Stagnation-Point Flow toward a Vertical, Nonlinearly Stretching Sheet with Prescribed Surface Heat Flux

An analysis is carried out to study the steady two-dimensional stagnation-point flow of an incompressible viscous fluid towards a stretching vertical sheet. It is assumed that the sheet is stretched nonlinearly, with prescribed surface heat flux. This problem is governed by three parameters: buoyancy, velocity exponent, and velocity ratio. Both assisting and opposing buoyant flows are considered.The governing partial differential equations are transformed into a system of ordinary differential equations and solved numerically by finite difference Keller-box method. The flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Dual solutions are found in the opposing buoyant flows, while the solution is unique for the assisting buoyant flows.


Introduction
The study of fluid flow and heat transfer due to a stretching surface has significant application in the industrial processes, for example, in polymer sheet extrusion from a die, drawing of plastic films, and manufacturing of glass fiber.The quality of the final product greatly depends on the heat transfer rate at the stretching surface as explained by Karwe and Jaluria [1,2].
Different from the flow induced by a stretching horizontal plate (see Crane [3], Weidman and Magyari [4], and Weidman and Ali [5]), the effect of the buoyancy force could not be neglected for the vertical plate.There are several works that reported the flow and heat transfer characteristics that are brought about by the buoyancy force [6][7][8][9][10][11]. Ramachandran et al. [12] studied the effect of buoyancy force on the stagnation point flows past a vertically heated surface at rest and found that dual solutions exist in the buoyancy opposing flow region.In the present paper, in addition to the flow under the influence of buoyancy force as discussed by Ramachandran et al. [12], we discuss the consequent flow and heat transfer characteristics that are also brought about by the stretching sheet with power-law velocity variation.It is worth mentioning that the problems of the stagnation-point flow toward a stretching sheet have been considered by many authors [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], by considering various flow configurations as well as surface heating conditions.

Problem Formulation
Consider a mixed convection stagnation-point flow towards a vertical nonlinearly stretching sheet immersed in an incompressible viscous fluid, as shown in Figure 1.The Cartesian coordinates (, ) are taken such that the -axis is measured along the sheet oriented in the upwards or downwards direction and the -axis is normal to it.It is assumed that the wall stretching velocity is given by   =   and the far field inviscid velocity distribution in the neighborhood of the stagnation point (0, 0) is given by  ∞ () =   ,  ∞ () = −  .The surface heat flux is in the form of   () =  (5−3)/2 (see Merkin and Mahmood [28]), where , , , and  are constants.This   () ensured that the   buoyancy parameter is independent of .For the assisting flow, as shown in Figure 1(a), the -axis points upwards in the same direction of the stretching surface such that the external flow and the stretching surface induce flow and heat transfer in the velocity and thermal boundary layers, respectively.On the other hand, for the opposing flow, as shown in Figure 1(b), the -axis points vertically downwards in the same direction of the stretching surface such that the external flow and the stretching surface also induce flow and heat transfer, respectively, in the velocity and thermal boundary layers.The steady boundary layer equations, with Boussinesq approximation, are subject to the boundary conditions where  and  are the velocity components along the and -axes, respectively,  is the acceleration due to gravity,  is the thermal diffusivity of the fluid,  is the kinematic viscosity,  is the coefficient of thermal expansion,  is the fluid density, and  ∞ is the far field ambient constant temperature.The continuity equation ( 1) can be satisfied automatically by introducing a stream function  such that  = / and  = −/.The momentum and energy equations are transformed by the similarity variables into the following nonlinear ordinary differential equations: The transformed boundary conditions are where  = /.Here primes denote differentiation with respect to ,  =   /Re 5/2  is the buoyancy or mixed convection parameter, Pr = / is the Prandtl number,   =    4 /( 2 ) is the local Grashof number, and Re  =  ∞ / is the local Reynolds number.We note that  is a constant, with  > 0 corresponds to the assisting flow and  < 0 denotes the opposing flow whilst  = 0 is for forced convective flow.For the forced convection flow ( = 0), the corresponding temperature problem possesses a larger similarity-solution domain than the mixed convection ( ̸ = 0) one.The reason is that in the forced convection case, the existence of similarity solutions does not require the restriction of the applied wall heat flux to the special form   () =  (5−3)/2 .Namely, in the forced convection case, the much weaker assumption   () =   suffices for the similarity reduction of the problem, where the flux exponent  does not depend on the velocity exponent  in any way.
The main physical quantities of interest are the values of   (0), being a measure of the skin friction, and the nondimensional wall temperature (0).Our main aim is to find how the values of   (0) and (0) vary in terms of the parameters  and .

Results and Discussion
Equations ( 6) subject to the boundary conditions (7) are integrated numerically using a finite difference scheme known as the Keller box method [30].Numerical results are presented for different physical parameters.To conserve space, we consider the Prandtl number as unity throughout this paper.The results presented here are comparable very well with those of Ramanchandran et al. [12].For no buoyancy effects  = 0 and  = 1, comparison of the values of   (0) was made with those of Wang [29] as presented Table 1, which shows a favourable agreement.
Figures 2 and 3 show the skin friction coefficient   (0) against buoyancy parameter  for some values of velocity exponent parameter  when velocity ratio parameter is  = 0.5 and  = 1.Two branches of solutions are found.The solid lines are the upper branch solutions and the dash lines are the lower branch solutions.With increasing , the range of  for which the solution exists increases.Also from both figures of the upper branch solutions, the skin friction is higher for the assisting flow ( > 0) compared to the opposing flow ( < 0).This implies that increasing the buoyancy parameter  increases the skin friction coefficient   (0) whilst for  = 1, the values of   (0) as shown in Figure 3 are positive for  > 0 and negatives for  < 0. Physically, this means that positive   (0) implies the fluid exerts a drag force on the sheet and negative implies the reverse.Similarly this also happens for  = 0.5 but at different values of .
As seen in Figures 2 and 3, there exists a critical value of velocity ratio   such that for  <   there will be no solutions, for   <  < 0 there will be dual solutions, and when  > 0, the solution is unique.Our numerical computations presented in Figure 2 show that for the velocity ratio  = 0.5,   = −8.331,−2.677, and −0.7411 for  = 2, 1, and 0.5, respectively.On the other hand, for the velocity ratio  = 1 shown in Figure 3,   = −14.98,−4.764, and −1.301 for  = 2, 1, and 0.5, respectively.The dual solutions exhibit the normal forward flow behavior and also the reverse flow where   () < 0. From these two results, it seems that an increase in velocity ratio parameter  leads to an increase of the critical values of |  |.This increases the dual solutions range of ( 6)- (7).
Figures 4 and 5 display the variations of the wall temperature (0) against the buoyancy parameter , for some values of  when  = 0.5 and  = 1, respectively.Both figures clearly show that the wall temperature increases as  increases for the upper branch solutions.For the lower branch solutions, the wall temperature becomes unbounded as  → 0 − .The velocity and temperature profiles for  = 0.5 when  = −0.5 are presented in Figures 6 and 7, respectively.Figure 6 shows that the velocity increases as  increases for the upper branch solutions, while opposite trend is observed for the lower branch solutions.In Figure 7, for the upper branch solutions, it is seen that an increase in  tends to decrease the temperature.Besides that, the temperature is higher for the lower branch solution than the upper branch solution at all points, near and away from the solid surface.It can be seen from Figures 6 and 7 that all profiles approach the far field boundary conditions (7) asymptotically, thus supporting the numerical results obtained.Finally, the typical streamlines for  = 1,  = 1, and  = −2 for both solution branches are shown in Figures 8 and 9.

Conclusions
The problem of mixed convection stagnation-point flow towards a nonlinearly stretching vertical sheet immersed in an incompressible viscous fluid was investigated numerically.The effects of the velocity exponent parameter , buoyancy parameter , and velocity ratio parameter  on the fluid flow and heat transfer characteristics were discussed.It was found that for the assisting flow, the solution is unique, while dual solutions were found to exist for the opposing flow up to a certain critical value   .Moreover, increasing the velocity exponent parameter  is to increase the range of the buoyancy parameter  for which the solution exists.

Figure 1 :
Figure 1: Physical model and coordinate system.

Figure 2 :
Figure 2: Variation of the skin friction coefficient   (0) with  for various values of  when  = 0.5.

Figure 3 :
Figure 3: Variation of the skin friction coefficient   (0) with  for various values of  when  = 1.

Nomenclature
, , : Constants :Dimensionless stream function :Acceleration due to gravity   : Local Grashof number : Thermalconductivity : Velocity exponent parameter Pr: Prandtl number   : Surface heat flux Re  : Local Reynolds number : Fluidtemperature  ∞ : Ambient temperature , : Velocity components along the and -directions, respectively  ∞ : Free stream velocity   : Stretching velocity , : Cartesian coordinates along the surface and normal to it, respectively.Greek Letters : Thermal diffusivity : Thermal expansion coefficient : Velocity ratio parameter : Similarity variable : Dimensionless temperature : Buoyancy parameter : Kinematic viscosity : Fluid density Ψ: Stream function.Subscripts : Condition at the solid surface ∞: Condition far away from the solid surface.Superscript  : Differentiation with respect to