Unsteady Hydro-Magnetic Heat and Mass Transfer Flow of a Non-Newtonian Power-Law Fluid past a Flat Plate in the Presence of Homogeneous Chemical Reaction

This paper investigates the flow, heat andmass transfer of a power law fluid from a vertical plate in presence of a magnetic field. The resulting non-linear partial differential equations governing the flow together with the boundary conditions are reduced to non-dimensional form. The governing equations are discretized using implicit finite difference scheme and solved numerically. The velocity, temperature and concentration profile are presented graphically while the skin friction, local Nusselt number and the Sherwood number are presented in tabular form for different values of parameters of the problem.


Introduction
The fluids which are encountered in chemical and allied processing applications are known as non-Newtonianfluids.The study of non-Newtonian fluid flows has considerable interest for their numerous engineering applications.During the past four decades the study of non-Newtonian fluids has gained interest because of their numerous technological applications, including manufacturing of the plastic sheets, performance of lubricants and

Formulation of the Problem
There exist different types of non-Newtonian fluids but the simplest and most common type is the power-law fluid for which the rheological equation of the state between stress components and strain rate components defined by Vujanovic is where, P is the pressure, ij δ is the Kronecker delta, K and n are the consistency and flow behavior indices of the fluid respectively.When n > 1 the fluid is described as dilatant, n < 1 as pseudo-plastic and when n = 1 it is known as the Newtonian fluid.Consider the unsteady free convection heat and mass transfer flow of a two-dimensional, viscous, incompressible, electrically conducting and chemically reactive non-Newtonian power-law fluid along an infinite nonconducting vertical flat plate in the presence of a uniform magnetic field B 0 applied in a transverse direction to fluid flow.Let x′-axis be along the plate in upward direction, y′-axis is normal to it & z′-axis is normal to x′y′-plane.Initially, at time 0 t′ ≤ , the fluid and plate are at rest and at a uniform temperature T ∞ ′ .When 0 t′ > the plate is maintained at constant temperature T ∞ ′ and constant species concentration w C′ .Since the plate is of infinite extent in x′ direction and is electrically non-conducting, except pressure all other physical quantities are functions of y′ and t′ only.The governing equations describing the model are ( ) where g is acceleration due to gravity, α represents the thermal diffusivity, β T is coefficient of thermal expansion of fluid, β C is volumetric coefficient of expansion or contraction, k is thermal conductivity of the fluid, ρ is fluid density, n is power law index, u′ & v′ are stream wise and transverse velocity respectively.
Similarly x′ and y′ are stream wise and transverse co-ordinate.T' is temperature of the fluid and t′ is time, D is the coefficient of mass diffusivity, k c is the rate of chemical reaction, T ∞ ′ and C ∞ ′ are the free stream temper- ature and concentration of the fluid respectively, and w T ′ and w C′ are the temperature and concentration at the wall respectively.
The initial and boundary conditions are 0, The dimensionless variables are defined as follows: where, and l is the suitable length scale.Substituting the above non-dimensional variables into Equations ( 2)-( 5) yield the following dimensionless equations where is the Reynold number, ( ) is the chemical reaction parameter, p C is the specific heat at constant pressure, v µ ρ = is the kinematic viscosity and µ is the constant viscosity of the fluid in boundary layer region.Accordingly, the initial and boundary conditions will be reduced to 0, 0, 0, 0, and 0, 0, 0, at 0, 0, 0, 1, 1 at 0, 0 0, 0, 0, 0 at , 0 The special significance of this type of flow with heat and mass transfer situation are the skin-friction coefficient C f , the local Nusselt number N u and Sherwood number S h .These physical quantities are defined in nondimensional form, respectively, as follows:

Solution of the Problem
The Equations ( 7)- (10) are solved by implicit finite difference method.For discretization in space and time a uniform mesh of step x ∆ and y ∆ along x & y direction respectively and time t ∆ are employed so that the grid points are ( ) ( ) , , , ,  .The discre- tized form of Equation ( 7), ( 8), ( 9) and ( 10) are obtained respectively as, .
The Steps (II)-(V) are repeated until the relative errors of two consecutive values of , , , , , , , u v T C are less than a given tolerance.

Results & Discussion
The non-linear governing Equations ( 7)- (10) with the boundary conditions (11) are solved using finite difference method.The velocity, temperature, and concentration of the fluid for different Reynold numbers are shown in Figures 1(a)-(c In It is evident from Figure 5 that the presence of transverse magnetic field has a retarding effect on velocity field.But from Figures 6-8 it is observed that with an increase in G r , G m or N the velocity increases.
For the physical interest in view we found the influence of power law index, magnetic parameter, Prandtl number, Reynold number, Schmidt number, chemical reaction parameter, thermal Grashof number and modified concentration does not change with it.It is evident from the table that with an increase in R e the skin friction decreases while a reverse effect is seen in case of Nusselt number N u and Sherwood number S h is very much affected by Reynold number.

Conclusions
Unsteady free convective heat and mass transfer in the flow of a two dimensional viscous incompressible electrically conducting and chemically reactive non-Newtonian power-law fluid along an infinite non-conducting vertical flat plate in the presence of uniform magnetic field are studied.It is found that, • With an increase in R e velocity, temperature & concentration of the fluid decrease.
• With the increasing value of chemical reaction parameter fluid velocity & concentration decrease near the plate, but the species concentration shows reverse characteristics as depicted in the skin friction table.
• Magnetic field has a retarding effect on the fluid flow while the thermal radiation has a reverse effect on it.
The solutions obtained are well agreed with the Newtonian case and they give improved results, taking into consideration of the behaviour of the magnetic field.This method well suits for other non-Newtonian fluid flow problems.
).The velocity, temperature and concentration decrease as R e increases.The velocity and temperature of the fluid for different Prandtl numbers are shown in Figure 2(a) and Figure 2(b).Prandtl number increases the viscous diffusivity of the fluid at the surface which enhanced the velocity of the fluid near the surface as depicted in Figure 2(a), increase in P r implies flow of liquid with low thermal diffusivity and high viscous stress, which increases thermal boundary layer thickness near the surface as shown in Figure 2(b).

Figure 3 (
a) and Figure 3(b) we have seen as chemical reaction parameter increases the velocity and species concentration decrease, but it is reverse in the case of Schmidt number.The velocity and species concentration increase as S c increases it is reflected through Figure 4(a) and Figure 4(b).

Table 1 .
Skin friction, Nusselt number and Sherwood number.