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

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 parameters λ1, λ2, chemical reaction parameterK, suction/ injection parameter fw, velocity slip parameter a convective heat transfer parameter γ , magnetic parameterM, Prandtl number Pr and Schmidt number, Sc. Comparison with results from the open literature shows a very good agreement.


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

Mathematical Formulations of the Problem
Consider a continuous moving permeable horizontal flat plate which moves with a nonuniform velocity U 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 T f x which provides a heat transfer coefficient h f 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
The boundary conditions are 49 where v w 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, β T : volumetric thermal coefficient, β C : volumetric concentration coefficient, g: acceleration due to gravity, k 0 : reaction rate, n: order of chemical reaction, and N 1 x : velocity slip factor with dimension velocity −1 .
Introducing stream function ψ, dimensionless temperature function θ and concentration function φ are defined by

2.8
The boundary conditions 2.2 become 2.9

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: x * e εα

3.2
We now find the relationship among the exponents α i 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
Using 3.1 in boundary conditions 2.9 and equating various exponents of e, we get Without loss of generality, we may put α 4 α 5 0. Solving 3.3 and 3.4 we have the following relationship among the exponents

3.5
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 .

Similarity Transformations
Using the relationships in 3.5 , we have from 3.1 x * e ε 7−5n

3.6
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
Solving the system of first order linear differential equations in 3.7 , we get where β i i 1, 2, 3, . . .8 are real constants.To get dimensionless form of the transformations, we define the following dimensionless transformations: where f η , θ η and φ η , represent dimensionless velocity, temperature, and concentration functions, respectively.B 0 , ΔT 0 , ΔC 0 , U 0 , v 0 and h 0 are constants.Using 3.9 , 2.8 become subject to the boundary conditions where fw v 0 L/ν √ Re is the suction/injection parameter, γ Lh 0 / Re k is the convective heat transfer parameter, λ 1 Gr/Re 2.5 is the thermal buoyancy parameter, λ 2 Gc/Re 2.5 is the concentration buoyancy parameter, M σB 2 0 L 2 /ρν Re is the magnetic parameter, K L 2 k 0 ΔC n−1 0 /ν Re is the reaction parameter, Gr g β c ΔT 0 L 3 /ν 2 is the thermal Grashof number, Gc g β c ΔT 0 L 3 /ν 2 is the solutal Grashof number, and a ν √ Re N 0 /L is the velocity slip parameter.Note that similarity solutions will exist if N 1 x N 0 x 2−2n / 7−5n , N 0 is a constant.
The physical quantities of interest are the local skin friction coefficient C fx , local Nusselt number Nu x rate of heat transfer , and local Sherwood number Sh x rate of mass transfer , which are defined as where τ w is the wall shear stress, q w is the wall heat flux, m w is the quantity of mass transfer through the unit area of the surface, which are given by τ w μ ∂u ∂y y 0 , q w k ∂T ∂y y 0 , m w D ∂C ∂y y 0 .

3.13
It can be shown that physical quantities are proportional to −f 0 , −θ 0 and −φ 0 .

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    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.Figures 5 a and 5 b depict the variation of the buoyancies λ 1 , λ 2 , 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 λ 1 , λ 2 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

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, λ 1 , and λ   Velocity of the plate and reference velocity u, v: Velocity components in x-direction and y-direction x, y: Coordinate along and normal to plate.

Figure 1 :
Figure 1: Physical configuration and coordinate system the problem.

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 Figure2 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.Figures 3 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 λ 1 and λ 2 , 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 λ 1 , λ 2 , 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 λ 1 and λ 2 , values of φ η decrease.

Figure 2 :
Figure 2: Effects of different parameters on the dimensionless velocity.

λ 1 =Figure 5 :
Figure 5: Effects of different parameters on the dimensionless shear stress.

Figure 6 :KM 1 f 1 ,Figure 7 :
Figure 6: Effects of different parameters on the dimensionless rate of heat transfer.

Subscript∞:
Condition at infinityw: Condition at the wall.

Table 1 :
Comparison of results for f 0 and −φ 0 for different values of parameters.Pr 0.72, Pr 0.72, λ 1 1, λ 2 1, K 1 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 Table1and found a good agreement.The influence of different parameters on the dimensionless velocity, temperature, and concentration is shown in Figures2, 3, and 4, respectively.Figure2a exhibits the influence of thermal buoyancy parameter λ 1 and suction/injection parameter fw, whilst Figure2 bshows the effects of solutal buoyancy parameter λ 2 and suction/injection parameter fw, on the dimensionless fluid velocity f η profiles.We further observed that velocity increases with buoyancy parameters λ 1 and λ 2 .