Mixed Convection Boundary Layers with Prescribed Temperature in the Unsteady Stagnation Point Flow toward a Stretching Vertical Sheet

Abstract

Mixed convection boundary layer caused by time-dependent velocity and the surface temperature in the two-dimensional unsteady stagnation point flow of an in-compressible viscous fluid over a stretching vertical sheet is studied. The transformed nonlinear boundary layer equations are solved numerically using the shooting technique in cooperation with Runge-Kutta-Fehlberg (RKF) method. Different step sizes are used ranging from 0.0001 to 1. Numerical results for the skin friction coefficient and local Nusselt number are presented for both assisting and opposing flows. It is found that the dual solutions exist for the opposing flow, whereas the solution is unique for the assisting flow. Important features of the flow characteristics are displayed graphically. Comparison with the existing results for the steady case show an excellent agreement.

1. Introduction

Stagnation point flow, describing the fluid motion near the stagnation region of a solid surface, exists in both cases of fixed or moving body in a fluid. Flow in incompressible viscous fluid on stretching sheet plays an important role in engineering applications such as extrusion of polymers, continuous casting, cooling of metallic plates, glass fiber production, hot rolling, paper production, wire drawing, and aerodynamic extrusion of plastic sheets [1–6]. Sakiadis [7] studied the boundary layer flow problem on a continuously moving surface with constant velocity. Crane [8] obtained the exact analytical solution for the two-dimensional stretching surface in a quiescent fluid. The effects of a variable surface temperature and linear surface stretching were examined by Grubka and Bobba [9]. Similar problem to that of Grubka and Bobba [9] for the case of power-law surface velocity and three different thermal boundary conditions were considered by Ali [10] and extended to the stretching surface subject to suction or injection [11]. Meanwhile, Ingham [12] investigated the existence of the dual solutions of the boundary layer equations of a continuously moving vertical plate with temperature inversely proportional to the distance up to the plate. Wang [13, 14] studied the behavior of liquid film on an unsteady stretching surface where the similarity transformation was employed to transform the governing partial differential equations to a nonlinear ordinary differential equation with an unsteadiness parameter. The boundary layer flow due to a stretching surface in vertical direction in steady, viscous, and incompressible fluid whereby the buoyancy forces are taken into consideration was discussed by Daskalakis [15], Lin and Chen [16], Ali and Al-Yousef [17], Chamkha [18], and Ishak et al. [19]. Vajravelu [20] studied the flow and heat transfer characteristics in a viscous fluid over a nonlinearly stretching sheet. Elbashbeshy and Bazid [21, 22] analyzed the heat transfer over an unsteady stretching surface. Tsai et al. [23] studied the nonuniform heat source or sink effect on the flow and heat transfer from an unsteady stretching sheet through a quiescent fluid numerically. Ishak and Nazar [24] extended Grubka and Bobba [9] and Ali’s [10] work to the axisymmetric laminar boundary layer flow along a continuous stretching cylinder immersed in an incompressible viscous fluid. Patil et al. [25] obtained the numerical results for two-dimensional mixed convection flow along a vertical semi-infinite power-law stretching sheet. Bachok et al. [26] adopted the RKF method to solve the free convection boundary layer flow near a continuously moving vertical plate subject to suction and injection, whereas Bachok and Ishak [27] discussed the variations of skin friction coefficient and heat transfer rate at the surface subject to the nonlinear stretching/shrinking sheet immersed in a viscous fluid. Aziz and Hashim [28] investigated the effects of viscous dissipation on flow and heat transfer in a thin liquid film on an unsteady stretching sheet using homotopy analysis method. Chamkha and Ahmed [29] presented the effects of heat generation/absorption and chemical reaction on unsteady magnetohydrodynamics flow heat and mass transfer near a stagnation point of a three-dimensional porous body in the presence of a uniform magnetic field. Suali et al. [30] studied the behavior of unsteady stagnation point flow of an incompressible viscous fluid by considering both stretching and shrinking sheets with suction and injection. Very recently, Liu and Megahed [31] analyzed the effects of internal heat generation on the flow and heat transfer on an unsteady stretching sheet in the presence of thermal radiation and variable heat flux by means of homotopy perturbation method.

In the present paper, we investigate the behavior of the mixed convection boundary layer flow with prescribed temperature in the unsteady stagnation point flow over a stretching vertical sheet. The momentum and energy equations are solved numerically, and the characteristics of the flow are analyzed.

2. Basic Equation

Consider two-dimensional incompressible viscous fluid near the unsteady stagnation point flow on a stretching vertical surface. Assume that the fluid occupies the half plane (𝑦>0) and the vertical surface located at the Cartesian coordinate (𝑥,𝑦) with origin 𝑂, and the 𝑥-axis is along the direction of the surface and the 𝑦-axis is normal to it. Two equal forces are impulsively applied along the 𝑥-axis of the vertical stretching sheet where 𝑢_𝑤(𝑥,𝑡), 𝑇_𝑤(𝑥,𝑡), 𝑇_∞(𝑥,𝑡), and 𝑢_∞(𝑥,𝑡) are denoted as the velocity of the stretching sheet, prescribed temperature, uniform temperature of the ambient fluid, and the free stream velocity to the boundary layer, respectively. Both assisting and opposing flows are taken into consideration, and their configuration is shown in Figure 1. Under this assumption together with the Boussinesq and boundary layer approximations, the governing equations can be written as follows: 𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦=0,𝜕𝑢𝜕𝑡+𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢=𝜕𝑦𝜕𝑢_∞𝜕𝑡+𝑢_∞𝜕𝑢_∞𝜕𝜕𝑥+𝜈²𝑢𝜕𝑦²±𝑔𝛽𝑇−𝑇_∞,𝜕𝑇𝜕𝑡+𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝜕𝑦=𝛼²𝑇𝜕𝑦², (1) where 𝑢 and 𝑣 are velocity components in 𝑥 and 𝑦 direction, 𝑔, 𝑇, 𝛼, 𝜈, and 𝛽 are gravity acceleration, fluid temperature, thermal diffusivity, kinematic viscosity, and thermal expansion coefficient, and the sign ± corresponds to the assisting and opposing buoyant flows, respectively, subjected to the boundary conditions Nik Long et al. [32]: 𝑣=0,𝑢=𝑢_𝑤(𝑥,𝑡),𝑇=𝑇_𝑤at𝑦=0;𝑢⟶𝑢_∞(𝑥,𝑡),𝑇⟶𝑇_∞as𝑦⟶∞ (2) and 𝑢_𝑤(𝑥,𝑡), 𝑢_∞(𝑥,𝑡), and 𝑇_𝑤 are denoted as 𝑢_𝑤(𝑥,𝑡)=𝑏𝑥1−𝜆𝑡,𝑢_∞(𝑥,𝑡)⟶𝑎𝑥,𝑇1−𝜆𝑡_𝑤=𝑇_∞+𝑐𝑥,1−𝜆𝑡 (3) where 𝑎, 𝑏, and 𝑐 are positive constants. Equation (1) describes the unsteady flow over a stretching surface.

(a) Assisting flow

(b) Opposite flow

Figure 1 

Physical model and coordinate system.

Introduce the following similarity transformations: Ψ𝑥,𝜂𝑦,𝑡=𝑎𝜈1−𝜆𝑡^1/2𝜂,𝜂=𝑎𝑥𝑓𝑦,𝑡𝜈(1−𝜆𝑡)^1/2𝜃𝜂=𝑦,𝑇−𝑇_∞𝑇_𝑤,−𝑇∞ (4) where 𝜂 is similarity variable and Ψ(𝑥,𝑦) is stream function defined as 𝑢=𝜕Ψ𝜕𝑦,𝑣=−𝜕Ψ.𝜕𝑥 (5) Substituting (4) into (5) yields 𝑢=𝑎𝑥𝑓1−𝜆𝑡^′𝜂,𝑣=−𝑎𝜈1−𝜆𝑡^1/2𝑓𝜂, (6) where the prime denotes the differentiation with respect to 𝜂. Substituting (4) into (1) leads to the following nonlinear ordinary differential equations: 𝑓^′′′𝜂+𝑓𝜂𝑓^′′𝜂−𝑓^′𝜂²+1+𝐴1−𝑓^′𝜂−12𝜂𝑓^′′𝜂±𝜆𝑄²1𝜃=0,𝜃Pr^′′𝜂+𝑓𝜂𝜃^′𝜂−𝑓^′𝜂𝜃𝜂𝜃𝜂+1−𝐴2𝜃^′𝜂𝜂𝑦,𝑡=0 (7)(8) and the boundary condition (2) becomes 𝑓(0)=0,𝑓^′(0)=𝑏𝑎,𝑓^′(∞)𝜃(0)(∞)=1,=1,𝜃=0, (9) where 𝑄=𝑏/𝑎 is the ratio of stretching velocity parameter, 𝐴=𝜆/𝑎 is the unsteadiness parameter, Pr=𝜈/𝛼 is the Prandtl number and the constant 𝜆, (𝜆≥0) is the buoyancy parameter defined as 𝜆=Gr_𝑥/Re²_𝑥, Gr_𝑥=𝑔𝛽(𝑇_𝑤−𝑇_∞)𝑥³/𝜈² is local Grashof number, and Re_𝑥=𝑢_𝑤𝑥/𝜈 is the local Reynolds number. The buoyancy parameter, 𝜆, can be written as 𝑇𝜆=𝑔𝛽_𝑤−𝑇_∞𝑥𝑢²_𝑤. (10) The solution of (7) subject to the boundary conditions (9) is given by 𝑓𝜂=𝜂, (11) where 𝜆=0 and 𝑏/𝑎=1.

The quantities of physical interest are the skin friction coefficient, 𝐶_𝑓, and the Local Nusselt number, Nu_𝑥, which are defined by 𝐶_𝑓=𝜏_𝑤𝜌𝑢²_𝑤,Nu_𝑥=𝑥𝑞_𝑤𝑘𝑇_𝑤−𝑇_∞, (12) where 𝜌 is the fluid density. The wall shear stress, 𝜏_𝑤, and the heat flux, 𝑞_𝑤, are defined as 𝜏_𝑤=𝜇𝜕𝑢𝜕𝑦_𝑦=0,𝑞_𝑤=−𝑘𝜕𝑇𝜕𝑦_𝑦=0, (13) where 𝜇 and 𝑘 are the dynamic viscosity and the thermal conductivity, respectively. The relation between 𝜇 and 𝜌 is defined as 𝜈=𝜇/𝜌. Using the similarity variables (4), we obtain 12𝐶_𝑓Re_𝑥^1/2=𝑓^′′(0),Nu_𝑥Re_𝑥^1/2=−𝜃^′(0) (14) which represent the skin friction coefficient and heat transfer rate at the surface, respectively.

3. Result and Discussion

The transformed nonlinear ordinary differential equations (7) subject to the boundary condition (9) are solved numerically using RKF method with shooting technique. The numerical results are given to carry out a parametric study showing the influence of the nondimensional parameters; that is, unsteadiness parameter, 𝐴, Prandtl number, Pr, and buoyancy parameter, 𝜆. For the validation of the numerical results, the case of 𝐴=0 (steady flow) is considered and compared with those of Ishak et al. [33]. The quantitative comparison is tabulated in Table 1, and it is found to be in excellent agreement.

Table 1 

Value of 𝑓^′′(0) and −𝜃^′(0) for 𝑏/𝑎=1 and 𝜆=1 at various Pr.

Figures 2 and 3 present the variations of skin friction coefficient, 𝑓^′′(0), and local Nusselt number, −𝜃^′(0), with respect to buoyancy parameter, 𝜆, for different values of Pr, and 𝐴=1. For 𝐴=𝜆, the variations are displayed in Figures 4 and 5. It is observed that for the assisting flow, an increase in 𝜆 would increase, 𝑓^′′(0) and −𝜃^′(0) for all cases (Figures 2 to 5), while an increase in Pr would decrease 𝑓^′′(0) (Figures 2 and 4) and increase −𝜃^′(0) (Figures 3 and 5). For the opposing flow, as 𝜆 and Pr increase the 𝑓^′′(0) decreases and −𝜃^′(0) increases for the first solution, but opposite behavior can be seen for the second solution. It is also noted that for the opposing flow, the dual solutions exist for 𝜆<𝜆_𝑐 and no solutions for 𝜆>𝜆_𝑐. For 𝜆=𝜆_𝑐, there exist a unique solution; the boundary layer separation occurs.

(a) Assisting flow

(b) Opposing flow

Figure 2 

Comparison of skin friction, 𝑓′′(0) for different Pr when 𝐴=𝑏/𝑎=1.

(a) Assisting flow

(b) Opposing flow

Figure 3 

Comparison of local Nusselt number, −𝜃′(0) for different Pr when 𝐴=𝑏/𝑎=1.

(a) Assisting flow

(b) Opposing flow

Figure 4 

Comparison of skin friction, 𝑓′′(0) for different Pr when 𝐴=𝜆.

(a) Assisting flow

(b) Opposing flow

Figure 5 

Comparison of local Nusselt number, −𝜃′(0) for different Pr when 𝐴=𝜆.

The effects of unsteadiness parameter, 𝐴 and Pr, on 𝑓^′′(0) and −𝜃^′(0) for fix 𝜆=0.5 and 𝑏/𝑎=1.25 are shown in Figures 6 and 7. For the assisting flow, as 𝐴 and Pr increase, the 𝑓^′′(0) decreases and −𝜃^′(0) increases. While for the opposing flow, as 𝐴 increases the 𝑓^′′(0) and −𝜃^′(0) decrease for the first solution, and for the second solution, 𝑓^′′(0) and −𝜃^′(0) decrease until they archive their minimum values, 𝑓^′′_min(0) and −𝜃^′_min(0) and then they increase until 𝐴=𝐴_𝑐. For 𝐴>𝐴_𝑐, there exist no solution. An increase in Pr would increase 𝑓^′′(0) but only give little effect on −𝜃^′(0) for the first solution, and 𝑓^′′(0) and −𝜃^′(0) decrease for the second solution.

(a) Assisting Flow

(b) Opposing Flow

Figure 6 

Comparison of skin friction, 𝑓′′(0) for different Pr when 𝜆=0.5 and 𝑓′(0)=1.25.

(a) Assisting flow

(b) Opposing flow

Figure 7 

Comparison of local Nusselt number, −𝜃′(0) for different Pr when 𝜆=0.5 and −𝜃(0)=1.25.

Figures 8 and 9 exhibit the variations of the skin friction coefficient, 𝑓^′′(0) and local Nusselt number, −𝜃^′(0) for some value of 𝑏/𝑎. As 𝐴 and 𝑏/𝑎 increase, the 𝑓^′′(0) decreases and −𝜃^′(0) increases for the assisting flow. While for the opposing flow as 𝐴 increases 𝑓^′′(0) and −𝜃^′(0) decreases for the first solution and 𝑓^′′(0) increases and −𝜃^′(0) decreases for the second solution. For the increasing values of 𝑏/𝑎, 𝑓^′′(0) decreases and −𝜃^′(0) increases for both first and second solutions.

(a) Assisting flow

(b) Opposing flow

Figure 8 

Comparison of Skin friction, 𝑓′′(0) for different 𝑏/𝑎 when Pr=0.72 and 𝜆=1.

(a) Assisting flow

(b) Opposing flow

Figure 9 

Comparison of local Nusselt number, −𝜃′(0) for different 𝑏/𝑎 when Pr=0.72 and 𝜆=1.

The velocity and temperature profile for different values of 𝐴 and 𝜆 when Pr=0.72 and 𝑏/𝑎=1 are displayed in Figures 10 and 11. The existance of the dual solutions for the opposing flow are clearly displayed. It is seen that for the assisting flow, the velocity profile increases to its maximum value, 𝑓^′_max(𝜂), then decreases monotonically to a constant value. For the opposing flow, the velocity profile shows the opposite trend. The temperature profile decreases monotonically with 𝜂 and becomes zero outside the boundary layer which satisfies the condition at infinity, 𝜃(∞)=0. These properties support the validation of the present results. It is also noticed that an increase in values of 𝐴 and 𝜆 will increase the heat transfer rate at the surface. For the second solution, the temperature profile increases to attain its maximum value 𝜃_max(𝜂), then decreases gradually to a constant.

(a) Assisting flow

(b) Opposing flow

Figure 10 

Velocity profile, 𝑓′(𝜂) for different 𝐴=𝜆 when Pr=0.72 and 𝑏/𝑎=1.

(a) Assisting flow

(b) Opposing flow

Figure 11 

Temperature profile, 𝑓′(𝜂) for different 𝐴=𝜆 when Pr=0.72 and 𝑏/𝑎=1.

4. Conclusion

In this study, the numerical solutions for both assisting and opposing flows of the mixed convection boundary layer caused by time-dependent velocity and surface temperature in two-dimensional unsteady stagnation point flow over a stretching vertical sheet are obtained. The similar transformation is advocated to reduce the governing partial differential equations into nonlinear ordinary differential equations which are solved numerically using RKF method via the shooting technique. Comparison with the existing results showed the excellent agreement. Dual solutions are found to exist for the opposing flow whereas the solution is unique for the assisting flow. The boundary layer separation occurs at 𝜆=𝜆_𝑐 for the opposing flow. The effect of buoyancy parameter, 𝜆, the unsteadiness parameter, 𝐴, and Prandtl number, Pr, of the fluid has been displayed graphically and discussed in details.

Acknowledgment

This project is supported by the University Putra Malaysia for the Research University Grant scheme, Project no. 05-02-12-1834RU.