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Lie Group Analysis and Similarity Solutions for Mixed Convection Boundary Layers in the Stagnation-Point Flow toward a Stretching Vertical Sheet
An analysis for the mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet is carried out via symmetry analysis. By employing Lie group method to the given system of nonlinear partial differential equations, we can obtain information about the invariants and symmetries of these equations. This information can be used to determine the similarity variables that will reduce the number of independent variables in the system. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using fifth-order Improved Runge-Kutta Method (IRK5) coupled with shooting method. The features of the flow and heat transfer characteristics are analyzed and discussed in detail. Both cases of assisting and opposing flows are considered. This paper' results in comparison with known results are excellent.
Over the course of the past several decades, there has been a considerable amount of effort to investigate the process of flow and heat transfer of a viscous and incompressible fluid over a continuously moving surface through a quiescent fluid. The proliferation of research on this particular phenomenon has been sparked by its vast array of pragmatic applications to a myriad of manufacturing processes. Examples of such processes include the extrusion of polymers, continuous casting, cooling of metallic plates, glass fiber production, hot rolling, paper production, wire drawing, aerodynamic extrusion of plastic sheets, crystal glowing, and others. The significance of studying heat transfer and flow field is that it is essential in determining the degree of quality of the end results of processes such as the ones explicated by Karwe and Jaluria . Sakiadis  was the first to explore the flow induced by a semi-infinite horizontally moving wall in an ambient fluid. Crane  subsequently examined the flow over a linearly stretching sheet in an ambient fluid and came up with a solution which bore its likeness in closed analytical form for the steady two-dimensional problem. Numerous authors, such as Carragher and Crane , Elbashbeshy and Bazid , P. S. Gupta and A. S. Gupta , Magyari and Keller [7, 8], Magyari et al. , Liao and Pop , and Nazar et al. , looked into this problem by considering its various facts, such as uniform heat flux, permeability of the surface, flow, and heat transfer unsteadiness. Pop , Andersson , Takhar and Nath , and Nazar et al.  have directed their attention to other physical characteristics such as magnetic field, fluid viscoelasticity, suction, and three-dimensional flow. In contrast, the stretching vertical plate has suffered a paucity of research. The problems dealt with by Chen [16, 17], Lin and Chen , Ali and Al-Yousef [19, 20], Ali [21, 22], and Abo-Eldahab  shall be addressed in this class. Of additional noteworthy interest is the unsteady boundary layer flow and heat transfer over a stretching vertical sheet, a phenomenon that has recently been treated by Ishak et al.  in a paper. As of late, Mahapatra and Gupta [25, 26] carried out research on the heat transfer in the steady two-dimensional stagnation point flow of a viscous and incompressible Newtonian and viscoelastic fluids over a horizontal stretching sheet considering the case of constant surface temperature.
2. Mathematical Formulation of the Heat Transfer in Steady Laminar Flow over a Moving Surface
Consider the steady, two-dimensional flow of a viscous and incompressible fluid near the stagnation point on a stretching vertical surface placed in the plane of a Cartesian system of coordinates () with the -axis along the sheet as shown in Figure 1. The fluid occupies the half plane (). It is assumed that the velocity and the temperature of the stretching sheet is proportional to the distance from the stagnation-point, where with being the uniform temperature of the ambient fluid. The velocity of the flow external to the boundary layer is . Under these assumptions along with the Boussinesq and boundary layer approximations, the system of equations, which model the boundary layer flow are given by where and are the velocity components along - and -axes, respectively, is the fluid temperature, is the gravity acceleration, , , and are the thermal diffusivity, kinematic viscosity, and thermal expansion coefficient, respectively, and the “+” and “−” signs in (2) correspond to assisting buoyant flow and to opposing buoyant flow, respectively. We shall assume that the boundary conditions of (1)–(3) are where , , and are positive constants. The continuity equation can be satisfied by introducing a stream function , such that
The boundary conditions (4) will be as
3. Main Results
3.1. Solution of the Problem by the Lie Point Symmetries
At first, we derive the similarity solutions using Lie group method under which (6) and the boundary conditions (7) are invariant, and then we use these symmetries to determine similarity variables. Now, consider the one-parameter , Lie group infinitesimal transformation in given by here is the group parameter, and is vector filed. A system of partial differential equations (6) is said to admit a symmetry generated by the vector filed as Equivalently, we can obtain by solving subject to initial conditions, . If is left invariant by the transformation , then the solutions , , and are invariant under the symmetry (9) if Assume The vector as (9) is a Lie point symmetry vector filed for, (6) if where is the third prolongation of . The components , , , , , , , and can be determined from the following expressions Here, and are introduced as the following total derivatives: Form (13) we have the system of linear differential equations as follows: Replacing the functions , , , , , , , and given by the relation (15) and eliminating any dependence between partial differential derivatives of the functions , , and , we obtain the new partial differential equations corresponding to (6) (see the Appendix).
Looking at this conditions as a polynomial in the partial derivatives of the functions , , and and identifying with the polynom zero, we obtain the PDE system of , , , , and . The general solution of this PDE system is where , and consequently the infinitesimal generator of the symmetry group is From which, the system of nonlinear (6) has the sixteen-parameter Lie group of point symmetries generated by For up to , respectively, the characteristic has the components as follows: Equation (21) shows that no solutions are invariant under the groups generated by and up to . For , the characteristic has the components Therefore, the general solutions of the invariant surface conditions (11) by using the boundary conditions (7) are as follows: Substitution from (23) into (6) yields For simplifying we can use and with . Therefore, we have where is the Prandtl number and is the buoyancy parameter with is the local Grashof number and is the local Reynolds number. Equations (25) and (26) subject to the boundary (7) become When and , the solution of (25) subject to boundary condition (27) is given by
3.2. Numerical Results
Equations (25) and (26) subject to boundary conditions (27) have been solved numerically using the shooting method coupled with fifth-order Improved Runge-Kutta Method (IRK5) . For the validation of the Lie group method used in this study, the case when the buoyancy term in (25) is absent has been also considered and compared with the results reported by Mahapatra and Gupta [25, 26], Nazar et al. , and Ishak et al. . This comparison is shown in Table 1. It is seen that the present values of are in very good agreement with those obtained by Mahapatra and Gupta [25, 26], Nazar et al. , and Ishak et al. . Therefore, it can be concluded that the present Lie group method can be used with great confidence to study the problem discussed in this paper.
The values of the skin friction coefficient and for various Pr when and are tabulated in Table 2, for both cases of assisting and opposing flows. The values of are positive in all cases discussed in this study. Also, the effects of on the skin friction coefficient are found to be more significant for fluids having smaller , since the viscosity is less than the fluids with larger .
The resulting profiles of dimensionless velocity and dimensionless temperature are shown in Figures 2 and 3 for various values of , , and . From Figure 2, it is seen that for assisting flow, the velocity increases at the beginning until it achieves a certain value then decreases until the value becomes constant, that is unity at the outside of the boundary layer. From Figure 2, it can be seen that when , the flow has a boundary layer structure, and the thickness of the boundary layer decreases with increase in . According to Mahapatra and Gupta [25, 26], it can be explained as follows: for fixed value of corresponding to the stretching of the surface, an increase in in relation to implies an increase in straining motion near the stagnation region resulting in increased acceleration of the external stream, and this leads to thinning of the boundary layer with increase in .
The opposite trend occurs for opposing flow. From Figure 3, it is observed that the temperature of the fluid decreases, as the distance from the surface increases, for both cases of assisting and opposing flow, for all values of , , and until it achieves a constant value, namely, zero. This is not surprising, since the fluid receives the heat from the surface, and then the heat energy is changed into other energy forms such as kinetic energy.
The skin friction coefficient and are shown in Figures 4, 5, 6, and 7. Figures 4 and 6 suggest that an assisting buoyancy flow produces an increase in the skin friction coefficient, while an opposing buoyant flow gives rise to a decrease in the skin friction coefficient. This is because, the fluid velocity increases when the buoyancy force increases and hence increases the wall shear stress, which increases the skin friction coefficient. Figure 4 shows that all curves intersect at a point where ; that is, when the buoyancy force is zero. This is because (25) and (26) are uncoupled when ; in other words, the solutions to the flow field are not affected by the thermal field in which the buoyancy force is lacking. Also in this case, the value of remains constant, namely, zero. This value agreed with the exact solution (25), which implies , for all . Moreover, for assisting flow, it can be seen that decreases when increases for a fixed value of . This is because when increases, the viscosity increases and slows down the flow hence reduces the surface shear stress and thus reduces the skin friction coefficient . The opposite trends can be observed for opposing flow. In addition, from Figure 7 the effects of can be examined; that is, increasing enhances the rate of heat transfer, since increasing of will cause the increasing of viscosity then reduces the thermal conductivity, and thus increases.
The resulting profiles of dimensionless velocity and dimensionless temperature are shown in Figures 8 and 9 for various values of . Figure 8 shows that the velocity profiles increases and decreases for assisting flow and opposing flow, respectively, when increases. In Figure 9, it is observed that, for a particular value of , the temperature profiles is slightly increased, as the buoyancy parameter is increased, for the case of assisting flow. The opposite trend occurs for opposing flow. This is clear from the fact that assisting buoyant flow produces a favorable pressure gradient that enhances the momentum transport, which in turn increases the surface heat transfer rate.
The values of and are shown in Table 3 for , , and various . Table 3 shows that the functions and increases and decreases for assisting flow and opposing flow, respectively, when increases. The values of and are shown in Table 4 for , , and various . Table 4 shows that the functions and increase for both assisting flow and opposing flow when increases.
Lie group method is applicable to both linear and nonlinear partial differential equations, which leads to similarity variables that used to reduce the number of independent variables in partial differential equations. By determining the transformation group under which the given partial differential equations are invariant, we can obtain information about the invariants and symmetries of these equations. This information can be used to determine the similarity variables that will reduce the number of independent variables in the system. In this work, we have used Lie group method to obtain similarity reductions of nonlinear boundary layer equations (1)–(3), for the two-dimensional boundary layer equations of the liquid flow for the mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet. By determining the transformation group under which the given partial differential equations are invariant, we obtained the invariants and the symmetries of these equations. In turn, we used these invariants and symmetries to determine the similarity variables that reduced the number of independent variables. Therefore, the governing partial differential equations (1)–(3) are reduced to a set of two nonlinear ordinary differential equations (25) and (26). The resulting system of nonlinear ordinary differential equations (25) and (26) subjected to the boundary conditions (27) is solved numerically using the shooting method coupled with fifth-order Improved Runge-Kutta Method (IRK5). Effects of the parameters , , and of the fluid on the flow and heat transfer characteristics have been examined and discussed in detail. Our results are in complete agreement with those reported by Ishak et al. . Therefore, it can be concluded that the Lie group method can be used with great confidence to study the problem discussed in this paper.
List of Symbols
|:||Acceleration due to gravity ()|
|:||Dimensionless stream function|
|:||Local Grashof number|
|:||Local Reynolds number|
|:||Fluid temperature ()|
|:||Ambient temperature ()|
|:||Temperature of the stretching surface ()|
|:||Velocity components along the and directions, respectively,|
|:||Velocity of external flow ()|
|:||Velocity of the stretching surface ()|
|:||Cartesian coordinates along the surface and normal to it, respectively, ().|
|:||Thermal diffusivity ()|
|:||Thermal expansion coefficient ()|
|:||Kinematic viscosity ()|
|:||Condition at the stretching sheet|
|:||Condition at infinity.|
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