# Similarity Solutions of MHD Mixed Convection Flow with Variable Reactive Index, Magnetic Field, and Velocity Slip Near a Moving Horizontal Plate: A Group Theory Approach

Authors: W. A. Khan; Md. Jashim Uddin; A. I. Md.  Ismail
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The mixed convection of Newtonian fluid flow along a moving horizontal plate with higher-order chemical reaction, variable concentration reactant, and variable wall temperature and concentration is considered. Velocity slip and the thermal convective boundary conditions are applied at the plate surface. The governing partial differential equations are transformed into similarity equations via dimensionless similarity transformations developed by one-parameter continuous group method. The numerical solutions of the transformed ordinary differential equations are constructed for velocity, temperature and concentration functions, the skin friction factor, the rate of heat, and the rate of mass transfer using an implicit finite difference numerical technique. The investigated parameters are buoyancy parametersλ1, λ2, chemical reaction parameter K, suction/injection parameter fw, velocity slip parameter a convective heat transfer parameter γ, magnetic parameter M, Prandtl number Pr and Schmidt number, Sc. Comparison with results from the open literature shows a very good agreement.


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## Body

## 1. Introduction

A convection situation involving both free and forced convection is known as mixed convection  and has been an important topic because of its application in electronic equipment cooled by a fan and flows in the ocean and in the atmosphere ([1, 2]). In mixed convection flows, the forced and the free convection effects are of comparable magnitude. Thus, mixed convection occurs if the effect of buoyancy forces on a forced flow or vice versa is significant. The laminar mixed convection takes place in various applications in thermal engineering and science and has drawn attention from researchers in the past decades. Examples of these applications include atmospheric boundary layer flows, solar energy systems, boilers, compact heat exchangers, and cooling of electronic devices [3, 4]. Many studies exist for the mixed convection boundary layer flow about vertical/inclined/horizontal plate and wedge surfaces immersed in a viscous fluid. The details of mixed convection can be found in books by Chen and Armaly [5], Jaluria [6], Gebhart et al. [7], Bejan [8], Pop and Ingham [9] and in the articles in the literatures by Lloyd and Sparrow [10], Schneider [11], Wilks [12], Yao [13], Ramachandran et al. [14], Moulic and Yao, [15], Daskalakis et al. [16], Fan et al. [17], Magyari et al. [18], Hassanien et al. [19], Aydin and Kaya [20], Ishak [21], Moulic and Yao [22], Datta et al. [23], Ishak et al. [24], Magyari [25].In all these studies, conventional no slip boundary conditions were applied at the surface. However, fluid flows in micro/nanoscale such as micronozzles, micropumps, microturbines, microheat exchangers, microvalves, turbines, sensors, and actuators are important for micro and nanoscience and the conventional no slip boundary condition at the solid fluid interface must be replaced with the slip condition [26]. The slip condition states that the tangential component of the velocity at the surface is equal to the velocity gradient normal to the surface ([27, 28]). Slips may occur on a stationary/moving boundary when the fluid is in particulate form such as in the form of emulsions, suspensions, foams, and polymer solutions [29, 30]. An excellent review for the fundamentals, theory, and applications of these microscale systems may be found in the text by Duncan and Peterson [31] Gad-el-Hak [32]. Slips effect on flow field has been studied by various authors; Fang and Lee [33], Martin and Boyd [34], Matthews and Hill [35], Wang [36]. Cao and Baker [37] studied laminar mixed convection over an isothermal vertical plate with first-order momentum and thermal discontinuities at the wall. The hydrodynamic and thermal slip flow boundary layer over a flat plate with constant heat flux boundary condition has been investigated by Hak [27] who concluded that as the slip parameter increases, the slip velocity increases and the wall shear stress decreases.Due to the nonlinearities of the governing equations, no analytical solution is available and the nonlinear equations usually solved numerically subject to initial/boundary conditions. The solution of differential equations with/without initial/boundary conditions is presented by different authors such as Ames [38], Boyce [39], Na [40], Zauderer [41], Seshadri and Na [42], Kincaid and Cheney [43], Strauss [44], Aziz [45], and White and Subramanian [46].We study combined convective Newtonian fluid flow adjacent to a moving horizontal permeable flat plate with higher order chemical reaction taking into account velocity slip and convective surface boundary conditions. The governing boundary layer equations are converted into a two point boundary value problem using similarity transformations developed by group method and the transformed equations are solved by means of an implicit finite difference numerical method. Graphs are plotted for velocity, temperature and concentration profiles as well as the skin friction factor, the rate of heat transfer and the rate of mass transfer to exhibits the influences of the parameters involving in our problem.


## 2. Mathematical Formulations of the Problem

Consider a continuous moving permeable horizontal flat plate which moves with a non-uniform velocityU(x) as shown in Figure 1. A variable magnetic field of strength B(x) is applied to the normal direction of the plate. The bottom surface of the plate is heated by convection from a hot fluid of temperature Tf(x) which provides a heat transfer coefficient hf(x). The viscous dissipation in the energy equation is neglected. Assume that fluid properties are constant accepts in buoyancy term. We will further assume the magnetic Reynolds number; the electric field owing to the polarization of charges and Hall effects are negligible. Under the forgoing assumptions the governing equations in dimensional form can be written as [47, 48]
(2.1)∂u∂x+∂v∂y=0,u∂u∂x+v∂u∂y=-1ρ∂p∂x+ν∂2u∂y2-σB2(x)ρu-1ρ∂p∂y+gβT(T-T∞)+gβC(C-C∞)=0,u∂T∂x+v∂T∂y=α∂2T∂y2,u∂C∂x+v∂C∂y=D∂2C∂y2-k0(C-C∞)n.
The boundary conditions are [49]
(2.2)u=U(x)+N1(x)ν∂u∂y,v=-vw(x),-k∂T∂y=hf(x)[Tf(x)-Tw],C=Cw(x)aty=0,u→0,v→0,T→T∞,C→C∞asy→∞,
where vw(x): velocity normal to the plate, ν: coefficient of kinematic viscosity, ρ: density of the fluid, σ: electric conductivity, p: pressure, k: thermal  conductivity, α: thermal diffusivity, D: mass diffusivity of species in fluid, βT: volumetric thermal coefficient, βC: volumetric concentration coefficient, g: acceleration due to gravity, k0: reaction rate, n: order of chemical reaction, and N1(x): velocity slip factor with dimension (velocity)-1.Figure 1
Physical configuration and coordinate system the problem.Introducing stream functionψ, dimensionless temperature function θ and concentration function ϕ are defined by
(2.3)u=∂ψ∂y,v=-∂ψ∂x,θ=T-T∞Tf-T∞,ϕ=C-C∞Cw-C∞.
Using (2.3) we get from (2.1)
(2.4)∂ψ∂y∂2ψ∂x∂y-∂ψ∂x∂2ψ∂y2=-1ρ∂p∂x+ν∂3ψ∂y3-σB2(x)ρ∂ψ∂y,(2.5)-1ρ∂p∂y+gβTθΔT+βCϕΔC=0,(2.6)∂ψ∂y∂∂x(θΔT)-∂ψ∂x∂∂y(θΔT)=α∂2∂y2(θΔT),(2.7)∂ψ∂y∂∂x(ϕΔC)-∂ψ∂x∂∂y(ϕΔC)=D∂2∂y2(ϕΔC)-k0(ϕΔC)n.
Eliminating the pressure gradient terms from (2.4) and (2.5) by cross-differentiation, we get
(2.8)ν∂4ψ∂y4+∂ψ∂x∂3ψ∂y3-∂ψ∂y∂3ψ∂x∂y2-gβT(ΔTθ)x-gβT(ΔCϕ)x-σB2(x)ρψyy=0,α∂2∂y2(θΔT)+∂ψ∂x∂∂y(θΔT)-∂ψ∂y∂∂x(θΔT)=0,D∂2∂y2(ϕΔC)+∂ψ∂x∂∂y(ϕΔC)-∂ψ∂y∂∂x(ϕΔC)-k0(ϕΔC)n=0.
The boundary conditions (2.2) become
(2.9)∂ψ∂y=U(x)+νN1(x)∂2ψ∂y2,-∂ψ∂x=vw(x)∂θ∂y=-hf(x)k[1-θ(0)],ϕ=1aty=0,∂ψ∂y→0,∂ψ∂x→0,θ→0,ϕ→0asy→∞.


## 3. Application of Group Theory

We will now search group invariant solutions (similarity solutions) of (2.8) to (2.9) under a particular continuous one-parameter group. We define the following one-parameter group transformations Γ:
(3.1)x*=eεα1x,y*=eεα2y,ψ*=eεα3ψ,θ*=eεα4θ,ϕ*=eεα5ϕ,B*=eεα6BΔT*=eεα7ΔT,ΔC*=eεα8ΔC,U=eεα9U,vw*=eεα10vw,hf*=eεα11hf
Here ε is the parameter of the group and αi(i=1,2,3,…,11) are the arbitrary real numbers whose interrelationship will be determined by our investigation.  Transformations in (3.1) may be considered as point transformations which transform the coordinates
(3.2)(x,y,ψ,θ,ϕ,B,ΔT,ΔC,U,vw,hf)to(x*,y*,ψ*,θ*,ϕ*,B*,ΔT*,ΔC*,U*,vw*,hf*).
We now find the relationship among the exponents αi such that (2.8) with the boundary conditions in (2.9) are invariant (the structure of the equations remains same before and after the transformations) under the transformation group in (3.1). Substituting transformations in (3.1) into (2.8) and equating various exponents of e for constant conformally invariant, we get the following algebraic equations:
(3.3)4α2-α3=α1+3α2-2α3=α1-α4-α7=α1-α5-α8=2α2-α3-2α6,2α2-α4-α7=α1+α2-α3-α4-α7,2α2-α5-α8=α1+α2-α3-α5-α8=n(-α5-α8).
Using (3.1) in boundary conditions (2.9) and equating various exponents of  e, we get
(3.4)α2-α3=-α9,α1-α3=-α10,α2=-α11.
Without loss of generality, we may put  α4=α5=0.Solving (3.3) and (3.4) we have the following relationship among the exponents
(3.5)α1=(7-5n)4α8,α2=(1-n)2α8,α3=(5-3n)4α8,α4=0,α5=0,α7=α8,α9=(3-n)4α8,α6=α10=α11=(n-1)2α8.
It can be easily verified that (2.8) and the boundary conditions (2.9) are invariant under the transformations in (3.1) subject to the conditions in (3.5).


### 3.1. Similarity Transformations

Using the relationships in (3.5), we have from (3.1)
(3.6)x*=eε(7-5n)/4α8x,y*=eε(1-n)/2α8y,ψ*=eε(5-3n)/4α8ψ,θ*=θ,ϕ=ϕ,B*=eε(n-1)/2α8B,ΔT*=eεα8ΔT,ΔC*=eεα8ΔC,U*=eε(3-n)/4α8U,vw*=eεα8(n-1)/2vw,hf*=eεα8(n-1)/2hf.
Expanding each of the transformations in (3.6) in powers of ε, keeping the first order term and neglecting higher order terms, we get the following characteristic equations:
(3.7)4dx(7-5n)x=2dy(1-n)y=4dψ(5-3n)ψ=dθ0=dϕ0=2dB(n-1)B=d(ΔT)ΔT=d(ΔC)ΔC=4dU(n-3)U=2dvw(n-1)vw=2dhf(n-1)hf.
Solving the system of first order linear differential equations in (3.7), we get
(3.8)η*=β1yx(2n-2)/(7-5n),ψ*=β2x(5-3n)/(7-5n)f(η),θ*=θ,ϕ*=ϕ,B*=β3x(2n-2)/(7-5n),ΔT*=β4x4/(7-5n),ΔC*=β5x4/(7-5n),U*=β6x(3-n)/(7-5n),vw*=β7x(2n-2)/(7-5n),hf*=β8x(2n-2)/(7-5n),
where βi(i=1,2,3,…8) are real constants. To get dimensionless form of the transformations, we define the following dimensionless transformations:
(3.9)η=ReyL(xL)(2n-2)/(7-5n),ψ=νRe(xL)(5-3n)/(7-5n)f(η),θ=θ(η),ϕ=ϕ(η),B=B0(xL)(2n-2)/(7-5n),ΔT=(ΔT)0(xL)4/(7-5n),ΔC=(ΔC)0(xL)4/(7-5n),U=U0(xL)(3-n)/(7-5n),vw=v0(xL)(2n-2)/(7-5n),hf(x)=h0(xL)(2n-2)/(7-5n),
where f(η), θ(η) and ϕ(η), represent dimensionless velocity, temperature, and concentration functions, respectively. B0,  (ΔT)0,  (ΔC)0,  U0,  v0 and h0 are constants.Using (3.9), (2.8) become
(3.10)f′′′′+5-3n7-5nff′′′-n+17-5nf′f′′-Mf′-λ1[2n-27-5nηθ′+47-5nθ]-λ2[2n-27-5nηϕ′+47-5nϕ]=0,Pr-1θ′′+5-3n7-5nfθ′-47-5nf'θ=0,Sc-1ϕ′′+5-3n7-5nfϕ′-47-5nf′ϕ-Kϕn=0,
subject to the boundary conditions
(3.11)f(0)=7-5n5-3nfw,f′(0)=1+af′′(0),θ′(0)=-γ[1-θ(0)],ϕ(0)=1,f(∞)=f′(∞)=θ(∞)=ϕ(∞)=0,
where fw=v0L/νRe is the suction/injection parameter, γ=Lh0/Rek is the convective heat transfer parameter, λ1=Gr/Re2.5 is the thermal buoyancy parameter, λ2=Gc/Re2.5 is the concentration buoyancy parameter, M=σB02L2/ρνRe is the magnetic parameter, K=L2k0(ΔC)0n-1/νRe is the reaction parameter, Gr=gβc(ΔT)0L3/ν2 is the thermal Grashof number, Gc=gβc(ΔT)0L3/ν2 is the solutal Grashof number, and a=νReN0/L is the velocity slip parameter. Note that similarity solutions will exist if N1(x)=N0x(2-2n)/(7-5n),N0 is a constant.The physical quantities of interest are the local skin friction coefficientCfx, local Nusselt number Nux (rate of heat transfer), and local Sherwood number Shx (rate of mass transfer), which are defined as
(3.12)Cfx=-τWρU2,Nux=-xqwkΔT,Shx=-mwxDΔC,
where τw is the wall shear stress, qw is the wall heat flux, mw is the quantity of mass transfer through the unit area of the surface, which are given by
(3.13)τw=μ(∂u∂y)y=0,qw=k(∂T∂y)y=0,mw=D(∂C∂y)y=0.
It can be shown that physical quantities are proportional to -f″(0),  -θ′(0) and -ϕ′(0).


## 3.1. Similarity Transformations

Using the relationships in (3.5), we have from (3.1)
(3.6)x*=eε(7-5n)/4α8x,y*=eε(1-n)/2α8y,ψ*=eε(5-3n)/4α8ψ,θ*=θ,ϕ=ϕ,B*=eε(n-1)/2α8B,ΔT*=eεα8ΔT,ΔC*=eεα8ΔC,U*=eε(3-n)/4α8U,vw*=eεα8(n-1)/2vw,hf*=eεα8(n-1)/2hf.
Expanding each of the transformations in (3.6) in powers of ε, keeping the first order term and neglecting higher order terms, we get the following characteristic equations:
(3.7)4dx(7-5n)x=2dy(1-n)y=4dψ(5-3n)ψ=dθ0=dϕ0=2dB(n-1)B=d(ΔT)ΔT=d(ΔC)ΔC=4dU(n-3)U=2dvw(n-1)vw=2dhf(n-1)hf.
Solving the system of first order linear differential equations in (3.7), we get
(3.8)η*=β1yx(2n-2)/(7-5n),ψ*=β2x(5-3n)/(7-5n)f(η),θ*=θ,ϕ*=ϕ,B*=β3x(2n-2)/(7-5n),ΔT*=β4x4/(7-5n),ΔC*=β5x4/(7-5n),U*=β6x(3-n)/(7-5n),vw*=β7x(2n-2)/(7-5n),hf*=β8x(2n-2)/(7-5n),
where βi(i=1,2,3,…8) are real constants. To get dimensionless form of the transformations, we define the following dimensionless transformations:
(3.9)η=ReyL(xL)(2n-2)/(7-5n),ψ=νRe(xL)(5-3n)/(7-5n)f(η),θ=θ(η),ϕ=ϕ(η),B=B0(xL)(2n-2)/(7-5n),ΔT=(ΔT)0(xL)4/(7-5n),ΔC=(ΔC)0(xL)4/(7-5n),U=U0(xL)(3-n)/(7-5n),vw=v0(xL)(2n-2)/(7-5n),hf(x)=h0(xL)(2n-2)/(7-5n),
where f(η), θ(η) and ϕ(η), represent dimensionless velocity, temperature, and concentration functions, respectively. B0,  (ΔT)0,  (ΔC)0,  U0,  v0 and h0 are constants.Using (3.9), (2.8) become
(3.10)f′′′′+5-3n7-5nff′′′-n+17-5nf′f′′-Mf′-λ1[2n-27-5nηθ′+47-5nθ]-λ2[2n-27-5nηϕ′+47-5nϕ]=0,Pr-1θ′′+5-3n7-5nfθ′-47-5nf'θ=0,Sc-1ϕ′′+5-3n7-5nfϕ′-47-5nf′ϕ-Kϕn=0,
subject to the boundary conditions
(3.11)f(0)=7-5n5-3nfw,f′(0)=1+af′′(0),θ′(0)=-γ[1-θ(0)],ϕ(0)=1,f(∞)=f′(∞)=θ(∞)=ϕ(∞)=0,
where fw=v0L/νRe is the suction/injection parameter, γ=Lh0/Rek is the convective heat transfer parameter, λ1=Gr/Re2.5 is the thermal buoyancy parameter, λ2=Gc/Re2.5 is the concentration buoyancy parameter, M=σB02L2/ρνRe is the magnetic parameter, K=L2k0(ΔC)0n-1/νRe is the reaction parameter, Gr=gβc(ΔT)0L3/ν2 is the thermal Grashof number, Gc=gβc(ΔT)0L3/ν2 is the solutal Grashof number, and a=νReN0/L is the velocity slip parameter. Note that similarity solutions will exist if N1(x)=N0x(2-2n)/(7-5n),N0 is a constant.The physical quantities of interest are the local skin friction coefficientCfx, local Nusselt number Nux (rate of heat transfer), and local Sherwood number Shx (rate of mass transfer), which are defined as
(3.12)Cfx=-τWρU2,Nux=-xqwkΔT,Shx=-mwxDΔC,
where τw is the wall shear stress, qw is the wall heat flux, mw is the quantity of mass transfer through the unit area of the surface, which are given by
(3.13)τw=μ(∂u∂y)y=0,qw=k(∂T∂y)y=0,mw=D(∂C∂y)y=0.
It can be shown that physical quantities are proportional to -f″(0),  -θ′(0) and -ϕ′(0).


## 4. Results and Discussion

Equations (3.10) with boundary conditions (3.11) were solved using the dsolve command in MAPLE 14 with numeric option. Depending upon the nature of the ordinary differential equations, MAPLE uses a suitable scheme based on trapezoid or midpoint rule with Richardson extrapolation or deferred correction enhancement. In order to justify the accuracy of our numerical method, we compared our results with available data in Table 1 and found a good agreement. The influence of different parameters on the dimensionless velocity, temperature, and concentration is shown in Figures 2, 3, and 4, respectively. Figure 2(a) exhibits the influence of thermal buoyancy parameter λ1 and suction/injection parameter fw, whilst Figure 2(b) shows the effects of solutal buoyancy parameter λ2 and suction/injection parameter fw, on the dimensionless fluid velocity f'(η) profiles. We further observed that velocity increases with buoyancy parameters λ1 and λ2. Figure 2(c) describes the influence of magnetic field parameter M on velocity f'(η) profiles. It is found that as M increases, the fluid velocity decreases. The effect of velocity slip parameter on the dimensionless velocity is depicted in Figure 2(d). It is observed that an increase in the velocity slip reduces the dimensionless velocity at the surface. In all cases suction increases the velocity whilst injection decreases the velocity, as expected.Table 1
Comparison of results forf″(0) and −ϕ'(0) for different values of parameters.
Pr=0.72, Pr =0.72, λ1=1, λ2=1, K=1
Sc
Fan et al. [17]
Present results
f
″
(
0
)
ϕ
'
(
0
)
f
″
(
0
)
−ϕ'(0)
0.1
0.9098
0.13678
0.9099
0.1368
0.72
0.47511
0.28697
0.4751
0.287
5
0.23157
0.77288
0.2316
0.7729
Pr=0.72, Pr =0.72, Sc =0.72, λ2=1, K=1
λ
1
Fan et al. [17]
Present results
0
0.06194
0.29757
0.0619
0.2976
0.4
0.24155
0.29252
0.2415
0.2925
1
0.47511
0.28697
0.4751
0.287
Pr=0.72, Pr =0.72, λ1=1, Sc =0.72, K=1
λ
2
Fan et al. [17]
Present results
0
0.14425
0.29487
0.1442
0.2949
0.4
0.2835
0.29133
0.2835
0.2913
1
0.47511
0.28697
0.4751
0.287
Pr=0.72, Pr =0.72, λ1=1, λ2=1, Sc =0.72
K
Fan et al. [17]
Present results
0
0.53159
0.00015
0.5316
0.00015
1
0.47511
0.28697
0.4751
0.287
3
0.42378
0.61989
0.4238
0.6199Effects of different parameters on the dimensionless velocity.
(a)
(b)
(c)
(d)Effects of different parameters on the dimensionless temperature.
(a)
(b)
(c)
(d)Effects of different parameters on the dimensionless concentration.
(a)
(b)
(c)
(d)Figures3(a), 3(b), 3(c), and 3(d) display effects of various parameters on the dimensionless temperature function. The dimensionless temperature decreases with suction and increases with injection, as expected. Note that as we increase the values of λ1 and λ2, the values of θ(η)  decrease (Figures 3(a) and 3(b)). It is found that magnetic field increases temperature whilst Prandtl number decreases the temperature. Figures 4(a), 4(b), 4(c), and 4(d) exhibit influence of suction/injection fw, buoyancies λ1, λ2, the magnetic field M and the Schmidt number Sc on the dimensionless concentration ϕ(η) function. The dimensionless concentration decreases with the increase of fw. We also notice from Figures 4(a) and 4(b) as we increase the values of λ1 and λ2, values of ϕ(η) decrease. Figure 4(c) shows that the increasing values of magnetic field parameter lead to increase the concentration whilst Figure 4(d) shows that increasing values of Schmidt number Sc leads to decrease the concentration, as expected.Figures5(a) and 5(b) depict the variation of the buoyancies λ1, λ2, the slip a and the suction/injection fw parameters on the dimensionless shear stress. It is observed that suction enhanced the values of the dimensionless shear stress. We also see that the velocity slip a and the parameters λ1, λ2 increase the values of the dimensionless shear stress fall. The effect of Prandtl number Pr, convective heat transfer parameter γ, order of chemical reaction n, and the suction/injection parameter fw on the dimensionless rate of heat transfer is shown in Figures 6(a) and 6(b). From Figure 6(a) we can conclude that as Pr, γ and fw increase, the values of the dimensionless rate of heat transfer increase. An increase in the order of reaction causes decrease in the dimensionless rate of heat transfer (Figure 6(b)). The variation of the dimensionless rate of mass transfer with different parameters is shown in Figures 7(a) and 7(b). Like the dimensionless rate of heat transfer, the dimensionless rate of mass transfer increases with an increase in Sc, K, and fw (Figure 7(a)) and decreases with an increase in the order of reaction n.Effects of different parameters on the dimensionless shear stress.
(a)
(b)Effects of different parameters on the dimensionless rate of heat transfer.
(a)
(b)Effects of different parameters on the dimensionless rate of mass transfer.
(a)
(b)


## 5. Conclusions

MHD boundary layer equations for mixed convection of Newtonian fluids along a moving horizontal plate with velocity slip and thermal convective boundary condition are transformed into similarity equations using one-parameter continuous group method and then solved numerically using an implicit finite difference numerical method. Our analysis revealed that-f′′(0) increases with suction fw and decreases with a, λ1, and λ2 whilst the values of -θ'(0) enhance with fw, Pr, and γ. The values of  -ϕ'(0) enhance with fw, Sc and k.



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*Source: 183029-2012-10-02.xml*