Lie Group Analysis and Similarity Solutions for Mixed Convection Boundary Layers in the Stagnation-Point Flow toward a Stretching Vertical Sheet

Abstract

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.

1. Introduction

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 [1]. Sakiadis [2] was the first to explore the flow induced by a semi-infinite horizontally moving wall in an ambient fluid. Crane [3] 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 [4], Elbashbeshy and Bazid [5], P. S. Gupta and A. S. Gupta [6], Magyari and Keller [7, 8], Magyari et al. [9], Liao and Pop [10], and Nazar et al. [11], looked into this problem by considering its various facts, such as uniform heat flux, permeability of the surface, flow, and heat transfer unsteadiness. Pop [12], Andersson [13], Takhar and Nath [14], and Nazar et al. [15] 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 [18], Ali and Al-Yousef [19, 20], Ali [21, 22], and Abo-Eldahab [23] 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. [24] 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 𝑦=0 of a Cartesian system of coordinates 𝑂_𝑥𝑦 (𝑦=0) with the 𝑥-axis along the sheet as shown in Figure 1. The fluid occupies the half plane (𝑦>0). It is assumed that the velocity 𝑢_𝑤(𝑥) and the temperature 𝑇_𝑤(𝑥) of the stretching sheet is proportional to the distance 𝑥 from the stagnation-point, where 𝑇_𝑤(𝑥)>𝑇_∞ with 𝑇_∞ being the uniform temperature of the ambient fluid. The velocity of the flow external to the boundary layer is 𝑢_𝑒(𝑥). Under these assumptions along with the Boussinesq and boundary layer approximations, the system of equations, which model the boundary layer flow are given by 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑢_𝑒𝑑𝑢_𝑒𝜕𝑑𝑥+𝜈²𝑢𝜕𝑦²±𝑔𝛽𝑇−𝑇_∞,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝜕𝑦=𝛼²𝑇𝜕𝑦², (1)(2)(3) where 𝑢 and 𝑣 are the velocity components along 𝑥- and 𝑦-axes, respectively, 𝑇 is the fluid temperature, 𝑔 is the gravity acceleration, 𝛼, 𝜈, and 𝛽 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 𝑣(𝑥,0)=0,𝑢(𝑥,0)=𝑢_𝑤(𝑥)𝑇()=𝑐𝑥,𝑥,0=𝑇_𝑤(𝑥)=𝑇_∞𝑢()+𝑏𝑥,𝑥,∞=𝑢_𝑒(𝑥)=𝑎𝑥,𝑇(𝑥,∞)=𝑇_∞, (4) where 𝑎, 𝑏, and 𝑐 are positive constants. The continuity equation can be satisfied by introducing a stream function Ψ, such that 𝑢=𝜕Ψ𝜕𝑦,𝑣=−𝜕Ψ.𝜕𝑥 (5)

Figure 1 

Physical model and coordinate system.

Therefore, from (1)–(3) with (5) we have Ψ_𝑦Ψ_𝑦𝑥−Ψ_𝑥Ψ_𝑦𝑦+𝜈Ψ_𝑦𝑦𝑦−𝑢_𝑒(𝑥)𝑑𝑢𝑑𝑥^𝑒(𝑥)±𝛽𝑇−𝑇_∞Ψ=0,_𝑦𝑇_𝑥−Ψ_𝑥𝑇_𝑦−𝛼𝑇_𝑦𝑦=0. (6)

The boundary conditions (4) will be as Ψ_𝑥(𝑥,0)=0,Ψ_𝑦()𝑇()𝑥,0=𝑐𝑥,𝑥,0=𝑇_𝑤(𝑥)=𝑇_∞Ψ+𝑏𝑥,_𝑦(𝑥,∞)=𝑎𝑥,𝑇(𝑥,∞)=𝑇_∞. (7)

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 𝜀, Lie group infinitesimal transformation in (𝑥,𝑦;Ψ,𝑢_𝑒,𝑇) given by 𝑥^∗=𝑥+𝜀𝜉𝑥,𝑦;Ψ,𝑢_𝑒,𝑇+𝑂𝜀²=𝑒^𝜀𝑋𝑦𝑥,_∗=𝑦+𝜀𝛾𝑥,𝑦;Ψ,𝑢_𝑒,𝑇+𝑂𝜀²=𝑒^𝜀𝑋Ψ𝑦,_∗=Ψ+𝜀Φ𝑥,𝑦;Ψ,𝑢_𝑒,𝑇+𝑂𝜀²=𝑒^𝜀𝑋𝑢Ψ,_∗𝑒=𝑢_𝑒+𝜀𝑈𝑥,𝑦;Ψ,𝑢_𝑒,𝑇+𝑂𝜀²=𝑒^𝜀𝑋𝑢_𝑒,𝑇^∗=𝑇+𝜀Υ𝑥,𝑦;Ψ,𝑢_𝑒,𝑇+𝑂𝜀²=𝑒^𝜀𝑋𝑇, (8) here 𝜀 is the group parameter, and 𝑋 is vector filed. A system of partial differential equations (6) is said to admit a symmetry generated by the vector filed as 𝜕𝑋≡𝜉𝜕𝜕𝑥+𝛾𝜕𝜕𝑦+𝜙𝜕𝜕Ψ+𝑈𝜕𝑢_𝑒𝜕+Υ.𝜕𝑇 (9) Equivalently, we can obtain (𝑥^∗,𝑦^∗;Ψ^∗,𝑢^∗_𝑒,𝑇^∗) by solving 𝑑𝑥^∗𝑑𝜀=𝜉𝑥,𝑦;Ψ,𝑢_𝑒,𝑇,𝑑𝑦^∗𝑑𝜀=𝛾𝑥,𝑦;Ψ,𝑢_𝑒,𝑇,𝑑Ψ^∗𝑑𝜀=Φ𝑥,𝑦;Ψ,𝑢_𝑒,𝑇,𝑑𝑢^∗_𝑒𝑑𝜀=𝑈𝑥,𝑦;Ψ,𝑢_𝑒,𝑇,𝑑𝑇^∗𝑑𝜀=Υ𝑥,𝑦;Ψ,𝑢_𝑒,𝑇, (10) subject to initial conditions, (𝑥^∗,𝑦^∗;Ψ^∗,𝑢^∗_𝑒,𝑇^∗)|_𝜀=0≡(𝑥,𝑦;Ψ,𝑢_𝑒,𝑇). If 𝑋 is left invariant by the transformation (𝑥,𝑦;Ψ,𝑢_𝑒,𝑇)→(𝑥^∗,𝑦^∗;Ψ^∗,𝑢^∗_𝑒,𝑇^∗), then the solutions Ψ=Ψ(𝑥,𝑦), 𝑢_𝑒=𝑢_𝑒(𝑥), and 𝑇=𝑇(𝑥,𝑦) are invariant under the symmetry (9) if 𝜑_Ψ=𝑋Ψ−Ψ𝑥,𝑦_{Ψ=Ψ(𝑥,𝑦)}𝜑=0,_𝑢𝑒=𝑋𝑢_𝑒−𝑢_𝑒(𝑥)_{𝑢𝑒=𝑢𝑒(𝑥,𝑦)}𝜑=0,_𝑇=𝑋𝑇−𝑇𝑥,𝑦_{𝑇=𝑇(𝑥,𝑦)}=0. (11) Assume Π₁=Ψ_𝑦Ψ_𝑦𝑥−Ψ_𝑥Ψ_𝑦𝑦+𝜈Ψ_𝑦𝑦𝑦−𝑢_𝑒𝑢_𝑒𝑥±𝛽𝑇−𝑇_∞Π=0,₂=Ψ_𝑦𝑇_𝑥−Ψ_𝑥𝑇_𝑦−𝛼𝑇_𝑦𝑦=0. (12) The vector 𝑋 as (9) is a Lie point symmetry vector filed for, (6) if 𝑋^[3]Π_𝑗Π𝑗=0=0,𝑗=1,2, (13) where 𝑋^[3]𝜕≡𝜉𝜕𝜕𝑥+𝛾𝜕𝜕𝑦+Φ𝜕𝜕Ψ+𝑈𝜕𝑢_𝑒𝜕+Υ𝜕𝑇+Φ^𝑥𝜕𝜕Ψ_𝑥+Φ^𝑦𝜕𝜕Ψ_𝑦+Υ^𝑥𝜕𝜕𝑇_𝑥+Υ^𝑦𝜕𝜕𝑇_𝑦+Υ^𝑦𝑦𝜕𝜕𝑇_𝑦𝑦+𝑈^𝑥𝜕𝜕𝑢_𝑒𝑥+Φ^𝑦𝑥𝜕𝜕Ψ_𝑦𝑥+Φ^𝑦𝑦𝜕𝜕Ψ_𝑦𝑦+Φ^𝑦𝑦𝑦𝜕𝜕Ψ_𝑦𝑦𝑦 (14) is the third prolongation of 𝑋. The components Φ^𝑥, Φ^𝑦, Υ^𝑥, Υ^𝑦, 𝑈^𝑥, Φ^𝑦𝑥, Φ^𝑦𝑦, and Φ^𝑦𝑦𝑦 can be determined from the following expressions Φ^𝑥=𝐷_𝑥Φ−Ψ_𝑥𝐷_𝑥𝜉−Ψ_𝑦𝐷_𝑥Φ𝛾,_𝑦=𝐷_𝑦Φ−Ψ_𝑥𝐷_𝑦𝜉−Ψ_𝑦𝐷_𝑦Υ𝛾,_𝑥=𝐷_𝑥Υ−𝑇_𝑥𝐷_𝑥𝜉−𝑇_𝑦𝐷_𝑥Υ𝛾,_𝑦=𝐷_𝑦Υ−𝑇_𝑥𝐷_𝑦𝜉−𝑇_𝑦𝐷_𝑦Υ𝛾,_𝑦𝑦=𝐷_𝑦Υ^𝑦−𝑇_𝑦𝑥𝐷_𝑦𝜉−𝑇_𝑦𝑦𝐷_𝑦𝑈𝛾,_𝑥=𝐷_𝑥𝑈−𝑢_𝑒𝑥𝐷_𝑥𝜉−𝑢_𝑒𝑦𝐷_𝑥Φ𝛾,_𝑥𝑦=𝐷_𝑦Φ^𝑥−Ψ_𝑥𝑥𝐷_𝑦𝜉−Ψ_𝑥𝑦𝐷_𝑦Φ𝛾,_𝑥𝑦=𝐷_𝑥Φ^𝑦−Ψ_𝑦𝑥𝐷_𝑥𝜉−Ψ_𝑦𝑦𝐷_𝑥Φ𝛾,_𝑦𝑦=𝐷_𝑦Φ^𝑦−Ψ_𝑦𝑥𝐷_𝑦𝜉−Ψ_𝑦𝑦𝐷_𝑦Φ𝛾,_𝑦𝑦𝑦=𝐷_𝑦Φ^𝑦𝑦−Ψ_𝑦𝑦𝑥𝐷_𝑦𝜉−Ψ_𝑦𝑦𝑦𝐷_𝑦𝛾. (15) Here, 𝐷_𝑥 and 𝐷_𝑦 are introduced as the following total derivatives: 𝐷_𝑥≡𝜕_𝑥+Ψ_𝑥𝜕_Ψ+𝑢_𝑒𝑥𝜕_𝑢𝑒+𝑇_𝑥𝜕_𝑇+Ψ_𝑥𝑥𝜕_Ψ𝑥+𝑢_𝑒𝑥𝑥𝜕_{(𝑢𝑒)𝑥}+𝑇_𝑥𝑥𝜕_𝑇𝑥+Ψ_𝑥𝑦𝜕_Ψ𝑦+𝑇_𝑥𝑦𝜕_𝑇𝑦𝐷+⋯,_𝑦≡𝜕_𝑦+Ψ_𝑦𝜕_Ψ+𝑢_𝑒𝑦𝜕_𝑢𝑒+𝑇_𝑦𝜕_𝑇+Ψ_𝑦𝑦𝜕_Ψ𝑦+𝑢_𝑒𝑦𝑦𝜕_{(𝑢𝑒)𝑦}+𝑇_𝑦𝑦𝜕_𝑇𝑦+Ψ_𝑦𝑥𝜕_Ψ𝑥+𝑇_𝑦𝑥𝜕_𝑇𝑥+⋯. (16) Form (13) we have the system of linear differential equations as follows: −𝑈𝑢_𝑒𝑥±𝛽Υ−Φ^𝑥Ψ_𝑦𝑦+Φ^𝑦Ψ_𝑦𝑥−𝑈^𝑥𝑢_𝑒+Φ^𝑦𝑥Ψ_𝑦−Φ^𝑦𝑦Ψ_𝑥+𝜈Φ^𝑦𝑦𝑦=0,−Φ^𝑥𝑇_𝑦+Φ^𝑦𝑇_𝑥+Υ^𝑥Ψ_𝑦−Υ^𝑦Ψ_𝑥−𝛼Υ^𝑦𝑦=0. (17) Replacing the functions 𝜙^𝑥, 𝜙^𝑦, 𝜙^𝑦𝑥, 𝜙^𝑦𝑦, 𝜙^𝑦𝑦𝑦, 𝑈^𝑥, Υ^𝑥, and Υ^𝑦 given by the relation (15) and eliminating any dependence between partial differential derivatives of the functions Ψ, 𝑢_𝑒, and 𝑇, 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 Ψ, 𝑢_𝑒, and 𝑇 and identifying with the polynom zero, we obtain the PDE system of 𝜁, 𝛾, Φ, 𝑈, and Υ. The general solution of this PDE system is 𝜁=𝐾₀+𝐾₁𝑥,𝛾=𝐾₂,Φ=𝐾₃+𝐾₄𝑥+𝐾₅𝐾𝑦+¹2Ψ+𝐾₆𝑢_𝑒,𝑈=𝐾₇+𝐾₈𝑥+𝐾₉𝑦+𝐾₁₀Ψ+𝐾₁₁𝑢_𝑒+𝐾₁2𝑇,Υ=𝐾₁₂+𝐾₁₃𝑥+𝐾₁₄Ψ+𝐾₁₅𝑢_𝑒+𝐾₁₆𝑇, (18) where 𝐾₀,…,𝐾₁₆∈ℝ, and consequently the infinitesimal generator of the symmetry group 𝐺 is 𝑋=𝐾₀𝜕𝜕𝑥+𝐾₁𝑥𝜕+1𝜕𝑥2Ψ𝜕+1𝜕Ψ2𝑇𝜕𝜕𝑢_𝑒+𝐾₂𝜕,𝜕𝑦+𝐾₃𝜕𝜕Ψ+𝐾₄𝑥𝜕𝜕Ψ+𝐾₅𝑦𝜕𝜕Ψ+𝐾₆𝑢_𝑒𝜕𝜕Ψ+𝐾₇𝜕𝜕𝑢_𝑒,+𝐾₈𝑥𝜕𝜕𝑢_𝑒+𝐾₉𝑦𝜕𝜕𝑢_𝑒+𝐾₁₀Ψ𝜕𝜕𝑢_𝑒+𝐾₁₁𝑢_𝑒𝜕𝜕𝑢_𝑒+𝐾₁₂𝜕,𝜕𝑇+𝐾₁₃𝑥𝜕𝜕𝑇+𝐾₁₄Ψ𝜕𝜕𝑇+𝐾₁₅𝑢_𝑒𝜕𝜕𝑇+𝐾₁₆𝑇𝜕.𝜕𝑇 (19) From which, the system of nonlinear (6) has the sixteen-parameter Lie group of point symmetries generated by 𝑋₁≡𝜕𝜕𝑥,𝑋₂𝜕≡𝑥+1𝜕𝑥2Ψ𝜕+1𝜕Ψ2𝑇𝜕𝜕𝑢_𝑒,𝑋₃≡𝜕𝜕𝑦,𝑋₄≡𝜕𝜕Ψ,𝑋₅𝜕≡𝑥,𝑋𝜕Ψ₆𝜕≡𝑦𝜕Ψ,𝑋₇≡𝑢_𝑒𝜕𝜕Ψ,𝑋₈≡𝜕𝜕𝑢_𝑒,𝑋₉𝜕≡𝑥𝜕𝑢_𝑒,𝑋₁₀𝜕≡𝑦𝜕𝑢_𝑒,𝑋₁₁𝜕≡Ψ𝜕𝑢_𝑒,𝑋₁₂≡𝑢_𝑒𝜕𝜕𝑢_𝑒,𝑋₁₃≡𝜕𝜕𝑇,𝑋₁₄𝜕≡𝑥,𝑋𝜕𝑇₁₅𝜕≡Ψ𝜕𝑇,𝑋₁₆≡𝑢_𝑒𝜕𝜕𝑇,𝑋₁₇𝜕≡𝑇.𝜕𝑇 (20) For 𝑋₁ up to 𝑋₁₇, respectively, the characteristic 𝜑=(𝜑_Ψ,𝜑_𝑢𝑒,𝜑_𝑇) has the components as follows: 𝜑_𝑋1=−Ψ_𝑥,−𝑢_𝑒𝑥,−𝑇_𝑥,𝜑_𝑋2=−𝑥Ψ_𝑥+Ψ,−𝑥𝑢_𝑒𝑥+12𝑇,−𝑥𝑇_𝑥,𝜑_𝑋3=−Ψ_𝑦,0,−𝑇_𝑦,𝜑_𝑋4=(),𝜑1,0,0_𝑋5=(𝑥,0,0),𝜑_𝑋6=,𝜑𝑦,0,0_𝑋7=𝑢_𝑒,0,0,𝜑_𝑋8=(),𝜑0,1,0_𝑋9=(0,𝑥,0),𝜑_𝑋10=,𝜑0,𝑦,0_𝑋11=(0,Ψ,0),𝜑_𝑋12=0,𝑢_𝑒,𝜑,0_𝑋13=(0,0,1),𝜑_𝑋14=(),𝜑0,0,𝑥_𝑋15=(0,0,Ψ),𝜑_𝑋16=0,0,𝑢_𝑒,𝜑_𝑋17=(0,0,𝑇). (21) Equation (21) shows that no solutions are invariant under the groups generated by 𝑋₁ and 𝑋₃ up to 𝑋₁₇. For 𝑋₂, the characteristic 𝜑=(𝜑_Ψ,𝜑_𝑢𝑒,𝜑_𝑇) has the components 𝜑_Ψ=−𝑥Ψ_𝑥+Ψ,𝜑_𝑢𝑒=−𝑥𝑢_𝑒𝑥+12𝑇,𝜑_𝑇=−𝑥𝑇_𝑥. (22) Therefore, the general solutions of the invariant surface conditions (11) by using the boundary conditions (7) are as follows: Ψ𝑥,𝑦=𝑐𝑥𝐹𝑦,𝑢_𝑒(𝑥)𝑇=𝑇=𝑎𝑥,𝑥,𝑦_𝑤−𝑇_∞𝐻𝑦+𝑇_∞. (23) Substitution from (23) into (6) yields 𝑐𝑥𝑐𝐹^′𝑦²−𝑐𝐹𝑦𝐹^′′𝑦−𝜈𝐹^′′′𝑦−𝑎²𝑥±𝑔𝛽𝑇_𝑤−𝑇_∞𝐻𝑦𝑦𝐻=0,𝑐𝐹^′𝑦+𝛼𝐻^′′𝑦=0. (24) For simplifying we can use 𝐹(𝑦)=𝜈/𝑐𝑓(𝜂) and 𝐻(𝑦)=𝜃(𝜂) with 𝜂=𝑐/𝑣𝑦. Therefore, we have 𝑓^′′′𝜂+𝑓𝜂𝑓^′′𝜂−𝑓^′𝜂²+𝑎𝑐²𝜂1±𝜆𝜃=0,𝜃Pr^′′𝜂+𝑓𝜂𝜃^′𝜂=0, (25)(26) where Pr=𝜈/𝑐 is the Prandtl number and 𝜆=𝐺_𝑟𝑥/Re²_𝑥 is the buoyancy parameter with 𝐺_𝑟𝑥=𝑔𝛽(𝑇_𝑤−𝑇_∞)𝑥³/𝑣² is the local Grashof number and Re²_𝑥=𝑢_𝑤𝑥/𝑣 is the local Reynolds number. Equations (25) and (26) subject to the boundary (7) become𝑓(0)=0,𝑓^′(0)(0)𝑓=1,𝜃=1,_′(∞)=𝑎𝑐,𝜃(∞)=0. (27) When 𝜆=0 and 𝑎/𝑐=1, the solution of (25) subject to boundary condition (27) is given by 𝑓𝜂=𝜂. (28)

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) [27]. For the validation of the Lie group method used in this study, the case when the buoyancy term 𝜆𝜃 in (25) is absent has been also considered and compared with the results reported by Mahapatra and Gupta [25, 26], Nazar et al. [11], and Ishak et al. [28]. This comparison is shown in Table 1. It is seen that the present values of 𝑓^′′(0) are in very good agreement with those obtained by Mahapatra and Gupta [25, 26], Nazar et al. [11], and Ishak et al. [28]. 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.

Table 1 

Values of 𝑓^′′(0) for different values of 𝑎/𝑐 when the buoyancy force term 𝜆𝜃 in (25) is absent.

The values of the skin friction coefficient and −𝜃^′(0) for various Pr when 𝑎/𝑐=1 and 𝜆=1 are tabulated in Table 2, for both cases of assisting and opposing flows. The values of −𝜃^′(0) are positive in all cases discussed in this study. Also, the effects of 𝜆 on the skin friction coefficient are found to be more significant for fluids having smaller Pr, since the viscosity is less than the fluids with larger Pr.

Table 2 

Values of 𝑓^′′(0) and −𝜃^′(0) for 𝑎/𝑐=1, 𝜆=1 and various Pr.

The resulting profiles of dimensionless velocity 𝑓^′(𝜂) and dimensionless temperature 𝜃(𝜂) are shown in Figures 2 and 3 for various values of 𝑎/𝑐, 𝜆, and Pr. 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 𝑎/𝑐>1, the flow has a boundary layer structure, and the thickness of the boundary layer decreases with increase in 𝑎/𝑐. According to Mahapatra and Gupta [25, 26], it can be explained as follows: for 𝑎 fixed value of 𝑐 corresponding to the stretching of the surface, an increase in 𝑎 in relation to 𝑐 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 𝑎/𝑐.

Figure 2 

Velocity profiles for some values of 𝑎/𝑐 when Pr=1 and 𝜆=1.

Figure 3 

Temperature profiles for some values of 𝑎/𝑐 when Pr=1 and 𝜆=1; solid line: assisting flow and dash line: opposing flow.

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 𝑎/𝑐, 𝜆, and Pr 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 −𝜃^′(0) 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 𝜆=0; that is, when the buoyancy force is zero. This is because (25) and (26) are uncoupled when 𝜆=0; 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 𝑓^′′(0)=0 remains constant, namely, zero. This value agreed with the exact solution (25), which implies 𝑓^′′(𝜂)=0, for all 𝜂. Moreover, for assisting flow, it can be seen that 𝑓^′′(0) decreases when Pr increases for a fixed value of 𝜆. This is because when Pr increases, the viscosity increases and slows down the flow hence reduces the surface shear stress and thus reduces the skin friction coefficient 𝑓^′′(0). The opposite trends can be observed for opposing flow. In addition, from Figure 7 the effects of Pr can be examined; that is, increasing Pr enhances the rate of heat transfer, since increasing of Pr will cause the increasing of viscosity then reduces the thermal conductivity, and thus −𝜃^′(0) increases.

Figure 4 

Variation with 𝜆 of the skin friction coefficient for some values of 𝑎/𝑐 when Pr=1; solid line: assisting flow and dash line: opposing flow.

Figure 5 

Variation with 𝜆 of −𝜃′(0) for some values of 𝑎/𝑐 when Pr=1; solid line: assisting flow and dash line: opposing flow.

Figure 6 

Variation with 𝜆 of the skin friction coefficient for some values of Pr when 𝑎/𝑐=1; solid line: assisting flow and dash line: opposing flow.

Figure 7 

Variation with 𝜆 of −𝜃′(0) for some values of Pr when 𝑎/𝑐=1; solid line: assisting flow and dash line: opposing flow.

The resulting profiles of dimensionless velocity 𝑓^′(𝜂) and dimensionless temperature 𝜃(𝜂) are shown in Figures 8 and 9 for various values of 𝜆. Figure 8 shows that the velocity profiles increases and decreases for assisting flow and opposing flow, respectively, when 𝜆 increases. In Figure 9, it is observed that, for a particular value of Pr, the temperature profiles is slightly increased, as the buoyancy parameter 𝜆 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.

Figure 8 

Velocity profiles for some values of 𝜆 when Pr=1 and 𝑎/𝑐=1.

Figure 9 

Temperature profiles for some values of 𝜆 when Pr=1 and 𝑎/𝑐=1.

The values of 𝑓^′′(0) and −𝜃^′(0) are shown in Table 3 for 𝑎/𝑐=1, Pr=1, and various 𝜆. Table 3 shows that the functions 𝑓^′′(0) and −𝜃^′(0) increases and decreases for assisting flow and opposing flow, respectively, when 𝜆 increases. The values of 𝑓^′′(0) and −𝜃^′(0) are shown in Table 4 for 𝜆=1, Pr=1, and various 𝑎/𝑐. Table 4 shows that the functions 𝑓^′′(0) and −𝜃^′(0) increase for both assisting flow and opposing flow when 𝑎/𝑐 increases.

Table 3 

Values of 𝑓^′′(0) and −𝜃^′(0) for 𝑎/𝑐=1, Pr =1, and various 𝜆.

Table 4 

Values of 𝑓^′′(0) and −𝜃^′(0) for 𝜆=1, Pr =1, and various 𝑎/𝑐.

4. Conclusions

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 𝜆, Pr, and 𝑎/𝑐 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. [28]. Therefore, it can be concluded that the Lie group method can be used with great confidence to study the problem discussed in this paper.

Appendix

−𝑈𝑢_𝑒𝑥±𝛽Υ−Ψ_𝑦𝑦Φ_𝑥+Ψ_𝑥Φ_Ψ+𝑢_𝑒𝑥Φ_𝑢𝑒+𝑇_𝑥Φ_𝑇−𝜁_𝑥Ψ_𝑥−𝜁_ΨΨ_𝑥Ψ_𝑥−𝜁_𝑢𝑒(𝑢_𝑒)_𝑥Ψ_𝑥−𝜁_𝑇𝑇_𝑥Ψ_𝑥−𝛾_𝑥Ψ_𝑦−𝛾_ΨΨ_𝑥Ψ_𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥−𝛾_𝑇𝑇_𝑥Ψ_𝑦+Ψ_𝑦𝑥Φ_𝑦+Ψ_𝑦Φ_Ψ+𝑢_𝑒𝑦Φ_𝑢𝑒+𝑇_𝑦Φ_𝑇−𝜁_𝑦Ψ_𝑥−𝜁_ΨΨ_𝑥Ψ_𝑦−𝜁_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑥−𝜁_𝑇𝑇_𝑦Ψ_𝑥−𝛾_𝑦Ψ_𝑦−𝛾_ΨΨ_𝑦Ψ_𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑦−𝛾_𝑇𝑇_𝑦Ψ_𝑦−𝑢_𝑒𝑈_𝑥+Ψ_𝑥𝑈_Ψ+𝑢_𝑒𝑥𝑈_𝑢𝑒+𝑇_𝑥𝑈_𝑇−𝜁_𝑥𝑢_𝑒𝑥−𝜁_ΨΨ_𝑥𝑢_𝑒𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑥𝑢_𝑒𝑥−𝜁_𝑇𝑇_𝑥𝑢_𝑒𝑥−𝛾_𝑥𝑢_𝑒𝑦−𝛾_Ψ𝑢_𝑒𝑦Ψ_𝑥−𝛾_𝑢𝑒𝑢_𝑒𝑥𝑢_𝑒𝑦−𝛾_𝑇𝑇_𝑥𝑢_𝑒𝑦+Ψ_𝑦Φ_𝑦𝑥+Φ_ΨΨ_𝑦𝑥+Φ_Ψ𝑥Ψ_𝑦+Φ_𝑢𝑒𝑢_𝑒𝑦𝑥+Φ_𝑢𝑒𝑥𝑢_𝑒𝑦+Φ_𝑇𝑇_𝑦𝑥+Φ_𝑇𝑥𝑇_𝑦−𝜁_𝑦𝑥Ψ_𝑥−𝜁_𝑦Ψ_𝑥𝑥−𝜁_Ψ𝑥Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑥Ψ_𝑥Ψ_𝑦−𝜁_ΨΨ_𝑥𝑥Ψ_𝑦−𝜁_ΨΨ_𝑥Ψ_𝑦𝑥−𝜁_𝑢𝑒𝑥𝑢_𝑒𝑦Ψ_𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑦𝑥Ψ_𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑥𝑥−𝜁_𝑇𝑥𝑇_𝑦Ψ_𝑥−𝜁_𝑇𝑇_𝑦𝑥Ψ_𝑥−𝜁_𝑇𝑇_𝑦Ψ_𝑥𝑥−𝛾_𝑦𝑥Ψ_𝑦−𝛾_𝑦Ψ_𝑦𝑥−𝛾_Ψ𝑥Ψ_𝑦Ψ_𝑦−𝛾_ΨΨ_𝑦𝑥Ψ_𝑦−𝛾_ΨΨ_𝑦Ψ_𝑦𝑥−𝛾_𝑢𝑒𝑥𝑢_𝑒𝑦Ψ_𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑦𝑥Ψ_𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑦𝑥−𝛾_𝑦𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑦𝑇_𝑦𝑥Ψ_𝑦−𝛾_𝑦𝑇_𝑦Ψ_𝑦𝑥−𝜁_𝑥Ψ_𝑦𝑥−𝜁_ΨΨ_𝑥Ψ_𝑦−𝜁_𝑢𝑒𝑢_𝑒𝑥+𝜁_𝑥𝑇_𝑥−𝛾_𝑥Ψ_𝑦𝑦−𝛾_ΨΨ_𝑥Ψ_𝑦𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑥Ψ_𝑦𝑦−𝛾_𝑇𝑇_𝑥Ψ_𝑦𝑦−Ψ_𝑥Φ_𝑦𝑥+Φ_𝑦ΨΨ_𝑥+Φ_𝑦𝑢𝑒𝑢_𝑒𝑥+Φ_𝑦𝑇𝑇_𝑥+Φ_𝑇𝑇_𝑦𝑥+Φ_𝑇𝑥𝑇_𝑦+Φ_𝑇Ψ𝑇_𝑦Ψ_𝑥+Φ_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦−𝜁_𝑦𝑥Ψ_𝑥−𝜁_𝑦Ψ_𝑥𝑥−𝜁_𝑦ΨΨ_𝑥Ψ_𝑥−𝜁_𝑦𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥−𝜁_𝑦𝑇𝑇_𝑥Ψ_𝑥−𝜁_Ψ𝑥Ψ_𝑥Ψ_𝑦−𝜁_ΨΨ_𝑥𝑥Ψ_𝑦−𝜁_ΨΨ_𝑥Ψ_𝑦𝑥−𝜁_ΨΨΨ_𝑥Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑇𝑇_𝑥Ψ_𝑥Ψ_𝑦−𝛾_𝑦𝑥Ψ_𝑦−𝛾_𝑦𝑥Ψ_𝑦−𝛾_𝑦Ψ_𝑦𝑥−𝛾_𝑦ΨΨ_𝑥Ψ_𝑦−𝛾_𝑦𝑢𝑒𝑢_𝑒𝑥Ψ_𝑦−𝛾_𝑦𝑇𝑇_𝑥Ψ_𝑦−𝛾_Ψ𝑥Ψ_𝑦Ψ_𝑦−𝛾_ΨΨ_𝑦𝑥Ψ_𝑦−𝛾_ΨΨ_𝑦Ψ_𝑦𝑥−𝛾_ΨΨΨ_𝑥Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑢𝑒𝑢_𝑒𝑥Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑇𝑇_𝑥Ψ_𝑦Ψ_𝑦−𝛾_𝑇𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑇_𝑦𝑥Ψ_𝑦−𝛾_𝑇𝑇_𝑦Ψ_𝑦𝑥−𝛾_𝑇ΨΨ_𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑇𝑇_𝑥𝑇_𝑦𝑇_𝑦+𝜈Φ_𝑦𝑥𝑦+Φ_𝑦ΨΨ_𝑥𝑦+Φ_𝑦Ψ𝑦Ψ_𝑥+Φ_𝑦𝑢𝑒𝑢_𝑒𝑥𝑦+Φ_{𝑦𝑢𝑒𝑦}𝑢_𝑒𝑥+Φ_𝑦𝑇𝑇_𝑥𝑦+Φ_𝑦𝑇𝑦𝑇_𝑥+Φ_𝑇𝑇_𝑦𝑥𝑦+Φ_𝑇𝑦𝑇_𝑦𝑥+Φ_𝑇𝑥𝑇_𝑦𝑦+Φ_𝑇𝑥𝑦𝑇_𝑦+Φ_𝑇Ψ𝑦𝑇_𝑦Ψ_𝑥+Φ_𝑇Ψ𝑇_𝑦𝑦Ψ_𝑥+Φ_𝑇Ψ𝑇_𝑦Ψ_𝑥𝑦+Φ_{𝑇𝑢𝑒𝑦}𝑢_𝑒𝑥𝑇_𝑦+Φ_𝑇𝑢𝑒𝑢_𝑒𝑥𝑦𝑇_𝑦+Φ_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦𝑦−𝜁_𝑦𝑥𝑦Ψ_𝑥−𝜁_𝑦𝑥Ψ_𝑥𝑦−𝜁_𝑦𝑦Ψ_𝑥𝑥−𝜁_𝑦Ψ_𝑥𝑥𝑦−𝜁_𝑦Ψ𝑦Ψ_𝑥Ψ_𝑥−𝜁_𝑦ΨΨ_𝑥𝑦Ψ_𝑥−𝜁_𝑦ΨΨ_𝑥Ψ_𝑥𝑦−𝜁_{𝑦𝑢𝑒𝑦}𝑢_𝑒𝑥Ψ_𝑥−𝜁_𝑦𝑢𝑒𝑢_𝑒𝑥𝑦Ψ_𝑥−𝜁_𝑦𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥𝑦−𝜁_𝑦𝑇𝑦𝑇_𝑥Ψ_𝑥−𝜁_𝑦𝑇𝑇_𝑥𝑦Ψ_𝑥−𝜁_𝑦𝑇𝑇_𝑥Ψ_𝑥𝑦−𝜁_Ψ𝑥𝑦Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑥Ψ_𝑥𝑦Ψ_𝑦−𝜁_Ψ𝑥Ψ_𝑥Ψ_𝑦𝑦−𝜁_Ψ𝑥𝑦Ψ_𝑥𝑥Ψ_𝑦−𝜁_ΨΨ_𝑥𝑥𝑦Ψ_𝑦−𝜁_ΨΨ_𝑥𝑥Ψ_𝑦𝑦−𝜁_Ψ𝑦Ψ_𝑥Ψ_𝑦𝑥−𝜁_ΨΨ_𝑥𝑦Ψ_𝑦𝑥−𝜁_ΨΨ_𝑥Ψ_𝑦𝑥𝑦−𝜁_ΨΨ𝑦Ψ_𝑥Ψ_𝑥Ψ_𝑦−𝜁_ΨΨΨ_𝑥𝑦Ψ_𝑥Ψ_𝑦−𝜁_ΨΨΨ_𝑥Ψ_𝑥𝑦Ψ_𝑦−𝜁_ΨΨΨ_𝑥Ψ_𝑥Ψ_𝑦𝑦−𝜁_{Ψ𝑢𝑒𝑦}𝑢_𝑒𝑥Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑢𝑒𝑢_𝑒𝑥𝑦Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥𝑦Ψ_𝑦−𝜁_Ψ𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥Ψ_𝑦𝑦−𝜁_Ψ𝑇𝑦𝑇_𝑥Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑇𝑇_𝑥𝑦Ψ_𝑥Ψ_𝑦−𝜁_Ψ𝑇𝑇_𝑥Ψ_𝑥𝑦Ψ_𝑦−𝜁_Ψ𝑇𝑇_𝑥Ψ_𝑥Ψ_𝑦𝑦−𝜁_𝑇𝑥𝑦𝑇_𝑦Ψ_𝑥−𝜁_𝑇𝑥𝑇_𝑦𝑦Ψ_𝑥−𝜁_𝑇𝑥𝑇_𝑦Ψ_𝑥𝑦−𝜁_𝑇𝑦𝑇_𝑦𝑥Ψ_𝑥−𝜁_𝑇𝑇_𝑦𝑥𝑦Ψ_𝑥−𝜁_𝑇𝑇_𝑦𝑥Ψ_𝑥𝑦−𝜁_𝑇𝑦𝑇_𝑦Ψ_𝑥𝑥−𝜁_𝑇𝑇_𝑦𝑦Ψ_𝑥𝑥−𝜁_𝑇𝑇_𝑦Ψ_𝑥𝑥𝑦−𝜁_𝑇Ψ𝑦𝑇_𝑦Ψ_𝑥𝑥−𝜁_𝑇Ψ𝑇_𝑦𝑦Ψ_𝑥𝑥−𝜁_𝑇Ψ𝑇_𝑦Ψ_𝑥𝑥𝑦−𝜁_{𝑇𝑢𝑒𝑦}𝑢_𝑒𝑥𝑇_𝑦Ψ_𝑥−𝜁_𝑇𝑢𝑒𝑢_𝑒𝑥𝑦𝑇_𝑦Ψ_𝑥−𝜁_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦𝑦Ψ_𝑥−𝜁_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦Ψ_𝑥𝑦−𝜁_{𝑇𝑢𝑒𝑦}𝑇_𝑥𝑇_𝑦Ψ_𝑥−𝜁_𝑇𝑢𝑒𝑇_𝑥𝑦𝑇_𝑦Ψ_𝑥−𝜁_{𝑇𝑢𝑒𝑦}𝑇_𝑥𝑇_𝑦𝑦Ψ_𝑥−𝜁_{𝑇𝑢𝑒𝑦}𝑇_𝑥𝑇_𝑦Ψ_𝑥𝑦−𝛾_𝑦𝑥𝑦Ψ_𝑦−𝛾_𝑦𝑥Ψ_𝑦𝑦−𝛾_𝑦𝑦Ψ_𝑦𝑥−𝛾_𝑦Ψ_𝑦𝑥𝑦−𝛾_𝑦Ψ𝑦Ψ_𝑥Ψ_𝑦−𝛾_𝑦ΨΨ_𝑥𝑦Ψ_𝑦−𝛾_𝑦ΨΨ_𝑥Ψ_𝑦𝑦−𝛾_Ψ𝑥𝑦Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑥Ψ_𝑦𝑦Ψ_𝑦−𝛾_Ψ𝑥Ψ_𝑦Ψ_𝑦𝑦−𝛾_Ψ𝑦Ψ_𝑦𝑥Ψ_𝑦−𝛾_ΨΨ_𝑦𝑥𝑦Ψ_𝑦−𝛾_Ψ𝑦Ψ_𝑦Ψ_𝑦𝑥−𝛾_ΨΨ_𝑦𝑦Ψ_𝑦𝑥−𝛾_ΨΨ_𝑦Ψ_𝑦𝑥𝑦−𝛾_ΨΨ𝑦Ψ_𝑥Ψ_𝑦Ψ_𝑦−𝛾_ΨΨΨ_𝑥𝑦Ψ_𝑦Ψ_𝑦−𝛾_ΨΨΨ_𝑥Ψ_𝑦𝑦Ψ_𝑦−𝛾_ΨΨΨ_𝑥Ψ_𝑦Ψ_𝑦𝑦−𝛾_{Ψ𝑢𝑒𝑦}𝑢_𝑒𝑥Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑢𝑒𝑢_𝑒𝑥𝑦Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑢𝑒𝑢_𝑒𝑥Ψ_𝑦𝑦Ψ_𝑦−𝛾_Ψ𝑢𝑒𝑢_𝑒𝑥Ψ_𝑦Ψ_𝑦𝑦−𝛾_Ψ𝑇𝑦𝑇_𝑥Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑇𝑇_𝑥𝑦Ψ_𝑦Ψ_𝑦−𝛾_Ψ𝑇𝑇_𝑥Ψ_𝑦𝑦Ψ_𝑦−𝛾_Ψ𝑇𝑇_𝑥Ψ_𝑦Ψ_𝑦𝑦−𝛾_𝑇𝑥𝑦𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑥𝑇_𝑦𝑦Ψ_𝑦−𝛾_𝑇𝑥𝑇_𝑦Ψ_𝑦𝑦−𝛾_𝑇𝑦𝑇_𝑦𝑥Ψ_𝑦−𝛾_𝑇𝑇_𝑦𝑥𝑦Ψ_𝑦−𝛾_𝑇𝑇_𝑦𝑥Ψ_𝑦𝑦−𝛾_𝑇𝑦𝑇_𝑦Ψ_𝑦𝑥−𝛾_𝑇𝑇_𝑦𝑦Ψ_𝑦𝑥−𝛾_𝑇𝑇_𝑦Ψ_𝑦𝑥𝑦−𝛾_𝑇Ψ𝑦Ψ_𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑇ΨΨ_𝑥𝑦𝑇_𝑦Ψ_𝑦−𝛾_𝑇ΨΨ_𝑥𝑇_𝑦𝑦Ψ_𝑦−𝛾_𝑇ΨΨ_𝑥𝑇_𝑦Ψ_𝑦𝑦−𝛾_{𝑇𝑢𝑒𝑦}𝑢_𝑒𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑢𝑒𝑢_𝑒𝑥𝑦𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦𝑦Ψ_𝑦−𝛾_𝑇𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦𝑦Ψ_𝑦−𝛾_𝑇𝑇𝑦𝑇_𝑥𝑇_𝑦𝑇_𝑦−𝛾_𝑇𝑇𝑇_𝑥𝑦𝑇_𝑦𝑇_𝑦−𝛾_𝑇𝑇𝑇_𝑥𝑇_𝑦𝑦𝑇_𝑦−𝛾_𝑇𝑇𝑇_𝑥𝑇_𝑦𝑇_𝑦𝑦−𝜁_𝑦Ψ_𝑦𝑦𝑥−𝜁_ΨΨ_𝑦Ψ_𝑦𝑦𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑦𝑦𝑥−𝜁_𝑇𝑇_𝑦Ψ_𝑦𝑦𝑥−𝛾_𝑦Ψ_𝑦𝑦𝑦−𝛾_ΨΨ_𝑦Ψ_𝑦𝑦𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑦𝑦𝑦−𝛾_𝑇𝑇_𝑦Ψ_𝑦𝑦𝑦Υ=0,_𝑥Ψ_𝑦+Υ_ΨΨ_𝑥Ψ_𝑦+Υ_𝑢𝑒𝑢_𝑒𝑥Ψ_𝑦+Υ_𝑇𝑇_𝑥Ψ_𝑦−𝜁_𝑥𝑇_𝑥Ψ_𝑦−𝜁_Ψ𝑇_𝑥Ψ_𝑥Ψ_𝑦−𝜁_𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥Ψ_𝑦−𝜁_𝑇𝑇_𝑥𝑇_𝑥Ψ_𝑦−𝛾_𝑥𝑇_𝑦Ψ_𝑦−𝛾_Ψ𝑇_𝑦Ψ_𝑥Ψ_𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑥𝑇_𝑦Ψ_𝑦−𝛾_𝑇𝑇_𝑥𝑇_𝑦Ψ_𝑦−Υ_𝑦Ψ_𝑥−Υ_ΨΨ_𝑦Ψ_𝑥−Υ_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑥−Υ_𝑇𝑇_𝑦Ψ_𝑥+𝜁_𝑦𝑇_𝑥Ψ_𝑥+𝜁_Ψ𝑇_𝑥Ψ_𝑦Ψ_𝑥+𝜁_𝑢𝑒𝑢_𝑒𝑦𝑇_𝑥Ψ_𝑥+𝜁_𝑇𝑇_𝑦𝑇_𝑥Ψ_𝑥+𝛾_𝑦𝑇_𝑦Ψ_𝑥+𝛾_Ψ𝑇_𝑦Ψ_𝑦Ψ_𝑥+𝜁_𝑢𝑒𝑢_𝑒𝑦𝑇_𝑦Ψ_𝑥+𝜁_𝑇𝑇_𝑦𝑇_𝑦Ψ_𝑥−Φ_𝑥𝑇_𝑦−Φ_Ψ𝑇_𝑦Ψ_𝑥−Φ_𝑢𝑒𝑇_𝑦𝑢_𝑒𝑥−Φ_𝑇𝑇_𝑦𝑇_𝑥+𝜁_𝑥Ψ_𝑥𝑇_𝑦+𝜁_ΨΨ_𝑥Ψ_𝑥𝑇_𝑦+𝜁_𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥𝑇_𝑦+𝜁_𝑇𝑇_𝑥Ψ_𝑥𝑇_𝑦+𝛾_𝑥Ψ_𝑦𝑇_𝑦+𝛾_ΨΨ_𝑥Ψ_𝑦𝑇_𝑦+𝛾_𝑢𝑒𝑢_𝑒𝑥Ψ_𝑥𝑇_𝑦+𝛾_𝑇𝑇_𝑥Ψ_𝑦𝑇_𝑦+Φ_𝑦𝑇_𝑥+Φ_Ψ𝑇_𝑥Ψ_𝑦+Φ_𝑢𝑒𝑇_𝑥𝑢_𝑒𝑦+Φ_𝑇𝑇_𝑥𝑇_𝑦−𝜁_𝑦Ψ_𝑥𝑇_𝑥−𝜁_ΨΨ_𝑥Ψ_𝑦𝑇_𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑥𝑇_𝑥−𝜁_𝑇𝑇_𝑦Ψ_𝑥𝑇_𝑥−𝛾_𝑦Ψ_𝑦𝑇_𝑥−𝛾_ΨΨ_𝑦Ψ_𝑦𝑇_𝑥−𝛾_𝑢𝑒𝑢_𝑒𝑦Ψ_𝑦𝑇_𝑥−𝛾_𝑇𝑇_𝑦Ψ_𝑦𝑇_𝑥−𝛼Υ_𝑦𝑦+Υ_Ψ𝑦Ψ_𝑦+Υ_ΨΨ_𝑦𝑦+Υ_𝑢𝑒𝑦𝑢_𝑒𝑦+Υ_𝑢𝑒𝑢_𝑒𝑦𝑦+Υ_𝑇𝑦𝑇_𝑦+Υ_𝑇𝑇_𝑦𝑦−𝜁_𝑦𝑦𝑇_𝑥−𝜁_𝑦𝑇_𝑥𝑦−𝜁_Ψ𝑦𝑇_𝑥Ψ_𝑦−𝜁_Ψ𝑇_𝑥𝑦Ψ_𝑦−𝜁_Ψ𝑇_𝑥Ψ_𝑦𝑦−𝜁_𝑢𝑒𝑦𝑢_𝑒𝑦𝑇_𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑦𝑦𝑇_𝑥−𝜁_𝑢𝑒𝑢_𝑒𝑦𝑇_𝑥𝑦−𝜁_𝑇𝑦𝑇_𝑦𝑇_𝑥−𝜁_𝑇𝑇_𝑦𝑦𝑇_𝑥−𝜁_𝑇𝑇_𝑦𝑇_𝑥𝑦−𝛾_𝑦𝑦𝑇_𝑦−𝛾_𝑦𝑇_𝑦𝑦−𝛾_Ψ𝑦𝑇_𝑦Ψ_𝑦−𝛾_Ψ𝑇_𝑦𝑦Ψ_𝑦−𝛾_Ψ𝑇_𝑦Ψ_𝑦𝑦−𝜁_𝑢𝑒𝑦𝑢_𝑒𝑦𝑇_𝑦−𝜁_𝑢𝑒𝑢_𝑒𝑦𝑦𝑇_𝑦−𝜁_𝑢𝑒𝑢_𝑒𝑦𝑇_𝑦𝑦−𝜁_𝑇𝑦𝑇_𝑦𝑇_𝑦−𝜁_𝑇𝑇_𝑦𝑦𝑇_𝑦−𝜁_𝑇𝑇_𝑦𝑇_𝑦𝑦−𝜁_𝑦𝑇_𝑦𝑥−𝜁_Ψ𝑇_𝑦𝑥Ψ_𝑦−𝜁_𝑢𝑒𝑢_𝑒𝑦𝑇_𝑦𝑥−𝜁_𝑇𝑇_𝑦𝑇_𝑦𝑥−𝛾_𝑦𝑇_𝑦𝑦−𝛾_Ψ𝑇_𝑦𝑦Ψ_𝑦−𝛾_𝑢𝑒𝑢_𝑒𝑦𝑇_𝑦𝑦−𝛾_𝑇𝑇_𝑦𝑇_𝑦𝑦=0. (A.1)

List of Symbols

𝑎,𝑏,and𝑐:	Constants
𝑔:	Acceleration due to gravity (ms−2)
𝑓:	Dimensionless stream function
Gr𝑥:	Local Grashof number
Pr:	Prandtl number
Re𝑥:	Local Reynolds number
𝑇:	Fluid temperature (K)
𝑇∞:	Ambient temperature (K)
𝑇𝑤(𝑥):	Temperature of the stretching surface (K)
𝑢,𝑣:	Velocity components along the 𝑥 and 𝑦 directions, respectively,
𝑢𝑒(𝑥):	Velocity of external flow (ms−1)
𝑢𝑤(𝑥):	Velocity of the stretching surface (ms−1)
𝑥,𝑦:	Cartesian coordinates along the surface and normal to it, respectively, (m).

Greek Symbols

𝛼:	Thermal diffusivity (m2s−1)
𝛽:	Thermal expansion coefficient (K−1)
𝜂:	Pseudo-similarity variable
𝜃:	Dimensionless temperature
𝜆:	Buoyancy parameter
𝜈:	Kinematic viscosity (m2s−1)
Ψ:	Stream function.

Subscripts

𝑤:	Condition at the stretching sheet
∞:	Condition at infinity.