Scientia Iranica B (2014) 21(4), 1355{1366

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

MHD stagnation slip ow over an unsteady stretchingsurface in porous medium

Z. Abbasa;�, N. Muhammadb and G. Mustafaa

a. Department of Mathematics, the Islamia University of Bahawalpur, Bahawalpur 63100, Pakistanb. Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Pakistan

Received 26 April 2012; received in revised form 18 September 2013; accepted 14 December 2013

KEYWORDSViscous uid;Stagnation-point ow;Stretching sheet;Porous medium;Slip condition.

Abstract. An analysis has been carried out to study the heat transfer analysis andMHD stagnation-point ow of a viscous uid over an unsteady stretching sheet in aporous medium with slip condition. For the present problem, the governing equationsare transformed into a system of non-linear ordinary di�erential equations by means ofsimilarity transformations. This system is solved both analytically by an analytic technique,namely the Homotopy Analysis Method (HAM), and numerically using a shooting methodwith Runge-Kutta algorithm. The in uences of the involved parameters on the ow andtemperature �elds are graphically illustrated and analyzed. The results obtained by meansof both methods are compared and found in excellent agreement.© 2014 Sharif University of Technology. All rights reserved.

1. Introduction

The study of boundary layer ow and heat transfercharacteristics due to a continuously stretching heatedsurface placed in a porous medium has considerableattention in many engineering and industrial applica-tions. The interest is due to many practical applica-tions in thermal engineering and geophysics which canbe formulated or approximated as transport phenom-ena in porous medium. These type of ows appear in awide range of industrial disciplines, as well as in manynatural circumstances such as geothermal extraction,storage of radioactive nuclear waste material, groundwater ows, oil recovery processes, food processing,industrial and agricultural water distribution, thermalinsulation engineering, packed-bed reactors, castingand welding of manufacturing processes, soil pollutionand the dispersion of chemical contaminants in variousprocesses in the chemical industry. Similar situationsexist during the manufacture of plastic and rubber

*. Corresponding author. Tel.: +92 62 9255480E-mail address: za qau@yahoo.com (Z. Abbas)

sheets where it is often necessary to blow a gaseousmedium through a not-yet solidi�ed material, andwhere the stretching forces may be varying with time(see Pal [1]). The work on this problem has beeninitiated by Sakiadis [2,3], who discussed the boundarylayer ow generated by a continuous solid surfacemoving with a constant velocity. Crane [4] studied the ow of a viscous uid over a linearly stretching surfaceand obtained an exact analytic solution. Additionallymany authors [5-10] investigated the Crane's problemfor various aspects of the ow and/or heat transferanalysis.

On the other side, in many practical problems, themotion of the continuous stretching surface may startimpulsively from rest and the transient or unsteadyaspects become more interesting. However, to thebest of our knowledge Wang [11] initially discussedthe ow of a liquid �lm over an unsteady stretchingsheet. Recently, Elbashbeshy and Bazid [12] presentedthe similarity solution of boundary layer ow andheat transfer due to an unsteady stretching sheet.Sharidan et al. [13] investigated the unsteady boundarylayers over a stretching sheet for special distribution of

1356 Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366

stretching velocity and surface temperature or surfaceheat ux. Tsai et al. [14] discussed the non-uniformheat source/sink e�ect on the ow and heat transferover an unsteady stretching sheet through a quiescent uid medium extending to in�nity. The problem forunsteady stretching surface condition and heat transferhas been studied by several authors such as Abd El-Aziz [15], Mukhopadhyay [16], Ishak et al. [17], Hayatet al. [18] and Ziabakhsh et al. [19]. Very recently,Mukhopadhyay [20] discussed the e�ects of slip andmixed connective ow past a porous unsteady stretch-ing surface numerically. Elbashbeshy and Emam [21]discussed the e�ects of the thermal radiation and heattransfer over an unsteady stretching surface embeddedin a porous medium in the presence of heat sourceor sink. Sharma [22] studied the e�ects of viscousdissipation and heat source on unsteady boundarylayer ow and heat transfer past a stretching sheetin a porous medium using the element free Galerkinmethod. Ibrahim and Shanker [23] presented thenumerical study of unsteady MHD boundary layer owand heat transfer due to a stretching sheet in thepresence of heat source/sink. Khan et al. [24] presenteda mathematical model for the unsteady stagnation-point ow of a linear viscoelastic uid bounded bya stretching/shrinking sheet. Recently, Reddy andReddy [25] discussed mass transfer and MHD e�ectson an unsteady porous stretching sheet in a porousmedium with variable heat ux in the presence of heatsource.

Having in mind all the above studies, the aim ofthe present paper is to study the combined e�ects ofMHD stagnation-point ow and heat transfer due toan unsteady stretching sheet embedded in a porousmedium with velocity/thermal slip condition. The con-servation equations of mass, momentum and energy areconverted into a non-linear boundary value problem bymeans of suitable stream function. The resultant non-linear equations along with the boundary conditionsare then solved numerically using shooting methodwith Runge-Kutta integration scheme and analyticallyusing a powerful technique the homotopy analysismethod [26-37]. Comparisons of the present resultswith previously published works in the literature aregiven and found in excellent agreement.

2. Flow equations

We consider an unsteady, two-dimensional MHD stag-nation point ow of a viscous uid in a porous mediumover an unsteady stretching sheet in the region y > 0as shown in Figure 1. The x axis is taken along thesurface, while the y axis is perpendicular to the surface.At time t = 0; the surface is stretched with the velocityU!(x; t) along the x axis, keeping the origin is �xed. Itis also assumed that the uid is electrically conducting

Figure 1. Flow geometry and coordinate system.

and the magnetic �eld B0 is applied in the y direction.The induced magnetic �eld is neglected due to asmall magnetic Reynolds number assumption, whereno external electric �eld is applied. The velocity of the ow external to the boundary layer is Ue(x; t) and thetemperature at the surface is T!(x; t); where T! > T1,with T1 being the temperature of the ambient uid.Under these assumptions along with the boundary layerapproximations, the governing equations for the owand energy are given as:@u@x

+@v@y

= 0; (1)

@u@t

+ u@u@x

+ v@u@y

=@Ue@t

+ Ue@Ue@x

+ �@2u@y2 +

�B20

�(Ue � u) +

��k(t)

(Ue � u) ; (2)

@T@t

+ u@T@x

+ v@T@y

= �@2T@y2 ; (3)

where u and v are the velocity components in the x andy directions, respectively, � is the kinematic viscosity,� is the uid density, � is the electrical conductivityof the uid, � is porosity of the medium, � is thethermal di�usivity, t is the time, T is the temperatureof the uid, and k(t) is the permeability of the porousmedium. Here we assume k(t) is of the form:

k(t) = k1(1� �t); (4)

where k1 is the initial permeability.The relevant boundary conditions for the present

problem are:

u = U!(x; t) +N1�@u@y; v = 0;

T = T!(x; t) +D1@T@y

at y = 0; (5)

u! Ue(x; t); T ! T1 as y !1: (6)

Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366 1357

Here:

U! =bx

(1� �t) ; Ue =dx

(1� �t) ;

T! (x; t) = T1 +cx

�(1� �t) ;where b � 0; d � 0; c � 0 and � � 0 are constants (with�t < 1) and have dimensions (time)�1, N1 = N

p1� �t

is the velocity slip parameter, and D1 = Dp

1� �tis the thermal slip parameter, both are changed withtime, and N and D arethe initial values of velocity andthermal slip parameters, having dimension (velocity)�1

and length, respectively. The no-slip condition can beobtained for N = 0 and D = 0; respectively.

We de�ne the following similarity transforma-tions:

�=rb�

(1��t)� 12 y; =

pb�x(1��t)� 1

2 f(�); (7)

� (�) =T � T1T! � T1 ;

T! = T1 + Tref

bx2 (1� �t)�3

2

2�

!; (8)

where Tref is taken as a constant reference temperatureand the stream function (x; y) is de�ned by u =@ =@y and v = �@ =@x, such that the continuityequation (Eq. (1)) is automatically satis�ed.

Using Eqs.(7) and (8), Eqs. (2) and (3) become:

f 000�f 02 + ff 00 �A�f 0 + 1

2�f 00

�+M2 ("� f 0)

+ � ("� f 0) +A"+ "2 = 0; (9)

�1Pr �00 � 2f 0� + f�0 � A

2(3� + ��0) = 0; (10)

and the corresponding boundary conditions are:

f = 0; f 0 = 1 + �f 00; � = 1 + �0 at � = 0;

f 0 ! "; � ! 0 as � !1; (11)

where A = �=b is the unsteadiness parameter, Pr =�=� is the Prandtl number, " = d=b is the ratio ofthe external ow rate to the stretching rate, M2 =�B2

0 (1� �t) =�b is the local magnetic parameter, � =��=kb is the porosity parameter, � = N

pb� is the

velocity slip parameter, = Dpb=� is the thermal slip

parameter and the primes indicate the di�erentiationwith respect to �. It is worth mentioning that wecould recover the no-slip condition by taking � = 0and = 0.

It should be mentioned here that in the paperof Elbashbeshy and Bazid [12] the sign of the term2f 0� is positive in their energy equation due to theincorrect de�nition of �T = T! � T1 and hencean exact comparison is not possible. According tothem �T = T! � T1 = b

2�x2 (1 � �t)�32 but the

correct value is bx2

2� (1� �t)�32 . Due to this error, some

physically unrealistic phenomena in the velocity andtemperature �elds are encountered for speci�c valuesof the unsteadiness parameter. Abd El-Aziz [15] alsomentioned this error in his paper. In the presentwork when M; �; "; � and are taken to be zerothen Eqs. (9) and (10) with the boundary conditions(Eq. (11)) reduce to those of Abd El-Aziz [15].

The skin-friction coe�cient, Cf , and the localNusselt number, Nux, are given by:

Cf =�!�U2

!; Nux =

xq!km (T! � T1)

; (12)

where km is thermal conductivity, �! is the shear stressat the wall and q! is the heat ux at wall, which arede�ned as:

�!=��@u@y

�y=0

; and q!=�km�@T@y

�y=0

:(13)

With the help of Eqs. (7), (8) and (13), Eq. (12) yields:

NuxpRex

= ��0(0); orp

RexCf = f 00(0); (14)

where Rex = xU!=� is the local Reynolds number.

3. Solution of the problem

3.1. Numerical solutionTo �nd the numerical solution, we use the moste�ective shooting method with fourth order Runge-Kutta integration scheme (see Na [38]) to solve theboundary value problem. The non-linear equations(Eqs. (9) and (10)) subject to boundary conditions(Eq. (11)) are transformed into a system of �ve �rstorder di�erential equations as:

df0

d�= f1; (15)

df1

d�= f2; (16)

df2

d�=� ff2 + (f1)2 +Af1 +

12A�f2

�M2 ("� f1)� � ("� f1)�A"� "2; (17)

1358 Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366

d�0

d�= �1; (18)

d�1

d�= Pr

�2f1� � f�1 +

32A� +

12A��1

�; (19)

and the boundary conditions are:

f(0) = 0; f1(0) = 1 + �f2(0); f1(1) = ";

�(0) = 1 + �1(0); �(1) = 0: (20)

Here, f0 = f(�) and �0 = �(�): A boundary valueproblem is �rst converted into an initial value problemby appropriately guessing the missing conditions, f2(0)and �1(0): The resultant initial value problem is solvedby shooting method for a set of parameters appearingin the governing equations with a known values of f2(0)and �1(0):

3.2. Homotopy analysis solutionFor the series solutions of Eqs. (9) and (10), using ho-motopy analysis method (HAM), it is straight forwardthat the velocity and the temperature �elds f (�) and� (�) can be expressed by the set of base functions:�

�k exp (�n�)�� k � 0; n � 0

; (21)

in the form:

f (�) = a00;0 +

1Xn=0

1Xk=0

akm;n�k exp (�n�) ; (22)

� (�) =1Xn=0

1Xk=0

bkm;n�k exp (�n�) ; (23)

where akm;n and bkm;n are the coe�cients. By rule ofsolution expressions of f(�) and �(�), with the help ofboundary conditions (Eq. (11)) one can choose f0 (�)and �0 (�):

f0 (�) = "� +(1� ") (1� e��)

1 + �; (24)

�0 (�) =e��

1 + ; (25)

as the initial guess approximations of f(�) and �(�)and the auxiliary linear operators:

Lf (f) =d3fd�3 � df

d�; (26)

L�(f) =d2fd�2 � f; (27)

which have the following properties:

Lf [C1 + C2 exp(�) + C3 exp(��)] = 0; (28)

L� [C4 exp(�) + C5 exp(��)] = 0; (29)

where Ci, (i = 1� 5) are arbitrary constants. If ~f and~� denote the non-zero auxiliary parameters then thezeroth-order deformation problems are constructed as:

(1� q)Lfh bf (�; q)� f0 (�)

i=

q~fNfh bf (�; q)

i; (30)

(1� q)L�hb� (�; q)� �0 (�)

i=

q~�N�h bf (�; q) ; b� (�; q)

i; (31)

bf (0; q) = 0;

d bf (0; q)d�

= 1 + �d2 bf (0; q)d�2 ;

d bf (1; q)d�

= "; (32)

b� (0; q) = 1 + db� (0; q)d�

;

b� (1; q) = 0; (33)

where q 2 [0; 1] is an embedding parameter and thenonlinear operators Nf and N� are:

Nf� bf��; q

��=@3 bf��; q�

@�3 ��@ bf��; q�

@�

�2

+ bf��; q�@2 bf��; q�@�2 �A

@ bf��; q�@�

� 12A�

@2 bf��; q�@�2

+M2�"�

@ bf��; q�@�

�

+ ��"�

@ bf��; q�@�

�+A"+ "2;

(34)

Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366 1359

N�� bf��; q

�; b���; q

��=�1Pr �00 � 2

@ bf��; q�@�

���; q�

+ f��; q�@b���; q�

@�� 3

2A���; q�

� 12A�

@b���; q�@�

: (35)

For q = 0 and q = 1, the above zeroth-orderdeformation equations (Eqs. (30) and (31)) have thesolutions:bf (�; 0) = f0 (�) ; bf (�; 1) = f (�) ; (36)b� (�; 0) = �0 (�) ; b� (�; 1) = � (�) : (37)

Expanding bf (�; q) and b� (�; q) in Taylor's series withrespect to q, we have:

bf (�; q) = f0 (�) +1Xm=1

fm (�) qm; (38)

b� (�; q) = �0 (�) +1Xm=1

�m (�) qm; (39)

where:

fm (�) =1m!

@m bf (�; q)@qm

�����q=0

;

�m (�) =1m!

@mb� (�; q)@qm

�����q=0

: (40)

Note that the zeroth-order deformation equations(Eqs. (30) and (31)) contain two auxiliary parameters~f and ~�. The convergence of the series (30) and (31)depends on these parameters. Assuming that ~f and ~�are selected such that the above series are convergentat q = 1, then using Eqs. (36) and (37), the seriessolutions are:

f (�) = f0 (�) +1Xm=1

fm (�) ; (41)

� (�) = �0 (�) +1Xm=1

�m (�) : (42)

Di�erentiate the zeroth-order deformation equations(Eqs. (30) and (31)) m times with respect to q, thensetting q = 0, and �nally dividing them by m!, we

obtain the mth-order deformations equations as:

Lf [fm (�)� �mfm�1 (�)] = ~fRfm (�) ; (43)

L� [�m (�)� �m�m�1 (�)] = ~�R�m (�) ; (44)

fm (0) = 0; f 0m(0) = �f 00m (0) ; fm (1) = 0; (45)

�m (0) = �0m (0) ; �m (1) = 0; (46)

where:

Rfm���

=f 000m�1 �A�f 0m�1 +

12�f 00m�1

��M2f 0m�1 � �f 0m�1

+m�1Xk=0

�fm�1�kf 00k � f 0m�1�kf 0k

�+ (1� �m)(M2"+ �"+A"+ "2); (47)

R�m (�) =�1Pr �00m�1 � 3

2A�m�1 �A�2�0m�1

+m�1Xk=0

�fm�1�k�0k � 2f 0m�1�k�k

�; (48)

�m =

(0; m � 1;1; m > 1:

(49)

If we suppose f?m (�) and �?m (�) as the special solutionsof Eqs. (43) and (44), then from Eqs. (43) and (44), thegeneral solutions are given by:

fm (�) = f?m (�) + C1 + C2 exp (�) + C3 exp (��) ;(50)

�m (�) = �?m (�) + C4 exp (�) + C5 exp (��) ; (51)

where the integral constants Ci, (i = 1� 5) are de-termined from the boundary conditions (Eqs. (45)and (46)) as:

C2 = C4 = 0;

C3 =

@f?m(�)@�

����=0� � @f2?

m (�)@�2

����=0

1 + �;

C1 = �C3 � f?m (0) ;

C5 = ��?m (�)j�=0 � @�?m(�)

@�

����=0

1 + : (52)

In this way, it is easy to solve the linear non-homogeneous Eqs. (43) and (44) by using Mathematicaone after the other in the order m = 1; 2; 3:::::::

1360 Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366

3.3. Convergence of the HAM solutionAs proved by Liao [26], as long as a solution series givenby the homotopy analysis method converges, it mustbe one of the solutions. Therefore, it is important toensure that the solutions series are convergent. Theseries solutions (41) and (42) contain the non-zeroauxiliary parameters ~f and ~�, which can be chosenproperly by plotting the so-called ~-curves to ensurethe convergence of the solutions series and rate ofapproximation of the HAM solution. To see the rangefor admissible values of ~f and ~�, ~-curves of f 00(0)and �0(0) are shown in Figure 2, for 20th order ofapproximation when A = 0:2; M = 0:5; � = 0:2;Pr = 0:5; " = 0:2 and � = = 0:2. From this �gure itcan be seen that ~-curves have a parallel linessegmentthat correspond to the regions �1:1 � ~f � �0:2 and�1:15 � ~� � �0:2; respectively. Table 1 is made toshow the convergence and comparison of HAM solutionfor various order of approximations with numerical

Figure 2. The h-curves of f 00(0) and �0(0) at the 20thorder of approximation; �lled circles are the numericalvalues with A = 0:2, M = 0:5, � = 0:2, Pr = 0:5, " = 0:2, = 0:2 and � = 0:2.

Table 1. Convergence and comparison of HAM solutionfor di�erent order of approximation with numerical resultswhen A = 0:2, M = 0:5, � = 0:2, Pr = 0.5, " = 0:2,� = 0:2 and = 0:2.

Order ofapproximations

�f 00(0) ��0(0)

1 0.815741 0.8164545 0.834424 0.7695239 0.834491 0.76637215 0.834491 0.76600520 0.834491 0.76598723 0.834491 0.76598525 0.834491 0.76598530 0.834491 0.765985

Numerical results 0.834519 0.765986

results when A = 0:2; M = 0:5; � = 0:2; Pr = 0:5;" = 0:2 and � = = 0:2.

3.4. Results and discussionThe system of Eqs. (9) and (10) with boundary con-ditions (11) has been solved both analytically usingHomotopy Analysis Method (HAM) and numericallyusing shooting method [38] with Runge-Kutta algo-rithm. Figures 3-12 are plotted in order to analyzethe in uences of the various involving physical param-eters, for example, an unsteadiness parameter A; themagnetic parameter M; the porosity parameter �; thevelocity slip parameter �; the ratio of external ow rateto the stretching rate "; the Prandtl number Pr andthe thermal slip parameter on the velocity f 0(�) andtemperature �(�) distributions. The numerical valuesof the skin-friction coe�cient �f 00(0) and the rate ofheat transfer at the wall (the local Nusselt number)��0(0) for various values of parameters are given inTables 2-5.

Figure 3 shows the e�ects of an unsteadiness

Figure 3. The velocity pro�le f 0(�) verses � for variousvalues of unsteadiness parameter A: dashed lines arenumerical solution and �lled circle are HAM solution at12th order of approximation with M = 0:2, � = 0:1," = 0:1 and � = 0:2.

Table 2. A comparison of the values of ��0(0) withRefs. [12,14] for several values of A and Pr withM = � = " = � = = 0.

Pr A Ref. [12] Ref. [15] Present resultsHAM Numerical

0.1 0.8 0.2707 0.4517 0.4517 0.453571 0.6348 1.6728 1.6728 1.672010 1.2552 5.70503 5.7059 5.704940.1 1.2 0.3576 0.5087 0.5086 0.50301 0.9491 1.818 1.818 1.81810 2.4177 6.12067 6.0612 6.120130.1 2.0 0.4991 0.604013 0.60376 0.604781 1.4086 2.07841 2.0784 2.0781710 3.9814 6.88506 6.63421 6.88176

Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366 1361

parameter A on the velocity component f 0(�) whenM = 0:2; � = 0:1; " = 0:1 and � = 0:2. Both thevelocity and the boundary layer thickness are decreasedas an unsteadiness parameter A increases. Figure 4elucidates the in uence of the magnetic parameter Mon the velocity f 0(�) when A = 0:2; � = 0:1; " = 0:1and � = 0:2. It is noted from this �gure that thevelocity decreases by increasing the values of magneticparameter M . This is because for the present problem,the magnetic force acts as a resistance to the ow.The boundary layer thickness is also decreased as Mincreases. The change in the velocity �eld f 0(�) fordi�erent values of porosity parameter � can be seen

Table 3. Numerical values of skin friction coe�cient,�f 00(0), and the local Nusselt number, ��0(0), for severalvalues of A, M and � with " = 0:5, = � = 0:2 and Pr =0.5.

�f 00(0) ��0(0)A M � HAM Numerical HAM Numerical0.2 0.5 0.5 0.60449 0.60449 0.83901 0.839010.8 0.64108 0.64109 0.95431 0.954251.2 0.66376 0.66376 1.0201 1.02012.0 0.70545 0.70545 1.1330 1.13300.8 0 0.62073 0.62075 0.95657 0.95651

0.5 0.64108 0.64109 0.95431 0.954251.0 0.69538 0.69548 0.94847 0.948411.5 0.76970 0.76970 0.94099 0.940992.0 0.85193 0.85193 0.93334 0.933340.5 0 0.59900 0.59904 0.95904 0.95900

0.5 0.64108 0.64109 0.95430 0.954251.0 0.67827 0.67829 0.95028 0.950211.5 0.71165 0.71165 0.94679 0.946792.0 0.74194 0.74195 0.94372 0.94370

in Figure 5. It is found that the velocity f 0(�) is adecreasing function of �: The boundary layer thicknessis decreased for large values of �: Figure 6 depictsthe variations of the velocity slip parameter � onthe velocity component f 0(�) when " = 0:1. It isobserved that the velocity is decreased by increasingthe values of the velocity slip parameter �: It is alsonoted that for � = 0 (no-slip condition), the values off 0 is equal to 1; which shows the standard conditionfor stretching ow at � = 0: Figure 7 shows thee�ects of the ratio of the external ow rate to the

Table 5. Numerical values of the local Nusselt number��0(0) for several values of A, Pr and withM = � = � = 0:2 and " = 0:5.

Pr A = 0:8 A = 1:2HAM Numerical HAM Numerical

0.1 0.2 0.46416 0.46410 0.50277 0.502710.3 0.76908 0.76901 0.82519 0.825110.7 1.1024 1.1021 1.1729 1.17241.0 1.2700 1.2700 1.3462 1.34531.5 1.4791 1.4761 1.5610 1.56042.0 1.6384 1.6315 1.7239 1.72113.0 1.8761 1.8704 1.9653 1.96475.0 2.1916 2.1901 2.2834 2.28010.7 0 1.4143 1.4149 1.5324 1.5329

0.5 0.82844 0.82854 0.86763 0.867961.0 0.58579 0.58586 0.60512 0.605261.5 0.45309 0.45317 0.46456 0.464612.0 0.36940 0.36983 0.37699 0.376993.0 0.26975 0.26979 0.27378 0.273845.0 0.17522 0.17584 0.17691 0.1761310.0 0.09339 0.09339 0.09387 0.09387

Table 4. Numerical values of skin friction coe�cient, �f 00(0), and the local Nusselt number, ��0(0), for several values of "and � with M = 0:2 and � = 0:2.

" � A = 0:8 A = 1:2 A = 2:0HAM Numerical HAM Numerical HAM Numerical

�f 00(0) ��0(0) �f 00(0) ��0(0) �f 00(0) ��0(0) �f 00(0) ��0(0) �f 00(0) ��0(0) �f 00(0) ��0(0)

0 0.2 1.0176 0.8710 1.0176 0.8716 1.0865 0.9470 1.0865 0.9473 1.2059 1.0724 1.2059 1.07260.5 0.5980 0.9591 0.5981 0.9590 0.6238 1.0240 0.6238 1.0240 0.6707 1.1357 0.6709 1.13571.0 0 1.0476 0 1.0472 0 1.1029 0 1.1025 0 1.2013 0 1.20191.5 0.7305 1.1300 0.7305 1.1304 0.7479 1.1781 0.7479 1.1780 0.7804 1.2653 0.7805 1.26672.0 1.5652 1.2057 1.5652 1.2056 1.5953 1.2483 1.5953 1.2480 1.6515 1.3264 1.6523 1.32690.5 0 0.8050 0.9831 0.8051 0.9831 0.8498 1.0467 0.8498 1.0461 0.9345 1.1562 0.9347 1.1561

0.5 0.4351 0.9386 0.4351 0.9386 0.4494 1.0052 0.4494 1.0050 0.4744 1.1195 0.4745 1.11901 0.3010 0.9206 0.3010 0.9201 0.3081 0.9889 0.3080 0.9889 0.3203 1.1060 0.3203 1.1068

1.5 0.2307 0.9106 0.2307 0.9106 0.2349 0.9801 0.2347 0.9801 0.2421 1.0989 0.2422 1.09872.0 0.1871 0.9043 0.1872 0.9041 0.1900 0.9746 0.1900 0.9746 0.1947 1.0946 0.1948 1.0949

1362 Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366

Figure 4. The velocity pro�le f 0(�) verses � for variousvalues of magnetic parameter M : dashed lines arenumerical solution and �lled circle are HAM solution at12th order of approximation with A = 0:2, � = 0:1," = 0:1 and � = 0:2.

Figure 5. The velocity pro�le f 0(�) verses � for variousvalues of porous medium �: dashed lines are numericalsolution and �lled circle are HAM solution at 12th orderof approximation with A = 0:2, M = 0:2, " = 0:1 and� = 0:2.

Figure 6. The velocity pro�le f 0(�) verses � for variousvalues of slip parameter �: dashed lines are numericalsolution and �lled circle are HAM solution at 12th orderof approximation with A = 0:2, M = � = 0:2 and " = 0:1.

Figure 7. The velocity pro�le f 0(�) verses � for variousvalues of stagnation point parameter ": solid/dashed linesare numerical solution and �lled circle are HAM solutionat 12th order of approximation with A = 0:2, M = 0:2 and� = 0:1.

Figure 8. The temperature pro�le �(�) verses � forvarious values of unsteadiness parameter A: dashed linesare numerical solution and �lled circle are HAM solutionat 12th order of approximation with M = 0:2, � = 0:1, Pr= 0.5, " = = 0:1 and � = 0:2.

stretching rate " on the velocity �eld f 0(�): solidlines for no-slip condition � = 0 and dashed linesfor slip condition � = 0:2, respectively. It is foundthat the velocity f 0(�) is increased for large values of" for both � = 0; � = 0:2 but this change in thevelocity in case of velocity slip parameter (� = 0:2)is smaller for " < 1 and larger for " > 1 near thewall when compared with the case of no-slip condition(� = 0).

Figure 8 gives the in uences of an unsteadinessparameter A on the temperature distribution �(�)when thermal slip parameter = 0:1: Both thetemperature pro�le and the thermal boundary layerthickness are decreased as A increases. Figure 9shows the change in the temperature �(�) for theseveral values of Prandtl number Pr: solid lines forno-thermal slip, = 0, and dashed lines for thermalslip, = 0:1. It is evident from this �gure that the

Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366 1363

Figure 9. The temperature pro�le �(�) verses � forvarious values of Prandtl number Pr: solid/dashed linesare numerical solution and �lled circle are HAM solutionat 12th order of approximation with A = 0:2, M = 0:2,� = 0:1, " = 0:1 and � = 0:2.

Figure 10. The temperature pro�le �(�) verses � forvarious values of magnetic parameter M: dashed lines arenumerical solution and �lled circle are HAM solution at12th order of approximation with A = 0:2," = � = = 0:1, Pr = 1.5 and � = 0:2.

temperature decreases by increasing the values of Prdue to the decreased thermal di�usivity: The thermalboundary layer thickness also decreases for large valuesof Prandtl number. Figure 10 gives the variations inthe temperature distribution �(�) for various valuesof a magnetic parameter M . One can see thatthe temperature is an increasing function of a mag-netic parameter M; and the thermal boundary layerthickness also increases as M is increased. Figure 11presents the e�ects of a porosity parameter, �, on thetemperature distribution, �. It is found from this �gurethat both the temperature and the thermal boundarylayer thickness are the increasing function of �: It isalso noticed from Figures 10 and 11 that for largevalues of M and �; the change in temperature is small,this is because both parameters have no in uence onthe energy equation, directly. The temperature �eld,�(�), for several values of thermal slip parameter, , is

Figure 11. The temperature pro�le �(�) verses � forvarious values of porous medium �: dashed lines arenumerical solution and �lled circle are HAM solution at12th order of approximation with A = 0:2, M = 0:2, Pr =1.5, " = = 0:1 and � = 0:2.

Figure 12. The temperature pro�le �(�) verses � forvarious values of slip parameter : dashed lines arenumerical solution and �lled circle are HAM solution at12th order of approximation with A = 0:2, M = 0:2,� = " = 0:1, Pr = 0.7 and � = 0:2.

shown in Figure 12. It is observed that as the thermalslip parameter increases, less heat is transformed fromthe sheet to the uid, therefore the temperature �(�)decreases by increasing the values of the thermal slipparameter, .

Table 2 shows the numerical values of the localNusselt number ��0(0) for various values of A and Prwhen M = " = � = = � = 0; both numerically andanalytically. In this table we have given a comparisonwith the existing numerical result of [13] and therelative error is also tabulated. One can easily seethat the error is very much smaller which is almostnegligible. Table 3 shows the numerical values of theskin-friction coe�cient, �f 00(0), and the local Nusseltnumber ��0(0) for various values of A; M and � when" = 0:5, = � = 0:2 and Pr = 0:5 both numericallyand analytically. It is noted that the magnitudes of�f 00(0) and ��0(0) are increased for large values of A:

1364 Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366

It can also be seen from this table that the magnitudesof the shear stress at the wall, �f 00(0), increases byincreasing the values of M and �, but the rate ofheat transfer at the wall decreases by increasing thevalues of both M and �: The numerical values of�f 00(0) and ��0(0) for several values of "; � and Aare given in Table 4. It is found that for �xed valuesof " and �; both the magnitude of �f 00(0) and ��0(0)are increased as the values of A increase. It is alsoseen that the magnitude of the skin-friction coe�cient,�f 00(0), decreases for " < 1 and it increases for " >1 for �xed values of A and �: On the other hand,the magnitude of �f 00(0) decreases as the velocityslip parameter, �, increases. However, the rate ofheat transfer at the wall ��0(0) is increased for largevalues of "; where as it decreases by increasing thevalues of �: Table 5 is made to show the numericalvalues of the local Nusselt number ��0(0) for di�erentvalues of Pr; and A when M = � = � = 0:2 and" = 0:5. It is observed that for the �xed values of A;the magnitude of the local Nusselt number increasesfor large values of Pr and decreases as increases. Itis also worth mentioning that the comparison of bothsolutions are given in these tables and found to be ingood agreement.

4. Concluding remarks

In the present investigation, the heat transfer andMHD stagnation point ow of a viscous uid over anunsteady stretching surface through a porous mediumwith ow/thermal slip conditions are studied. Asimilarity solution of the non-linear system of ordinarydi�erential equations is obtained, both analyticallyusing Homotopy Analysis Method (HAM), and numer-ically using shooting method. The e�ects of the variousemerging parameters on the velocity and temperaturedistributions are shown through graphs. The values ofskin friction coe�cient and local Nusselt number arealso given in tabular form. From this analysis, we havemade the following observations:

� Both the velocity f 0 and the boundary layer thick-ness are decreased by increasing � and M .

� The magnitude of the velocity f 0 decreases with anincrease in slip parameter �:

� The temperature � and the thermal boundary layerthickness decrease with an increase in Pr, while theyincrease with an increase in M and �:

� The temperature � decreases by increasing thevalues of thermal slip parameter, :

Acknowledgment

The authors are thankful to the anonymous reviewersfor their useful comments to improve this paper.

References

1. Pal, D. \Combined e�ects of non-uniform heatsource/sink and thermal radiation on heat trans-fer over an unsteady stretching permeable surface",Commn. Nonlinear. Sci. Num. Sim, 16, pp. 1890-1904(2011).

2. Sakiadis, B.C. \Boundary layer behavior on continuoussolid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric ow", AIChE. J., 7, pp.26-28 (1961).

3. Sakiadis, B.C. \Boundary layer behaviour on contin-uous solid surface: II. Boundary layer behavior oncontinuous at surface", AIChE. J., 7, pp. 221-225(1961).

4. Crane, L.J. \Flow past a stretching plane", Z AngewMath Phy., 21, pp. 645-647 (1970).

5. Tsou, F.K. and Sparrow, E.M. and Goldstein, R.J.\Flow and heat transfer in the boundary layer on acontinuous moving surface", Int J. Heat Mass Trans-fer., 10, pp. 219-235 (1967).

6. Vleggaar, J. \Laminar boundary layer behavior oncontinuous, accelerating surface", Chem Eng. Sci., 32,pp. 1517-1525 (1977).

7. Gupta, P.S. and Gupta, A.S. \Heat and mass transferon a stretching sheet with suction or blowing", Cana-dian J. Chem Eng., 55, pp. 744-746 (1977).

8. Soundalgekar, V.M. and Ramana, T.V. \Heat transferpast a continuous moving plate with variable temper-ature", Warme-Und Sto�ubertraguge., 14, pp. 91-93(1980).

9. Grubka, L.J. and Bobba, K.M. \Heat transfer char-acteristics of a continuous stretching surface withvariable temperature", J. Heat Transfer., 107, pp. 248-250 (1985).

10. Ali, M.E. \Heat transfer characteristics of a continuousstretching surface", Warme-Und Sto�ubertraguge, 29,pp. 227-234 (1994).

11. Wang, C.Y. \Liquid �lm on unsteady stretchingsheet", Quart. J. Appl. Math., 48, p. 601 (1990).

12. Elbashbeshy, E.M.A. and Bazid, M.A.A. \Heat trans-fer over an unsteady stretching surface", Heat MassTransfer, 41, pp. 1-4 (2004).

13. Sharidan, S., Mahmood, T. and Pop, I. \Similaritysolutions for the unsteady boundary layer ow andheat transfer due to a stretching sheet", Int. J. Appl.Mech. Eng., 11, pp. 647-654 (2006).

14. Tsai, R., Huang, K.H. and Huang, J.S. \Flow and heattransfer over an unsteady stretching surface with non-uniform heat source", Int. Com. Heat Mass Transfer,35, pp. 1340-1343 (2008).

15. Aziz, Abd El \Radiation e�ect on the ow and heattransfer over an unsteady stretching sheet", Int. Com.Heat Mass Transfer, 36, pp. 521-524 (2009).

Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366 1365

16. Mukhopadhyay, S. \E�ect of thermal radiation onunsteady mixed convection ow and heat transfer overa porous stretching surface in medium", Int J. HeatMass Transfer, 52, pp. 3261-3265 (2009).

17. Ishak, A., Nazar, R. and Pop, I. \Heat transfer over anunsteady stretching permeable surface with prescribedwall temperature", Nonlinear Anal: Real World Appl.,10, pp. 2909-2913 (2009).

18. Hayat, T., Qasim, M. and Abbas, Z. \Radiation andmass transfer e�ects on the magnetohydrodynamicunsteady ow induced by a stretching sheet", ZNA.,65a, pp. 231-239 (2010).

19. Ziabakhsh, Z., Domairry, G., Moza�ari, M. andMahbobifar, M. \Analytical solution of heat transferover an unsteady stretching permeable surface withprescribed wall temperature", J. Taiwan Institute ofChemical Engineers, 41, pp. 169-177 (2010).

20. Mukhopadhyay, S. \E�ect of slip on unsteady mixedconvective ow and heat transfer past a porous stretch-ing surface in medium", Nuclear Engin. and Design.,241, pp. 2660-2665 (2011).

21. Elabashbeshy, E.M.A. and Emam, T.G. \E�ects ofthermal radiations and heat transfer over an unsteadystretching surface embedded in a porous medium inthe presence of heat source or sink", Thermal Science,15, pp. 477-485 (2011).

22. Sharma, R. \E�ect of viscous dissipation and heatsource on unsteady boundary layer ow and heattransfer past a stretching surface embedded in a porousmedium using element free Galerkin method", Appl.Maths. Comput., 219, pp. 976-987 (2012).

23. Ibrahim, W. and Shankar, B. \Unsteady MHDboundary-layer ow and heat transfer due to stretchingin the presence of heat source or sink", Comp andFluids, 70, pp. 21-28 (2012).

24. Khan, Y., Hussain, A. and Faraz, N. \Unsteady lin-ear viscoelastic uid model over stretching /shrinkingsheet in the region of stagnation point ows", ScientiaIranica, 19, pp. 1541-1549 (2012).

25. Reddy, G.V.R. and Reddy, N.B. \Mass transfer andMHD e�ects on unsteady porous stretching surfaceembedded in a porous medium with variable heat uxin the presence of heat source", J. Appl. Comp. Sci.Math., 14, pp. 49-54 (2013).

26. Liao, S.J., Beyond Perturbation, Introduction to Ho-motopy Analysis Method, Boca Raton: Chapman &Hall/CRC Press (2003).

27. Liao, S.J. \A uniformly valid analytical solution of 2Dviscous ow past a semi-in�nite at plate", J. FluidMech., 385, p. 101 (1999).

28. Abbasbandy, S. and Zakaria, F.S. \Soliton solutionsfor the �fth-order KdV equation with the homotopyanalysis method", Nonlinear Dyn., 51, p. 83 (2008).

29. Abbasbandy, S. and Parkes, E.J. \Solitary smoothsolutions for the Camassa-Holm equation by meansof homotopy analysis method", Chaos, Solitons &Fractals, 36, p. 581 (2008).

30. Abbasbandy, S. \Homotopy analysis method for radi-ation equations", Int. Com. Heat Mass Transfer, 34,p. 380 (2007).

31. Abbas, Z., Sheikh, M. and Sajid, M. \Mass transferin two MHD viscoelastic uids over a shrinking sheetin porous medium with chemical reaction species", J.Porous Media, 16, pp. 619-636 (2013).

32. Abbas, Z., Wang, Y., Hayat, T. and Oberlack, M.\Mixed convection in the stagnation point ow of aMaxwell uid towards a vertical stretching surface",Nonlinear Analysis: Real World Appl., 11, pp. 3218-3228 (2010).

33. Javed, T., Ahmed, I., Abbas, Z. and Hayat, T.\Rotating ow of a micropolar uid induced by astretching surface", Zeitschrift fur Naturforschung A,65a, pp. 829-836 (2010).

34. Abbas, Z. and Hayat, T. \Stagnation slip ow and heattransfer over a non linear stretching sheet", Numer.Mech. for PDE's, 27, pp. 302-314 (2011).

35. Abbas, Z., Hayat, T., Sajid, M. and Asgher, S.\Unsteady ow of a second grade uid �lm over an un-steady stretching sheet", Mathematical and ComputerModelling, 48, pp. 518-526 (2008).

36. Abbas, Z., Wang, Y., Hayat, T. and Oberlack, M.\Hydromagnetic ow in a viscoelastic uid due tothe oscillatory stretching surface", Int. J. NonlinearMechanic, 43, pp. 783-793 (2008).

37. Hayat, T., Abbas, Z. and Pop, I. \Momentum andheat transfer over a continuously moving surface witha parallel free stream in a viscoelastic uid", NumericalMethods for PDE's, 26, pp. 305-319 (2010).

38. Na, T.Y. Computational Methods in EngineeringBoundary Value Problems, New York (1979).

Biographies

Zaheer Abbas is an assistant professor in De-partment of Mathematics, the Islamia University ofBahawalpur, Bahawalpur. He earned his PhD inapplied mathematics from Quaid-I-Azam University,Islamabad, in 2010. His research interests includeNewtonian and non-Newtonian uids ow, uid owin porous medium, heat and mass transfer, mag-netohydrodynamics and uid dynamics of Peristaltic ows.

Noor Muhammad at present is Lecturer in Depart-ment of Mathematics, COMSATS Institute of Infor-

1366 Z. Abbas et al./Scientia Iranica, Transactions B: Mechanical Engineering 21 (2014) 1355{1366

mation Technology, Attock. He attained his M.Philin Applied Mathematics from International IslamicUniversity, Islamabad, in 2011. His research interestsincludes the study of heat and mass transfer and uiddynamics.

Ghulam Mustafa received a PhD (2004) in Mathe-matics from the University of Science and Technologyof China, P.R. China. He also received his Post

Doctorate from Durham University, UK, sponsored byAssociation of Commonwealth Universities, UK, 2011.Currently, he is Visiting Fellow of Chinese Academyof Sciences at University of Science and Technology ofChina. He is an Associate Professor in the Departmentof Mathematics, the Islamia University of Bahawalpur,Pakistan. His research interests include geometricmodelling, applied approximation theory and solutionof di�erential equations.