International Journal of Heat and Mass Transfer 54 (2011) 302–307

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Effects of suction/blowing on steady boundary layer stagnation-point flow andheat transfer towards a shrinking sheet with thermal radiation

Krishnendu Bhattacharyya ⇑, G.C. LayekDepartment of Mathematics, The University of Burdwan, Burdwan 713104, West Bengal, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 April 2010Received in revised form 4 September 2010Accepted 10 September 2010Available online 15 October 2010

Keywords:Boundary layer flowStagnation-pointShrinking sheetSuction/blowingHeat transferThermal radiation

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.09.043

⇑ Corresponding author. Tel.: +91 9474634200; faxE-mail addresses: krish.math@yahoo.com, krish

Bhattacharyya).

In this paper, the effects of suction/blowing and thermal radiation on steady boundary layer stagnation-point flow and heat transfer over a porous shrinking sheet are investigated. The existence of dualsolutions, unique solution and non-existence of solution for self-similar equations of the flow and heattransfer are analyzed numerically. It is noted that the range of velocity ratio parameter where the solu-tion exists increases/decreases with increasing suction/blowing. With increasing suction, temperature atthe wall is found to increase (decrease) for the first (second) solution. Due to increasing Prandtl numberand thermal radiation parameter the thermal boundary layer thickness becomes thinner.

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1. Introduction a situation appears, where some other external force is needed,

The stagnation-point flow over a stretching plate is an impor-tant problem in fluid mechanics. The steady flow in the neighbour-hood of a stagnation-point was first studied by Hiemenz [1]. Chiam[2] investigated the flow around the stagnation-point over astretching sheet and concluded that when the stretching velocityof the plate is equal to the stagnation flow velocity in the inviscidfree stream, no boundary layer is formed near the plate. Mahapatraand Gupta [3] re-investigated the stagnation-point flow problemtowards a stretching plate taking different stretching and stagna-tion flow velocities. They found that two different types of bound-ary layer are formed near the stretching plate depending on theratio of the stretching and stagnation flow velocities. Layek et al.[4] studied mass and heat transfers for boundary layer stagna-tion-point flow towards a stretching sheet subjected to heatabsorption/generation and suction/blowing. Nadeem et al. [5] ob-tained solutions for boundary layer flow in the region of the stag-nation-point towards a stretching sheet using homotopy analysismethod (HAM).

Recently, the boundary layer flow near a shrinking sheet is gi-ven significant attention due to its various engineering applica-tions. Normally, the steady flow over a shrinking sheet is notoccurred. Because the vorticity generated due to the shrinkingsheet is not confined inside the boundary layer and consequently

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: +91 342 2530452.nendu.math@gmail.com (K.

which helps to confine the vorticity inside the boundary layerand only then the steady flow is possible. In confining the vorticity,the most suitable external force is the suction at the sheet. Theflow development around the shrinking sheet was demonstratedby Wang [6]. The existence and uniqueness of steady viscous flowdue to a shrinking sheet was established by Miklavcic and Wang[7] considering the suction effect and they concluded that for somespecific values of suction, dual solutions exist and also in somerange of suction, no boundary layer solution is possible. But forstagnation-point flow over a shrinking sheet, the vorticity gener-ated due to shrinking sheet remains confined within the boundarylayer by the stagnation flow velocity in some cases and in thisregard, Wang [8] investigated the stagnation flow problem to-wards a shrinking sheet and obtained dual solutions for some val-ues of the ratio of shrinking and stagnation flow rates. After that,Hayat et al. [9] gave an analytic solution of magnetohydrodynamic(MHD) rotating flow of a second grade fluid over a shrinking sur-face using HAM. A paper was published by Hayat et al. [10] tostudy the MHD flow and mass transfer of an upper-convected Max-well fluid past a porous shrinking sheet with chemical reactionspecies. A series solution of three-dimensional MHD and rotatingflow over a porous shrinking sheet was obtained by Hayat et al.[11] using HAM. Fang [12] studied the boundary layer flow overa continuously shrinking sheet with a power-law surface velocityand with mass transfer. Fang and Zhang [13] obtained a closed-form analytical solution for MHD viscous flow over a shrinkingsheet subjected to applied suction through the porous sheet. Theunsteady viscous flow over a continuously shrinking sheet with

Nomenclature

a straining rate parameterc stretching/shrinking rate parameterc/a velocity ratio parametercp specific heatf dimensionless stream functionf0 dimensionless velocityk1 absorption coefficientPr Prandtl numberqr radiative heat fluxR thermal radiation parameterS suction/blowing parameterT temperatureT1 free stream temperature

Tw temperature of the sheetu, v velocity componentsU stagnation flow velocityvw suction/blowing velocityg similarity variablej fluid thermal conductivityl coefficient of fluid viscosityt kinematic fluid viscosityq fluid densityw stream functionr Stefan–Boltzmann constanth dimensionless temperature

K. Bhattacharyya, G.C. Layek / International Journal of Heat and Mass Transfer 54 (2011) 302–307 303

mass suction was investigated by Fang et al. [14]. Noor et al. [15]found a series solution for MHD viscous flow due to a shrinkingsheet by Adomian decomposition method (ADM). An analytic solu-tion was developed for shrinking flow in a rotating frame of refer-ence by Hayat et al. [16]. The exact analytic solution of thermalboundary layer over a shrinking sheet with mass transfer was ob-tained by Fang and Zhang [17]. Very recently, Fang et al. [18] studiedthe viscous flow over a shrinking sheet taking a second order slip flowmodel and solved it analytically and Ishak et al. [19] extended theWang’s [8] problem for two-dimensional flow taking micropolar fluid.

A new dimension is added to the boundary layer flow and heattransfer by considering the thermal radiation effects. In the year1962, Viskanta and Grosh [20] investigated the effects of thermalradiation on boundary layer flow and the heat transfer over awedge in an absorbing and emitting media. Ali et al. [21] consid-ered the interaction of natural convection and radiation in bound-ary layer flow over horizontal surface. Elbashbeshy [22] discussedthe effect of thermal radiation on heat transfer over a stretchingsurface. Ouaf [23] obtained an exact solution of steady flow of anelectrically conducting fluid past a porous stretching sheet in pres-ence of radiation. Hayat et al. [24] described the influence of ther-mal radiation on two-dimensional Blasius flow of a second gradefluid and heat transfer. The radiation effects on the flow and heattransfer of a second grade fluid were discussed by Hayat et al.[25,26]. The radiation effects on the unsteady flow and heat trans-fer over a stretching sheet were investigated by El-Aziz [27],Mukhopadhyay [28], Hayat et al. [29] and Hayat and Qasim [30].

In this paper, an investigation of the effects of suction/blowingon the stagnation-point flow and heat transfer over a shrinkingsheet with thermal radiation is performed. The self-similar equa-tions are obtained by applying similarity transformations and arethen solved using shooting method. The results reveal that dualsolutions exist for different values of suction and blowing. The ef-fects of suction/blowing and the thermal radiation are demon-strated through their graphical representations and discussed atlength.

2. Analysis of the flow problem and heat transfer

Let us consider a steady laminar boundary layer flow in theneighbourhood of a stagnation-point of an incompressible Newto-nian fluid over a shrinking sheet. The equation of motion for steadytwo-dimensional boundary layer flow and the equations repre-senting temperature with heat radiation may be written in usualnotation as below:

@u@xþ @v@y¼ 0; ð1Þ

u@u@xþ v @u

@y¼ U

dUdxþ t

@2u@y2 ð2Þ

and u@T@xþ v @T

@y¼ j

qcp

@2T@y2 �

1qcp

@qr

@y; ð3Þ

where u and v are velocity components in x- and y-directions,respectively, t (=l/q) the kinematic fluid viscosity, q the fluid den-sity, l the coefficient of fluid viscosity, U(x) the stagnation flowvelocity in the inviscid free stream, T the temperature, j the ther-mal conductivity of the fluid, cp the specific heat, qr is the radiativeheat flux. Using Rosseland’s approximation for radiation [31], weobtain qr = �(4r/3k1)oT4/oy, where r is the Stefan–Boltzmann con-stant, k1 is the absorption coefficient. We presume that the temper-ature variation within the flow is such that T4 may be expanded in aTaylor’s series. Expanding T4 about T1 and neglecting higher orderterms we get, T4 ¼ 4T3

1T � 3T41.

Now Eq. (3) reduces to

u@T@xþ v @T

@y¼ j

qcp

@2T@y2 þ

16rT31

3k1qcp

@2T@y2 ð4Þ

The appropriate boundary conditions for the velocity componentsand temperature are given by

u ¼ cx; v ¼ vw at y ¼ 0; u! UðxÞ ¼ ax as y!1: ð5Þ

and T ¼ Tw at y ¼ 0; T ! T1 as y!1 ð6Þ

where c and a (>0) are the shrinking/stretching rate of the sheet andstagnation flow rate parameters, respectively, with c < 0 for shrink-ing, c > 0 for stretching. Tw is temperature of the sheet and T1 is thefree stream temperature, both are assumed to be constant withTw > T1. Here vw is a prescribed suction velocity (vw < 0) or blowingvelocity (vw > 0).

3. Method of solution

To find out the solution we first introduce the stream functionw(x, y) as

u ¼ @w@y

and v ¼ � @w@x

: ð7Þ

The continuity equation (1) is satisfied automatically and themomentum equation (2) and the temperature equation (4) takethe following forms using (7)

@w@y

@2w@x@y

� @w@x

@2w@y2 ¼ U

dUdxþ t

@3w@y3 ; ð8Þ

304 K. Bhattacharyya, G.C. Layek / International Journal of Heat and Mass Transfer 54 (2011) 302–307

@w@y

@T@x� @w@x

@T@y¼ j

qcp

@2T@y2 þ

16rT31

3k1qcp

@2T@y2 : ð9Þ

The boundary conditions (5) for the flow reduce to

@w@y¼ cx;

@w@x¼ �vw at y ¼ 0;

@w@y! ax as y!1: ð10Þ

Fig. 1. Skin friction coefficient f00(0) for several values of S (suction).

Fig. 2. Skin friction coefficient f00(0) for several values of S (blowing).

3.1. Similarity analysis

We now introduce the dimensionless variables for w and T asgiven below

w ¼ffiffiffiffiffiffi

atp

xf ðgÞ and T ¼ T1 þ ðTw � T1ÞhðgÞ: ð11Þ

The similarity variable is denoted by g and g = y(a/t)1/2.Using (11) we obtain following self-similar equations as

f 000 þ ff 00 � f 02 þ 1 ¼ 0 ð12Þ

and ð3Rþ 4Þh00 þ 3RPr f h0 ¼ 0; ð13Þ

where Pr (=lcp/j) and R ð¼ jk1=4rT31Þ are the Prandtl number and

the thermal radiation parameter, respectively.The boundary conditions (10) and (6) also reduce to

f ðgÞ ¼ S; f 0ðgÞ ¼ c=a at g ¼ 0; f 0ðgÞ ! 1 as g!1; ð14Þ

and hðgÞ ¼ 1 at g ¼ 0; hðgÞ ! 0 as g!1; ð15Þ

where S = �vw/(at)1/2, S > 0 (i.e. vw < 0) corresponds to suction andS < 0 (i.e. vw > 0) corresponds to blowing.

3.2. Numerical method for solution

The nonlinear differential equations (12) and (13) along withthe boundary conditions (14) and (15) are solved using shootingtechnique, by converting it into an initial value problem (IVP).The shoot values are selected using secant method so that theboundary conditions are matched at a suitable finite value ofg ?1, say g*. We set the following first-order system

f 0 ¼ z; z0 ¼ p; p0 ¼ z2 � fp� 1 ð16Þ

and h0 ¼ q; q0 ¼ �3RPrfq=ð3Rþ 4Þ ð17Þ

with the boundary conditions

f ð0Þ ¼ S; zð0Þ ¼ c=a; hð0Þ ¼ 1: ð18Þ

To solve (16) and (17) as an initial value problem (IVP) we mustneed values for p(0), i.e. f00(0) and q(0), i.e. h0(0) but no such valuesare given. The initial guess values for f00(0) and h0(0) are chosenand fourth order Runge–Kutta method is applied to obtain the solu-tion. We compare the calculated values of f0(g) and h(g) at g = g*(=25) with the given boundary conditions f0(g*) = 1 and h(g*) = 0and adjust the values f00(0) and h0(0) to give better approximationfor the solution. The step-size is taken as h = 0.01. The process is re-peated until we get the results correct up to the desired accuracy of10�5 level.

4. Results and discussion

The numerical computations have been carried out for variousvalues of the parameters involved in the system viz., velocity ratioparameter (c/a) and suction/blowing parameter (S), Prandtl num-ber (Pr), radiation parameter (R). For illustration of the results,computed values are plotted and the physical explanations aregiven.

Wang [8] obtained some interesting results in case of solution ofstagnation-point flow over a non-porous shrinking sheet. That

study revealed that there are dual solutions when the parameterc/a satisfies the inequality �1.2465 6 c/a 6 �1, no solution existswhen c/a < �1.2465 and unique solution is possible whenc/a > �1. In the present study the boundary layer stagnation-pointflow of a Newtonian fluid over a porous shrinking sheet is investi-gated in presence of suction/blowing. The analysis shows that theexistence of solutions depends on the suction/blowing parameterS in addition to the velocity ratio parameter c/a. The range of c/a,where solution exists is increased when suction increases and it de-creases when blowing increases. The dual solutions of the boundarylayer flow over a shrinking sheet are obtained when c/a satisfies theinequality e 6 c/a 6 �1, where e � �1.34552, �1.52225, �1.89256for S = 0.2, 0.5, 1, respectively. It is interesting to note that in caseof blowing through the porous sheet the dual solutions also existfor the range e 6 c/a 6 �1 where e � �1.16301, �1.06753 forS = �0.2, �0.5, respectively. But, it is also found that there existsonly one solution for all values of c/a > �1 and for all S, which en-sures the uniqueness of the solution. Furthermore, for a largeshrinking rate i.e., c/a < e, the self-similar boundary layer solutiondoes not exist. The graphs of skin friction coefficient f00(0) for differ-ent values of S against c/a are shown in Figs. 1 and 2. Table 1 pre-sents a comparison between the values of f00(0) obtained in thepresent study and that was obtained by Wang [8] and it is foundin excellent agreement. It is worth noting that for increase of blow-ing, the length of the interval [e, �1] where the dual solutions exist,becomes smaller and smaller and it vanishes above a critical valueof blowing parameter, S � �0.9696. Accordingly, there is no bound-ary layer solution when the shrinking velocity is greater than thestagnation flow velocity, i.e. c/a < �1, which can be observed from

Table 1Comparison of the values for f00(0) when S = 0.

c/a Present study Wang [8]

First solution Second solution First solution Second solution

�0.25 1.40224051 1.40224�0.50 1.49566972 1.49567�0.625 1.50715673�0.75 1.48929811 1.48930�1.00 1.32881689 0 1.32882 0�1.15 1.08223164 0.11670230 1.08223 0.116702�1.20 0.93247277 0.23364910�1.2465 0.58429146 0.55428565 0.55430

Fig. 5. Dual velocity profiles f0(g) for several values of S (blowing).

K. Bhattacharyya, G.C. Layek / International Journal of Heat and Mass Transfer 54 (2011) 302–307 305

the curve for S = �1 in Fig. 2. The above mentioned effects of suctionand blowing are compatible with the physics of the flow. Due toincrease in suction at the sheet, the generated vorticity remainsconfined for greater value of magnitude of c/a with c/a < �1, i.e. incase of larger shrinking rates than the stagnation flow rates andhence the range of existence of solution increases. But, the blowingacts in the opposite manner and hence the above range decreases.

The boundary layer velocity profiles for several values of c/a andS are plotted in Figs. 3–5. In case of dual solutions it is noted thatthe boundary layer thickness for second solution is larger than thatof the first solution. The first solution is physically stable becausethe second solution is coincided with the first solution whenc/a > �1. Also it is noticed from those figures that the value ofthe fluid velocity increases with increasing c/a for the first solutionand for the second solution, though initial behaviour is same as

Fig. 3. Dual velocity profiles f0(g) for several values of c/a.

Fig. 4. Dual velocity profiles f0(g) for several values of S (suction).

that of first solution, for slightly large values of g the behaviouris totally opposite, i.e. for increasing values of c/a the velocity de-creases. Further, for the first solution, with the increasing valuesof suction the fluid velocity increases and the velocity decreaseswith increasing suction in the second solution and the reversebehaviours are observed for both solutions in case of blowing.The graphs for f(g) are plotted in Figs. 6 and 7 for various valuesof c/a and S. In all cases, it is noted that the function f(g) initiallydecreases and for larger values of g there is a turn around and itbecomes increasing type, i.e. the region of reverse flow appears.This fact is obvious because the shrinking and stagnation flowvelocities are in opposite directions.

Fig. 6. Variation of f(g) for several values of c/a.

Fig. 7. Variation of f(g) for several values of S (suction).

Fig. 8. Temperature gradient at the sheet �h0(0) for several values of S (suction).

Fig. 9. Temperature gradient at the sheet – h0(0) for several values of Pr.

Fig. 10. Temperature gradient at the sheet – h0(0) for several values of R.

Fig. 11. Dual temperature profiles h(g) for several values of c/a.

Fig. 12. Dual temperature profiles h(g) for several values of S (suction).

Fig. 13. Dual temperature profiles h(g) for several values of S (blowing).

306 K. Bhattacharyya, G.C. Layek / International Journal of Heat and Mass Transfer 54 (2011) 302–307

The values of temperature gradient at the sheet �h0(0) whichare proportional to the rate of heat transfer from the sheet fordifferent values of suction parameter, Prandtl number and radi-ation parameter have shown in Figs. 8–10 against c/a, which ex-hibit the dual solutions of temperature distribution. Thetemperature profiles for different values of c/a, S, Pr and R aredepicted in Figs. 11–15. It is observed from the figures thatthe first solution for temperature distribution decreases withincreasing c/a and the second solution increases with increasingof c/a. Furthermore, for the increment of suction, temperature atfixed g decreases in first solution and increases in second

solution; whereas the effects of blowing is entirely oppositefor both solutions. From sketched figures it is understood thatfor the increment of Pr and R, the temperature boundary layerthickness becomes thinner and thinner for first solution as wellas second solution. Effect of suction on heat transfer is presentedin Fig. 8. From Fig. 10, it is seen that the value of �h0(0) in-creases with increasing thermal radiation parameter for the firstsolution and for the second solution, that value also increaseswith radiation parameter except a region closed to c/a = �1.The effect of the Prandtl number on the heat transfer is similaras that of the radiation parameter.

Fig. 14. Dual temperature profiles h(g) for several values of Pr.

Fig. 15. Dual temperature profiles h(g) for several values of R.

K. Bhattacharyya, G.C. Layek / International Journal of Heat and Mass Transfer 54 (2011) 302–307 307

5. Conclusions

The stagnation-point steady boundary layer flow over a porousshrinking sheet subjected to applied suction or blowing and theheat transfer with consideration of radiation effects are investi-gated. With the help of similarity transformations, the governingequations are reduced to self-similar nonlinear ordinary differen-tial equations which are then solved numerically using shootingmethod. The applied suction or blowing through the porous sheetaffects the existence of boundary layer solution as well as unique-ness of solution. For increase of suction value, the length of intervalof ratio of shrinking and stagnation flow rates, where dual solu-tions exist increases. But, the opposite effects in case of blowingare noted. The heat transfer from the sheet increases with increas-ing suction for the first solution. Due to the consideration of ther-mal radiation parameter the rate of heat transfer for first solutionincreases. The same behaviour is noticed for the variations of Pra-ndtl number.

Acknowledgements

The authors are very much thankful to the honourable review-ers for their constructive suggestions to improve the quality of thepaper. One of the authors (KB) gratefully acknowledges the finan-

cial support of National Board for Higher Mathematics (NBHM),DAE, Mumbai, India for pursuing this work.

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