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

Authors: Sarkhosh Seddighi Chaharborj; Fudziah Ismail; Yousof Gheisari; Reza Seddighi Chaharborj; Mohd Rizam Abu Bakar; Zanariah Abdul Majid
Journal: Abstract and Applied Analysis
 (2013)

---

## 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.


---

## Body

## 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 planey=0 of a Cartesian system of coordinates Oxy (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 uw(x) and the temperature Tw(x) of the stretching sheet is proportional to the distance x from the stagnation-point, where Tw(x)>T∞ with T∞ being the uniform temperature of the ambient fluid. The velocity of the flow external to the boundary layer is ue(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
(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=ueduedx+ν∂2u∂y2±gβ(T-T∞),(3)u∂T∂x+v∂T∂y=α∂2T∂y2,
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
(4)v(x,0)=0,u(x,0)=uw(x)=cx,T(x,0)=Tw(x)=T∞+bx,u(x,∞)=ue(x)=ax,T(x,∞)=T∞,
where a, b, and c are positive constants. The continuity equation can be satisfied by introducing a stream function Ψ, such that
(5)u=∂Ψ∂y,v=-∂Ψ∂x.Figure 1
Physical model and coordinate system.Therefore, from (1)–(3) with (5) we have
(6)ΨyΨyx-ΨxΨyy+νΨyyy-ue(x)ddxue(x)±β(T-T∞)=0,ΨyTx-ΨxTy-αTyy=0.The boundary conditions (4) will be as
(7)Ψx(x,0)=0,Ψy(x,0)=cx,T(x,0)=Tw(x)=T∞+bx,Ψy(x,∞)=ax,T(x,∞)=T∞.


## 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,ue,T) given by
(8)x*=x+εξ(x,y;Ψ,ue,T)+O(ε2)=eεXx,y*=y+εγ(x,y;Ψ,ue,T)+O(ε2)=eεXy,Ψ*=Ψ+εΦ(x,y;Ψ,ue,T)+O(ε2)=eεXΨ,ue*=ue+εU(x,y;Ψ,ue,T)+O(ε2)=eεXue,T*=T+εΥ(x,y;Ψ,ue,T)+O(ε2)=eεXT,
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
(9)X≡ξ∂∂x+γ∂∂y+ϕ∂∂Ψ+U∂∂ue+Υ∂∂T.
Equivalently, we can obtain (x*,y*;Ψ*,ue*,T*) by solving
(10)dx*dε=ξ(x,y;Ψ,ue,T),dy*dε=γ(x,y;Ψ,ue,T),dΨ*dε=Φ(x,y;Ψ,ue,T),due*dε=U(x,y;Ψ,ue,T),dT*dε=Υ(x,y;Ψ,ue,T),
subject to initial conditions, (x*,y*;Ψ*,ue*,T*)|ε=0≡(x,y;Ψ,ue,T). If X is left invariant by the transformation (x,y;Ψ,ue,T)→(x*,y*;Ψ*,ue*,T*), then the solutions Ψ=Ψ(x,y), ue=ue(x), and T=T(x,y) are invariant under the symmetry (9) if
(11)φΨ=X(Ψ-Ψ(x,y))|Ψ=Ψ(x,y)=0,φue=X(ue-ue(x))|ue=ue(x,y)=0,φT=X(T-T(x,y))|T=T(x,y)=0.
Assume
(12)Π1=ΨyΨyx-ΨxΨyy+νΨyyy-ue(ue)x±β(T-T∞)=0,Π2=ΨyTx-ΨxTy-αTyy=0.
The vector X as (9) is a Lie point symmetry vector filed for, (6) if
(13)X[3](Πj)|Πj=0=0,j=1,2,
where
(14)X[3]≡ξ∂∂x+γ∂∂y+Φ∂∂Ψ+U∂∂ue+Υ∂∂T+Φx∂∂Ψx+Φy∂∂Ψy+Υx∂∂Tx+Υy∂∂Ty+Υyy∂∂Tyy+Ux∂∂(ue)x+Φyx∂∂Ψyx+Φyy∂∂Ψyy+Φyyy∂∂Ψyyy
is the third prolongation of X. The components Φx, Φy, Υx, Υy, Ux, Φyx, Φyy, and Φyyy can be determined from the following expressions
(15)Φx=DxΦ-ΨxDxξ-ΨyDxγ,Φy=DyΦ-ΨxDyξ-ΨyDyγ,Υx=DxΥ-TxDxξ-TyDxγ,Υy=DyΥ-TxDyξ-TyDyγ,Υyy=DyΥy-TyxDyξ-TyyDyγ,Ux=DxU-(ue)xDxξ-(ue)yDxγ,Φxy=DyΦx-ΨxxDyξ-ΨxyDyγ,Φxy=DxΦy-ΨyxDxξ-ΨyyDxγ,Φyy=DyΦy-ΨyxDyξ-ΨyyDyγ,Φyyy=DyΦyy-ΨyyxDyξ-ΨyyyDyγ.
Here, Dx and Dy are introduced as the following total derivatives:
(16)Dx≡∂x+Ψx∂Ψ+(ue)x∂ue+Tx∂T+Ψxx∂Ψx+(ue)xx∂(ue)x+Txx∂Tx+Ψxy∂Ψy+Txy∂Ty+⋯,Dy≡∂y+Ψy∂Ψ+(ue)y∂ue+Ty∂T+Ψyy∂Ψy+(ue)yy∂(ue)y+Tyy∂Ty+Ψyx∂Ψx+Tyx∂Tx+⋯.
Form (13) we have the system of linear differential equations as follows:
(17)-U(ue)x±βΥ-ΦxΨyy+ΦyΨyx-Uxue+ΦyxΨy-ΦyyΨx+νΦyyy=0,-ΦxTy+ΦyTx+ΥxΨy-ΥyΨx-αΥyy=0.
Replacing the functions ϕx, ϕy, ϕyx, ϕyy, ϕyyy, Ux, Υx, and Υy given by the relation (15) and eliminating any dependence between partial differential derivatives of the functions Ψ, ue, 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 functionsΨ, ue, 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
(18)ζ=K0+K1x,γ=K2,Φ=K3+K4x+K5y+K12Ψ+K6ue,U=K7+K8x+K9y+K10Ψ+K11ue+K12T,Υ=K12+K13x+K14Ψ+K15ue+K16T,
where K0,…,K16∈ℝ, and consequently the infinitesimal generator of the symmetry group G is
(19)X=K0∂∂x+K1(x∂∂x+12Ψ∂∂Ψ+12T∂∂ue)+K2∂∂y,+K3∂∂Ψ+K4x∂∂Ψ+K5y∂∂Ψ+K6ue∂∂Ψ+K7∂∂ue,+K8x∂∂ue+K9y∂∂ue+K10Ψ∂∂ue+K11ue∂∂ue+K12∂∂T,+K13x∂∂T+K14Ψ∂∂T+K15ue∂∂T+K16T∂∂T.
From which, the system of nonlinear (6) has the sixteen-parameter Lie group of point symmetries generated by
(20)X1≡∂∂x,X2≡x∂∂x+12Ψ∂∂Ψ+12T∂∂ue,X3≡∂∂y,X4≡∂∂Ψ,X5≡x∂∂Ψ,X6≡y∂∂Ψ,X7≡ue∂∂Ψ,X8≡∂∂ue,X9≡x∂∂ue,X10≡y∂∂ue,X11≡Ψ∂∂ue,X12≡ue∂∂ue,X13≡∂∂T,X14≡x∂∂T,X15≡Ψ∂∂T,X16≡ue∂∂T,X17≡T∂∂T.
For X1 up to X17, respectively, the characteristic φ=(φΨ,φue,φT) has the components as follows:
(21)φX1=(-Ψx,-(ue)x,-Tx),φX2=(-xΨx+Ψ,-x(ue)x+12T,-xTx),φX3=(-Ψy,0,-Ty),φX4=(1,0,0),φX5=(x,0,0),φX6=(y,0,0),φX7=(ue,0,0),φX8=(0,1,0),φX9=(0,x,0),φX10=(0,y,0),φX11=(0,Ψ,0),φX12=(0,ue,0),φX13=(0,0,1),φX14=(0,0,x),φX15=(0,0,Ψ),φX16=(0,0,ue),φX17=(0,0,T).
Equation (21) shows that no solutions are invariant under the groups generated by X1 and X3 up to X17. For X2, the characteristic φ=(φΨ,φue,φT) has the components
(22)φΨ=-xΨx+Ψ,φue=-x(ue)x+12T,φT=-xTx.
Therefore, the general solutions of the invariant surface conditions (11) by using the boundary conditions (7) are as follows:
(23)Ψ(x,y)=cxF(y),ue(x)=ax,T(x,y)=(Tw-T∞)H(y)+T∞.
Substitution from (23) into (6) yields
(24)cx[c(F′(y))2-cF(y)F′′(y)-νF′′′(y)]-a2x±gβ(Tw-T∞)H(y)=0,cF(y)H′(y)+αH′′(y)=0.
For simplifying we can use F(y)=ν/cf(η) and H(y)=θ(η) with η=c/vy. Therefore, we have
(25)f′′′(η)+f(η)f′′(η)-(f′(η))2+(ac)2±λθ(η)=0,(26)1Prθ′′(η)+f(η)θ′(η)=0,
where Pr=ν/c is the Prandtl number and λ=Grx/Rex2 is the buoyancy parameter with Grx=gβ(Tw-T∞)x3/v2 is the local Grashof number and Rex2=uwx/v is the local Reynolds number. Equations (25) and (26) subject to the boundary (7) become(27)f(0)=0,f′(0)=1,θ(0)=1,f′(∞)=ac,θ(∞)=0.
When λ=0 and a/c=1, the solution of (25) subject to boundary condition (27) is given by
(28)f(η)=η.


### 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 off′′(0) for different values of a/c when the buoyancy force term λθ in  (25) is absent.
a
/
c
Mahapatra and Gupta [25]
Nazar et al. [11]
Ishak et al. [28]
Present work
0
-
1.0000
0.1
-
0.9694
-
0.9694
-
0.9694
-
0.9694
0.2
-
0.9181
-
0.9181
-
0.9181
-
0.9181
0.5
-
0.6673
-
0.6673
-
0.6673
-
0.6673
2
2.0175
2.0176
2.0175
2.0175
3
4.7293
4.7296
4.7294
4.7293
4
—
—
—
8.0004
10
—
—
—
36.2574The 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 off′′(0) and -θ′(0) for a/c=1, λ=1 and various Pr.
Pr
Buoyancy assisting flow
Buoyancy opposing flow
f
′′
(
0
)
-
θ
′
(
0
)
f
′′
(
0
)
-
θ
′
(
0
)
0
0.604
0.200
-
0.663
0.200
0.72
0.419
0.702
-
0.448
0.647
6.8
0.219
2.101
-
0.224
2.059
20
0.145
3.584
-
0.146
3.552
40
0.109
5.058
-
0.109
5.034
60
0.091
6.191
-
0.091
6.169
80
0.080
7.146
-
0.080
7.127
100
0.072
7.987
-
0.073
7.970
200
0.053
11.290
-
0.053
11.277
400
0.038
15.962
-
0.038
15.953
1000
0.024
25.234
-
0.024
25.228The resulting profiles of dimensionless velocityf′(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 ofa/c when Pr=1 and λ=1.Figure 3
Temperature profiles for some values ofa/c when Pr=1 and λ=1; solid line: assisting flow and dash line: opposing flow.The opposite trend occurs for opposing flow. From Figure3, 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 withλ 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 withλ of -θ′(0) for some values of a/c when Pr=1; solid line: assisting flow and dash line: opposing flow.Figure 6
Variation withλ 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 withλ 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 velocityf′(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 ofλ when Pr=1 and a/c=1.Figure 9
Temperature profiles for some values ofλ when Pr=1 and a/c=1.The values off′′(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 off′′(0) and -θ′(0) for a/c=1, Pr =1, and various λ.
λ
Buoyancy assisting flow
Buoyancy opposing flow
f
′′
(
0
)
-
θ
′
(
0
)
f
′′
(
0
)
-
θ
′
(
0
)
0
10.6673
1.2346
2.6696
2.1381
2
16.7304
1.3282
0.65818
2.1298
4
22.1328
1.3987
−1.4727
2.1207
6
27.1134
1.4562
−3.7690
2.1104
8
31.7902
1.5051
−6.3244
2.0982
10
36.2321
1.5478
−9.4295
2.0816Table 4
Values off′′(0) and -θ′(0) for λ=1, Pr =1, and various a/c.
a
/
c
Buoyancy assisting flow
Buoyancy opposing flow
f
′′
(
0
)
-
θ
′
(
0
)
f
′′
(
0
)
-
θ
′
(
0
)
0.0
6.8627
1.2032
1.0149
0.9356
0.5
8.3387
1.2788
3.3862
1.1005
1.0
11.379
1.3952
7.5631
1.2931
1.5
15.567
1.5207
12.692
1.4596
2.0
20.578
1.6420
18.479
1.6032


## 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,ue,T) given by
(8)x*=x+εξ(x,y;Ψ,ue,T)+O(ε2)=eεXx,y*=y+εγ(x,y;Ψ,ue,T)+O(ε2)=eεXy,Ψ*=Ψ+εΦ(x,y;Ψ,ue,T)+O(ε2)=eεXΨ,ue*=ue+εU(x,y;Ψ,ue,T)+O(ε2)=eεXue,T*=T+εΥ(x,y;Ψ,ue,T)+O(ε2)=eεXT,
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
(9)X≡ξ∂∂x+γ∂∂y+ϕ∂∂Ψ+U∂∂ue+Υ∂∂T.
Equivalently, we can obtain (x*,y*;Ψ*,ue*,T*) by solving
(10)dx*dε=ξ(x,y;Ψ,ue,T),dy*dε=γ(x,y;Ψ,ue,T),dΨ*dε=Φ(x,y;Ψ,ue,T),due*dε=U(x,y;Ψ,ue,T),dT*dε=Υ(x,y;Ψ,ue,T),
subject to initial conditions, (x*,y*;Ψ*,ue*,T*)|ε=0≡(x,y;Ψ,ue,T). If X is left invariant by the transformation (x,y;Ψ,ue,T)→(x*,y*;Ψ*,ue*,T*), then the solutions Ψ=Ψ(x,y), ue=ue(x), and T=T(x,y) are invariant under the symmetry (9) if
(11)φΨ=X(Ψ-Ψ(x,y))|Ψ=Ψ(x,y)=0,φue=X(ue-ue(x))|ue=ue(x,y)=0,φT=X(T-T(x,y))|T=T(x,y)=0.
Assume
(12)Π1=ΨyΨyx-ΨxΨyy+νΨyyy-ue(ue)x±β(T-T∞)=0,Π2=ΨyTx-ΨxTy-αTyy=0.
The vector X as (9) is a Lie point symmetry vector filed for, (6) if
(13)X[3](Πj)|Πj=0=0,j=1,2,
where
(14)X[3]≡ξ∂∂x+γ∂∂y+Φ∂∂Ψ+U∂∂ue+Υ∂∂T+Φx∂∂Ψx+Φy∂∂Ψy+Υx∂∂Tx+Υy∂∂Ty+Υyy∂∂Tyy+Ux∂∂(ue)x+Φyx∂∂Ψyx+Φyy∂∂Ψyy+Φyyy∂∂Ψyyy
is the third prolongation of X. The components Φx, Φy, Υx, Υy, Ux, Φyx, Φyy, and Φyyy can be determined from the following expressions
(15)Φx=DxΦ-ΨxDxξ-ΨyDxγ,Φy=DyΦ-ΨxDyξ-ΨyDyγ,Υx=DxΥ-TxDxξ-TyDxγ,Υy=DyΥ-TxDyξ-TyDyγ,Υyy=DyΥy-TyxDyξ-TyyDyγ,Ux=DxU-(ue)xDxξ-(ue)yDxγ,Φxy=DyΦx-ΨxxDyξ-ΨxyDyγ,Φxy=DxΦy-ΨyxDxξ-ΨyyDxγ,Φyy=DyΦy-ΨyxDyξ-ΨyyDyγ,Φyyy=DyΦyy-ΨyyxDyξ-ΨyyyDyγ.
Here, Dx and Dy are introduced as the following total derivatives:
(16)Dx≡∂x+Ψx∂Ψ+(ue)x∂ue+Tx∂T+Ψxx∂Ψx+(ue)xx∂(ue)x+Txx∂Tx+Ψxy∂Ψy+Txy∂Ty+⋯,Dy≡∂y+Ψy∂Ψ+(ue)y∂ue+Ty∂T+Ψyy∂Ψy+(ue)yy∂(ue)y+Tyy∂Ty+Ψyx∂Ψx+Tyx∂Tx+⋯.
Form (13) we have the system of linear differential equations as follows:
(17)-U(ue)x±βΥ-ΦxΨyy+ΦyΨyx-Uxue+ΦyxΨy-ΦyyΨx+νΦyyy=0,-ΦxTy+ΦyTx+ΥxΨy-ΥyΨx-αΥyy=0.
Replacing the functions ϕx, ϕy, ϕyx, ϕyy, ϕyyy, Ux, Υx, and Υy given by the relation (15) and eliminating any dependence between partial differential derivatives of the functions Ψ, ue, 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 functionsΨ, ue, 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
(18)ζ=K0+K1x,γ=K2,Φ=K3+K4x+K5y+K12Ψ+K6ue,U=K7+K8x+K9y+K10Ψ+K11ue+K12T,Υ=K12+K13x+K14Ψ+K15ue+K16T,
where K0,…,K16∈ℝ, and consequently the infinitesimal generator of the symmetry group G is
(19)X=K0∂∂x+K1(x∂∂x+12Ψ∂∂Ψ+12T∂∂ue)+K2∂∂y,+K3∂∂Ψ+K4x∂∂Ψ+K5y∂∂Ψ+K6ue∂∂Ψ+K7∂∂ue,+K8x∂∂ue+K9y∂∂ue+K10Ψ∂∂ue+K11ue∂∂ue+K12∂∂T,+K13x∂∂T+K14Ψ∂∂T+K15ue∂∂T+K16T∂∂T.
From which, the system of nonlinear (6) has the sixteen-parameter Lie group of point symmetries generated by
(20)X1≡∂∂x,X2≡x∂∂x+12Ψ∂∂Ψ+12T∂∂ue,X3≡∂∂y,X4≡∂∂Ψ,X5≡x∂∂Ψ,X6≡y∂∂Ψ,X7≡ue∂∂Ψ,X8≡∂∂ue,X9≡x∂∂ue,X10≡y∂∂ue,X11≡Ψ∂∂ue,X12≡ue∂∂ue,X13≡∂∂T,X14≡x∂∂T,X15≡Ψ∂∂T,X16≡ue∂∂T,X17≡T∂∂T.
For X1 up to X17, respectively, the characteristic φ=(φΨ,φue,φT) has the components as follows:
(21)φX1=(-Ψx,-(ue)x,-Tx),φX2=(-xΨx+Ψ,-x(ue)x+12T,-xTx),φX3=(-Ψy,0,-Ty),φX4=(1,0,0),φX5=(x,0,0),φX6=(y,0,0),φX7=(ue,0,0),φX8=(0,1,0),φX9=(0,x,0),φX10=(0,y,0),φX11=(0,Ψ,0),φX12=(0,ue,0),φX13=(0,0,1),φX14=(0,0,x),φX15=(0,0,Ψ),φX16=(0,0,ue),φX17=(0,0,T).
Equation (21) shows that no solutions are invariant under the groups generated by X1 and X3 up to X17. For X2, the characteristic φ=(φΨ,φue,φT) has the components
(22)φΨ=-xΨx+Ψ,φue=-x(ue)x+12T,φT=-xTx.
Therefore, the general solutions of the invariant surface conditions (11) by using the boundary conditions (7) are as follows:
(23)Ψ(x,y)=cxF(y),ue(x)=ax,T(x,y)=(Tw-T∞)H(y)+T∞.
Substitution from (23) into (6) yields
(24)cx[c(F′(y))2-cF(y)F′′(y)-νF′′′(y)]-a2x±gβ(Tw-T∞)H(y)=0,cF(y)H′(y)+αH′′(y)=0.
For simplifying we can use F(y)=ν/cf(η) and H(y)=θ(η) with η=c/vy. Therefore, we have
(25)f′′′(η)+f(η)f′′(η)-(f′(η))2+(ac)2±λθ(η)=0,(26)1Prθ′′(η)+f(η)θ′(η)=0,
where Pr=ν/c is the Prandtl number and λ=Grx/Rex2 is the buoyancy parameter with Grx=gβ(Tw-T∞)x3/v2 is the local Grashof number and Rex2=uwx/v is the local Reynolds number. Equations (25) and (26) subject to the boundary (7) become(27)f(0)=0,f′(0)=1,θ(0)=1,f′(∞)=ac,θ(∞)=0.
When λ=0 and a/c=1, the solution of (25) subject to boundary condition (27) is given by
(28)f(η)=η.


## 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 off′′(0) for different values of a/c when the buoyancy force term λθ in  (25) is absent.
a
/
c
Mahapatra and Gupta [25]
Nazar et al. [11]
Ishak et al. [28]
Present work
0
-
1.0000
0.1
-
0.9694
-
0.9694
-
0.9694
-
0.9694
0.2
-
0.9181
-
0.9181
-
0.9181
-
0.9181
0.5
-
0.6673
-
0.6673
-
0.6673
-
0.6673
2
2.0175
2.0176
2.0175
2.0175
3
4.7293
4.7296
4.7294
4.7293
4
—
—
—
8.0004
10
—
—
—
36.2574The 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 off′′(0) and -θ′(0) for a/c=1, λ=1 and various Pr.
Pr
Buoyancy assisting flow
Buoyancy opposing flow
f
′′
(
0
)
-
θ
′
(
0
)
f
′′
(
0
)
-
θ
′
(
0
)
0
0.604
0.200
-
0.663
0.200
0.72
0.419
0.702
-
0.448
0.647
6.8
0.219
2.101
-
0.224
2.059
20
0.145
3.584
-
0.146
3.552
40
0.109
5.058
-
0.109
5.034
60
0.091
6.191
-
0.091
6.169
80
0.080
7.146
-
0.080
7.127
100
0.072
7.987
-
0.073
7.970
200
0.053
11.290
-
0.053
11.277
400
0.038
15.962
-
0.038
15.953
1000
0.024
25.234
-
0.024
25.228The resulting profiles of dimensionless velocityf′(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 ofa/c when Pr=1 and λ=1.Figure 3
Temperature profiles for some values ofa/c when Pr=1 and λ=1; solid line: assisting flow and dash line: opposing flow.The opposite trend occurs for opposing flow. From Figure3, 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 withλ 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 withλ of -θ′(0) for some values of a/c when Pr=1; solid line: assisting flow and dash line: opposing flow.Figure 6
Variation withλ 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 withλ 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 velocityf′(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 ofλ when Pr=1 and a/c=1.Figure 9
Temperature profiles for some values ofλ when Pr=1 and a/c=1.The values off′′(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 off′′(0) and -θ′(0) for a/c=1, Pr =1, and various λ.
λ
Buoyancy assisting flow
Buoyancy opposing flow
f
′′
(
0
)
-
θ
′
(
0
)
f
′′
(
0
)
-
θ
′
(
0
)
0
10.6673
1.2346
2.6696
2.1381
2
16.7304
1.3282
0.65818
2.1298
4
22.1328
1.3987
−1.4727
2.1207
6
27.1134
1.4562
−3.7690
2.1104
8
31.7902
1.5051
−6.3244
2.0982
10
36.2321
1.5478
−9.4295
2.0816Table 4
Values off′′(0) and -θ′(0) for λ=1, Pr =1, and various a/c.
a
/
c
Buoyancy assisting flow
Buoyancy opposing flow
f
′′
(
0
)
-
θ
′
(
0
)
f
′′
(
0
)
-
θ
′
(
0
)
0.0
6.8627
1.2032
1.0149
0.9356
0.5
8.3387
1.2788
3.3862
1.1005
1.0
11.379
1.3952
7.5631
1.2931
1.5
15.567
1.5207
12.692
1.4596
2.0
20.578
1.6420
18.479
1.6032


## 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.



---

*Source: 269420-2013-03-12.xml*