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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.

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

movement of biological fluids. To explain the behavior of non-Newtonian fluid different models have been proposed. Among these the power law fluid has gained importance. The order of chemical reactions depends on several factors. One of the simplest chemical reactions is the first order reaction in which the rate of reaction is directly proportional to the species concentration. Now a days, due to the growing use of these non-Newtonian substances in various manufacturing and processing industries, considerable efforts have been directed towards understanding their friction and heat transfer characteristics. By the application of a magnetic field hydromagnetic techniques are used for the purification of molten metal. The problem of steady flow and heat transfer in power law fluid by free convection along a vertical plate has been investigated by many researchers. Vujanovic et al. [

The aim of the present work is to investigate the unsteady hydro magnetic non-Newtonian power law fluid past a flat plate with heat and mass transfer effect. The governing equations, describing the model are highly nonlinear coupled partial differential equations in nature. Hence closed form solutions are not possible. Suitable implicit finite difference scheme has been used to get the solution of the problem. Graphs have been plotted against various flow parameters to study the characteristics of velocity, temperature and concentration of the fluid.

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,

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 non- conducting 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

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,

The initial and boundary conditions are

The dimensionless variables are defined as follows:

where,

Equations (2)-(5) yield the following dimensionless equations

where

Accordingly, the initial and boundary conditions will be reduced to

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 non- dimensional form, respectively, as follows:

The Equations (7)-(10) are solved by implicit finite difference method. For discretization in space and time a uniform mesh of step

The above discretized Equations (15)-(18) are solved iteratively using the following algorithm.

Step I

Initialize

Step II

For

For

For

Step III

Step IV

Step V

The Steps (II)-(V) are repeated until the relative errors of two consecutive values of

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). The velocity, temperature and concentration decrease as R_{e} increases.

The velocity and temperature of the fluid for different Prandtl numbers are shown in _{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

In _{c} increases it is reflected through

It is evident from _{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

Grashof number on the skin friction C_{f}, local Nusselt number N_{u} and Sherwood number S_{h} is shown in the

n | M | P_{r} | R_{e} | S_{c} | K_{r} | G_{r} | G_{m} | C_{f} | N_{u} | S_{h} |
---|---|---|---|---|---|---|---|---|---|---|

1.5 | 2 | 1 | 1 | 0.6 | 1 | 5 | 5 | 0.4049 | - | - |

2 | 0.4046 | - | - | |||||||

4 | 0.4029 | - | - | |||||||

- | 4 | - | - | - | - | - | - | 0.3692 | - | - |

8 | 0.3222 | - | - | |||||||

- | - | 2 | - | - | - | - | - | 0.5662 | 3.8563 | - |

4 | 0.7651 | 2.7748 | - | |||||||

- | - | - | 3 | - | - | - | - | 0.1658 | 7.5284 | 8.9185 |

6 | 0.0868 | 8.5764 | 9.4168 | |||||||

- | - | - | - | 1.0 | - | - | - | 0.4709 | - | 6.6021 |

1.5 | 0.5386 | - | 5.8548 | |||||||

- | - | - | - | - | 2 | - | - | 0.3659 | - | 8.1591 |

3 | 0.3425 | - | 8.5691 | |||||||

- | - | - | - | - | - | 10 | - | 0.7066 | 5.2366 | - |

15 | 1.0146 | 5.2366 | - | |||||||

- | - | - | - | - | - | 5 | 10 | 0.5428 | - | 7.4954 |

20 | 0.8224 | - | 7.4954 |

C_{f} = Skin Friction :_{u} = Local Nusselt Number:_{h} = Local Nusselt Number:

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.

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.