative studies dealing with

such effects can be found in Turkyilmazoglu [19–21].Suction or injection (blowing) of a fluid through the bound-

ing surface can significantly change the flow field. In general,suction tends to increase the skin friction whereas injection

acts in the opposite manner. Injection or withdrawal of fluidthrough a porous bounding wall is of general interest in prac-tical problems involving boundary layer control applications

such as film cooling, polymer fiber coating, and coating ofwires. The process of suction and blowing has also its impor-tance in many engineering activities such as in the design of

thrust bearing and radial diffusers, and thermal oil recovery.Suction is applied to chemical processes to remove reactants.Blowing is used to add reactants, cool the surface, prevent cor-

rosion or scaling and reduce the drag.The radiative effects have important applications in physics

and engineering. The radiation heat transfer effects on differ-ent flows are very important in space technology and high

Please cite this article in press as: Mukhopadhyay S, Slip effects osheet with suction/blowing and thermal radiation, Ain Shams En

temperature processes. But very little is known about the ef-fects of radiation on the boundary layer. Thermal radiation ef-fects may play an important role in controlling heat transfer in

polymer processing industry where the quality of the finalproduct depends on the heat controlling factors to someextent. High temperature plasmas, cooling of nuclear reactors,

liquid metal fluids, magnetohydrodynamics (MHD) accelera-tors, power generation systems are some important applica-tions of radiative heat transfer from a vertical wall to

conductive gray fluids.Since no attempt has been made to analyze the effects of

partial slip on MHD boundary layer flow over an exponen-tially stretching surface with suction or injection, so it is con-

sidered in this article. Using similarity transformations, athird order ordinary differential equation corresponding tothe momentum equation and a second order differential equa-

tion corresponding to the heat equation are derived. Usingshooting method numerical calculations up to desired levelof accuracy were carried out for different values of dimension-

less parameters of the problem under consideration for thepurpose of illustrating the results graphically. The analysis ofthe results obtained shows that the flow field is influenced

appreciably by the slip parameter in the presence of magneticfield and suction or injection at the wall. Estimation of skinfriction which is very important from the industrial applicationpoint of view is also presented in this analysis. It is hoped that

the results obtained will not only provide useful informationfor applications, but also serve as a complement to theprevious studies.

2. Equations of motion

Consider the flow of an incompressible viscous electrically con-

ducting fluid past a flat sheet coinciding with the plane y= 0(see Fig. 1). The x-axis is directed along the continuous stretch-ing surface and points in the direction of motion while the

y-axis is perpendicular to the surface. The flow is confined toy> 0. Two equal and opposite forces are applied along thex-axis so that the wall is stretched keeping the origin fixed.

The flow is assumed to be generated by stretching of the elastic

boundary sheet from a slit with a large force such that thevelocity of the boundary sheet is an exponential order of the



Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing 3

flow directional coordinate x. The flow takes place in the upperhalf plane y > 0. A variable magnetic field BðxÞ ¼ B0e

x2L is

applied normal to the sheet, B0 being a constant. The continu-

ity, momentum and energy equations governing such type offlow are written as

@u

@xþ @v@y¼ 0; ð1Þ

u@u

@xþ t

@u

@y¼ m

@2u

@y2� rB2

qu; ð2Þ

u@T

@xþ t

@T

@y¼ j

qcp

@2T

@y2� 1

qcp

@qr@y

; ð3Þ

where u and t are the components of velocity respectively in

the x and y directions, m ¼ lq is the kinematic viscosity, q is

the fluid density (assumed constant), l is the coefficient of fluidviscosity, r is the electrical conductivity, qr is the radiative heat

flux, cp is the specific heat at constant pressure, j is the thermalconductivity of the fluid.

In writing Eq. (2), we have neglected the induced magneticfield since the magnetic Reynolds number for the flow is

assumed to be very small.Using Rosseland approximation for radiation [22] we can

write

qr ¼ �4r�

3k�@T4

@y; ð3aÞ

where r* is the Stefan–Boltzman constant, k* is the absorptioncoefficient.

Assuming that the temperature difference within the flow is

such that T4 may be expanded in a Taylor series and expandingT4 about T1, the free stream temperature and neglectinghigher orders we get T4 � 4T3

1T� 3T41. Therefore, the

Eq. (3) becomes

u@T

@xþ t

@T

@y¼ j

qcp

@2T

@y2þ 16r�T3

13qcpk

�@2T

@y2: ð4Þ

2.1. Boundary conditions

The appropriate boundary conditions for the problem are gi-

ven by

u ¼ UþNm@u

@y; t ¼ �VðxÞ; T ¼ Tw þD

@T

@y

at y ¼ 0; ð5Þ

u! 0; T! T1 as y!1: ð6Þ

Here U ¼ U0exL is the stretching velocity, Tw ¼ T1 þ T0e

x2L is

the temperature at the sheet, U0, T0 are the reference velocityand temperature respectively, N ¼ N1e

� x2L is the velocity slip

factor which changes with x, N1 is the initial value of velocityslip factor and D ¼ D1e

� x2L is the thermal slip factor which also

changes with x, D1 is the initial value of thermal slip factor.The no-slip case is recovered for N= 0= D. V(x) > 0 is the

velocity of suction and V(x) < 0 is the velocity of blowing,VðxÞ ¼ V0e

x2L, a special type of velocity at the wall is consid-

ered. V0 is the initial strength of suction.


2.2. Method of solution

Introducing the similarity variables as

g ¼ffiffiffiffiffiffiffiffiU0

2mL

re

x2Ly; u ¼ U0e

xLf0ðgÞ;

t ¼ �ffiffiffiffiffiffiffiffimU0

2L

re

x2LffðgÞ þ gf0ðgÞg; T ¼ T1 þ T0e

x2LhðgÞ ð7Þ

and upon substitution of (5) in Eqs. 2, 3, 3a, 4 the governing

equations reduce to

f 000 þ ff 00 � 2f 0 2 �M2f 0 ¼ 0; ð8Þ

1þ 4

3R

� �h 00 þ Prðfh 0 � f 0hÞ ¼ 0; ð9Þ

and the boundary conditions take the following form:

f0 ¼ 1þ kf 00; f ¼ S; h ¼ 1þ dh 0 at g ¼ 0 ð10Þ

and

f0 ! 0; h! 0 as g!1; ð11Þ

where the prime denotes differentiation with respect to g,

M ¼ffiffiffiffiffiffiffiffiffi2rB2

0L

qU0

qis the magnetic parameter, k ¼ N1

ffiffiffiffiffiffiU0m2L

qis the

velocity slip parameter, d ¼ D1

ffiffiffiffiffiffiU0

2mL

qis the thermal slip param-

eter and S ¼ V0ffiffiffiffiffiU0m2L

p > 0 (or <0) is the suction (or blowing)

parameter, R ¼ 4r�T31

jk� is the radiation parameter, Pr ¼ lcpj is

the Prandtl number.Eq. (9) can also be written as h 00 þ Prmðfh 0 � f 0hÞ ¼ 0 where

Prm ¼ m=j 1þ 43R

� �is the modified Prandtl number.

3. Numerical method for solution

The above Eqs. (8) and (9) along with the boundary conditionsare solved by converting them to an initial value problem. We

set

f 0 ¼ z; z 0 ¼ p; p 0 ¼ ½2z2 � fpþM2z�; ð12Þ

h 0 ¼ q; q 0 ¼ � 3Pr

4Rþ 3

� �ðfq� zhÞ; ð13Þ

with the boundary conditions

fð0Þ ¼ S;f 0ð0Þ ¼ 1þ kc; f 00ð0Þ ¼ c; hð0Þ ¼ 1þ db; h 0ð0Þ ¼ b:

ð14Þ

In order to integrate (12) and (13) as an initial value prob-lem one requires a value for p(0), i.e. f00ð0Þ and q(0), i.e. h0(0)but no such value is given at the boundary. The suitable guess

value for f00ð0Þ and h0(0) are chosen and then integration iscarried out.

The most important factor of shooting method is to choose

the appropriate finite value of g1. In order to determine g1 forthe boundary value problem stated by Eqs. (12)–(14), we startwith some initial guess value for some particular set of physical

parameters to obtain f00ð0Þ and h0(0). The solution procedure isrepeated with another large value of g1 until two successivevalues of f00ð0Þ and h0(0) differ only by the specified significant

digit. The last value of g1 is finally chosen to be the mostappropriate value of the limit g1 for that particular set of



η

λλλ

= 0.3

= 0.5

= 0.1S = 0.1, R = 0.1, M = 0.1,

= 0.1, Pr = 0.7δ

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

θ (η)

Figure 2b Variation of temperature h(g) with g for several values

of velocity slip parameter k.

η

λ = 0.1, S = 0.1

M = 0M = 0.3M = 0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6

/ ( )f η

Figure 3a Variation of horizontal velocity f0(g) with g for several

values of magnetic parameter M.

θ (η)

M = 0.6

M = 0

M = 0.3

λ = 0.1, S = 0.1, R = 0.1, δ = 0.1, Pr = 0.7

0.2

0.4

0.6

0.8

1

1.2

4 S. Mukhopadhyay

parameters. The value of g1 may change for another set ofphysical parameters. Once the finite value of g1 is determinedthen the integration is carried out. We compare the calculated

values for f0 and h at g = g1( = 10) (say) with the given bound-ary conditions f0(10) = 0 and h(10) = 0 and adjust theestimated values, f00ð0Þ and h0(0), to give a better approximation

for the solution.We take the series of values for f00ð0Þ and h0(0) and apply the

fourth order classical Runge–Kutta method with step-size

h= 0.01. The above procedure is repeated until we get theconverged results within a tolerance limit of 10�5.

4. Results and discussion

In order to analyze the results, numerical computation hasbeen carried out using the method described in the previous

section for various values of the velocity slip parameter ðkÞ,suction (/injection) parameter (S), magnetic parameter (M),radiation parameter (R), thermal slip parameter (d). For illus-trations of the results, numerical values are plotted in the Figs.

1a–7d.For the verification of accuracy of the applied numerical

scheme, comparisons of the present results corresponding to

the values of heat transfer coefficient [�h0(0)] for k ¼ 0; d ¼ 0and S = 0 (i.e. in absence of velocity slip, thermal slip andsuction at the boundary) are made with the available results

of Magyari and Keller [5], Bidin and Nazar [10], El-Aziz[11] and Ishak [12] (for some special cases) and presented inTable 1. The results are found in excellent agreement. As inthis paper, numerical solution is obtained with the help of

shooting method so our results differ slightly with the otherresearchers. One knows that such type of solution dependson the initial guesses which are not obviously be accurate so

the slight variation in the data presented in Table 1 is found.Let me first concentrate on the effects of velocity slip

parameter k on velocity and shear stress profiles in presence

of suction at the wall.In Fig. 2a, velocity profiles are shown for different values of

kðk ¼ 0:1; 0:3; 0:5Þ. The velocity curves show that the rate of

transport decreases with the increasing distance (g) of thesheet. In all cases the velocity vanishes at some large distancefrom the sheet (at g = 6). With the increasing k, the horizontalvelocity is found to decrease.

η

λλλ

= 0.1

= 0.3

= 0.5M = 0.1, S = 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6

/ ( )f η


values of velocity slip parameter k.

η

00 2 4 6 8 10


of magnetic parameter M.


When slip occurs, the flow velocity near the sheet is nolonger equal to the stretching velocity of the sheet. With the in-

crease in k, such slip velocity increases and consequently fluidvelocity decreases because under the slip condition, the pullingof the stretching sheet can be only partly transmitted to the

fluid. It is noted that k has a substantial effect on the solutions.Temperature profiles are presented in Fig. 2b for the variationof velocity slip parameter in presence of suction and magnetic

field. With the increasing k, the temperature is found to de-crease initially but after a certain distance from the sheet it in-creases with k (Fig. 2b).



η

λ = 0.1S = −0.5

S = −0.3S = 0

S = 0.3S = 0.5

M = 0.1,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7

/ ( )f η


values of suction/blowing parameter S.

η

S = 0.5

S = 0.3

S = 0

S = −0.5

S = −0.3

λ = 0.1, = 0.1, R

Pr = 0.7δ = 0.1, M = 0.1,

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

θ (η)


of suction /blowing parameter S.

η

δ

δδ

= 0.1δ = 1.5

= 2.5

= 4.0

R ,1.0= rP 7.0=

S = 0.1, = 0.1, λM = 0.1,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9

θ (η)

Figure 5 Variation of temperature h(g) with g for several values

of thermal slip parameter d.

R = 0.1

R = 0.3

R = 0.6

η

Pr = 0.7= 0.1,δ = 0.1, S = 0.1, λM = 0.1,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

θ (η)

Figure 6 Variation of temperature h(g) with g for several values

of thermal radiation parameter R.

S

λ = 0.3

λ = 0.1

M = 0M = 0.3

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.4 −0.2 0 0.2 0.4

// (0)f

Figure 7a Skin-friction coefficient f00ð0Þ against suction param-

eter S for two values of magnetic parameter M and velocity slip

parameter k.

R = 0.1, = 0.1, Pr = 0.7δ

λ = 0.1 M = 0M = 0.3

= 0.3λ

S

−0.2

−0.1

0

0.1

0.2

0.3

−0.4 −0.2 0 0.2 0.4

/ (0)θ

Figure 7b Heat transfer coefficient h0(0) against suction param-

eter S for two values of magnetic parameter M and velocity slip

parameter k.


Fig. 3a exhibits the nature of velocity field for the variationof magnetic parameter M. With increasing M, velocity is foundto decrease (Fig. 3a) but the temperature increases in this case

(Fig. 3b). The transverse magnetic field opposes the motion ofthe fluid and the rate of transport is considerably reduced. Thisis because with the increase in M, Lorentz force increases andit produces more resistance to the flow. As M increases, ther-

mal boundary layer thickness increases.Figs. 4a and 4b depict the effects of suction parameter S on

velocity and temperature profiles respectively in presence of

slip at the boundary for exponentially stretching sheet. It is ob-served that velocity decreases significantly with increasing suc-


tion parameter whereas fluid velocity is found to increase withblowing (Fig. 4a). It is observed that, when the wall suction(S > 0) is considered, this causes a decrease in the boundary

layer thickness and the velocity field is reduced. S = 0 repre-sents the case of non-porous stretching sheet. Opposite behav-ior is noted for blowing (S < 0).

From Fig. 4b, it is seen that temperature decreases withincreasing suction parameter whereas it increases due to blow-ing (Fig. 4b). Temperature overshoot is noted for blowing

(S < 0). This feature prevails up to certain heights and then



R

R

= 0.1

= 0.6

δ

rP 0.7=

S = 0.1, λ = 0.1, M = 0,

−0.0275

−0.027

−0.0265

−0.026

−0.0255

−0.025

−0.0245

−0.024

−0.0235

−0.023

−0.0225

0.5 1 1.5 2 2.5 3 3.5 4

/ (0)θ

Figure 7c Heat transfer coefficient h0(0) against thermal slip

parameter d for two values of radiation parameter R.

δ

Pr = 0.7

Pr = 0.5

= 0.1, R = 0.1 Sλ = 0.1, M = 0,

−0.0255

−0.025

−0.0245

−0.024

−0.0235

−0.023

−0.0225

0 0.5 1 1.5 2 2.5 3 3.5 4

/ (0)θ

Figure 7d Heat transfer coefficient h0(0) against thermal slip

parameter d for two values of Prandtl number Pr.

6 S. Mukhopadhyay

the process is slowed down and at a far distance from the walltemperature vanishes.

Fig. 5 depicts the effects of thermal slip parameter d on tem-

perature. Initially the temperature decreases with thermal slipparameter d but after a certain distance g from the sheet, suchfeature is smeared out. With the increase of thermal slip

parameter d, less heat is transferred to the fluid from the sheetand so temperature is found to decrease.

Table 1 Values of [�h0(0)] for several values of Prandtl number,

velocity and thermal slips and suction/blowing.

Pr R M Magyari and Keller [5] Bidin

1 0 0 0.9548 0.954

2 1.471

3 1.8691 1.869

5 2.5001

10 3.6604

1 1

0.5 0 0.676

1 0.531

1

2 0.5 0 1.073

1 0.862

3 0.5 1.380

1 1.121


Next, the effect of thermal radiation on temperature pro-files is presented in Fig. 6. It is found that temperature in-creases as the radiation parameter R increases [Fig. 6]. This

is in agreement with the physical fact that the thermal bound-ary layer thickness increases with increasing R. The effect ofradiation in the thermal boundary layer Eq. (4) is equivalent

with an increased thermal diffusivity, i.e. Pr= 1þ 43R

� �in Eq.

(9) can be considered as an effective Prandtl number which re-duces with increasing values of R.

Fig. 7a exhibits the nature of skin-friction coefficient ½f00ð0Þ�with suction/blowing parameter S for two values of velocityslip ðk ¼ 0:1; 0:3Þ. It is found that skin-friction coefficient de-creases with S whereas it increases with increasing magnetic

parameter M as well as with the higher values of slip velocityat the boundary.

Fig. 7b presents the behavior of heat transfer coefficient

[h0(0)] with suction/blowing parameter S for two values ofvelocity slip ðk ¼ 0:1; 0:3Þ. It is very clear that heat transfer de-creases with increasing values of suction, magnetic parameter

and also with increasing values of velocity slip. Fig. 7c displaysthe nature of heat transfer coefficient against thermal slipparameter d for two values of radiation parameter R. Heat

transfer increases with thermal slip parameter d but decreaseswith radiation parameter R. Rate of heat transfer increaseswith Prandtl number (Fig. 7d). An increase in Prandtl numberreduces the thermal boundary layer thickness. Prandtl number

signifies the ratio of momentum diffusivity to thermal diffusiv-ity. Fluids with lower Prandtl number will possess higher ther-mal conductivities (and thicker thermal boundary layer

structures) so that heat can diffuse from the sheet faster thanfor higher Pr fluids (thinner boundary layers). Hence Prandtlnumber can be used to increase the rate of cooling in conduct-

ing flows.But thermal slip parameter, Prandtl number and radiation

parameter have no effects on skin friction coefficient as the

momentum boundary layer equation is independent of h.

5. Conclusions

The present study gives the numerical solutions for steadyboundary layer MHD flow and radiative heat transfer overan exponentially porous stretching surface in presence of

radiation parameter and magnetic parameter in the absence of

and Nazar [10] El-Aziz [11] Ishak [12] Present study

7 0.9548 0.9548 0.9547

4 1.4715 1.4714

1 1.8691 1.8691 1.8691

2.5001 2.5001 2.5001

3.6604 3.6604 3.6603

0.8611 0.8610

5

5 0.5312 0.5311

0.4505 0.4503

5 1.0734

7 0.8626

7 1.3807

4 1.1213




magnetic field and partial slips at the boundary. Based on thepresent investigation, the following observations are made:

(i) The effect of suction parameter on a viscous incompress-ible fluid is to suppress the velocity field which in turncauses the enhancement of the skin-friction coefficient.

(ii) Due to increasing velocity slip, velocity decreases. Withthe increase in thermal slip parameter, temperaturedecreases.

(iii) Surface shear stress increases as the magnetic parameterincreases.

(iv) Magnetic parameter reduces the rate of transport.(v) Wall temperature increases with increasing magnetic

parameter.(vi) The temperature increases with increasing values of the

radiation parameter. This phenomenon is ascribed to a

higher effective thermal diffusivity.(vii) Thermal boundary layer thickness increases with the

increase in magnetic parameter as well as radiation

parameter.(viii) Prandtl number reduces the thermal boundary layer

thickness.

Acknowledgement

Thanks are indeed due to the reviewers for their constructive

suggestions which led to a definite improvement of the qualityof the manuscript.

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Swati Mukhopadhyay was born and brought

up in the district of Burdwan, West Bengal,

India. She obtained the B.Sc. Honours and

M.Sc. degrees in Mathematics from the

University of Burdwan. She joined M.U.C.

Women’s College, Burdwan as an Assistant

Professor in Mathematics in 2006. She was

awarded Ph.D. degree in Fluid Mechanics by

the University of Burdwan in 2007. Being

awarded BOYSCAST Fellowship by the

Department of Science and Technology,

Govt. of India in 2007–2008, she carried out her post-doctoral research

work in NTNU, Trondheim, Norway in 2008. She is serving the

Department of Mathematics, the University of Burdwan, West Bengal

as an Assistant Professor since 2010. Besides teaching she is actively

engaged in research in the field of Fluid mechanics particularly, in

bio-fluid dynamics, boundary layer flows, Newtonian/non-Newtonian

fluids, heat and mass transfer in porous/non-porous media. Her

research interest also covers the Nano fluid flow problems, existence

and uniqueness of solutions and other related matters.



Documents

Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation