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The present note consists, the effects of thermal diffusion and chemical reaction on MHD flow of dusty viscous incom-pressible, electrically conducting fluid between two vertical heated, porous, parallel plates with heat source/sink. The plate temperature is raised linearly with time and concentration level near the plate to Cw. The variable temperature and uniform mass diffusion taking into account the chemical reaction of first order. The series solution method is used to solve the mathematical equations. Effects of various parameters like chemical reaction (K), thermal diffusion (ST) and magnetic field (M) etc. on velocity profile, skin friction, concentration profile and temperature field are displayed graphically and discussed numerically for different physical parameters. The analysis developed here for thermal diffusion, bears good agreement with real life problems.

Many transport processes exist in nature and industrial application in which the transfer of heat and mass occurs simultaneously as a result of combined buoyancy effects of thermal diffusion and diffusion of chemical species. In the last few decades several efforts have been made to solve the problems on heat and mass transfer in view of their application to astrophysics, geophysics and engineering.

Chemical reaction can be codified either heterogeneous or homogeneous processes. Its effect depends on the nature of the reaction whether the reaction is heterogeneous or homogeneous. A reaction is of order n, if the reaction rate is proportional to the n^{th} power of concentration. In particular, a reaction is of first order, if the rate of reaction is directly proportional to concentration itself. Experimental and theoretical works on MHD flow with thermal diffusion and chemical reaction have been done extensively in various areas i.e. sustain plasma confinement for controlled thermo nuclear fusion, liquid metal cooling of nuclear reactions and electromagnetic casting of metals. Chambre and Yang [

Recently Kumar and Srivastava [

B_{0}_{:} Magnetic field

m: Magnetic field Parameter

: Species concentration in the field

P_{r} : Prandtl Number

C_{w}: Concentration of the plate

S_{c} : Schmidt Number

C_{0} : Initial uniform concentration at

T_{w} : Plate temperature

C_{p} : Specific heat at constant pressure

: Initial temperature

G : Acceleration due to gravity

t : Time

G_{r} : Thermal Grashof number

u,v: Velocities of dusty fluid and dust particle respectively

S : Source/sink parameter respectively in the x-direction

G_{m}: modified Grashof number

y : Co-ordinate axes as in normal to the plate

K : dimensionless chemical reaction parameter

A : Decay factor

K_{1} : Chemical reaction parameter

D_{T}: Thermal diffusion coefficient

We consider the effects of thermal diffusion and chemical reaction on the unsteady dusty flow of an incompressible, slightly conducting, visco-elastic fluid between two heated porous infinite parallel plates (distance 2 h apart) under the influence of uniform magnetic field normal to the flow field in presence of heat source/sink. We assume x-axis along the flow in the mid-way of the plates and y-axis perpendicular to it. Let u, v be the velocities of dusty fluid and dust particles respectively in the direction of x-axis. The present analysis is based on the following assumptions:

1) The flow is in the direction of x-axis and is driven by a constant pressure with negligible body forces.

2) The dust particles are non-conducting, solid, spherical, and equal in size, uniformly and symmetrically distributed in the flow field and their number density is constant throughout the motion.

3) There is no externally applied electric field and the induced magnetic field is negligible.

4) Initially, when, the channel, walls as well as dusty fluid are assumed to be at the same temperature T_{0}. The foreign mass is assumed to be present at low level and it is uniformly distributed such that it is everywhere C_{0.}_{ }

5) When t > 0, the temperature of the walls is instantaneously raised to T_{W} and the species concentration is raised to C_{W}.

6) There exists a chemical reaction in the mixture.

Under these assumptions and Boussinesq’s approximation with concentration, the equations governing the flow are:

….(1)

(where the symbols have their usual meaning), at t = 0, the temperature and concentration level changes according to the following laws:

The initial and boundary conditions relevant to the problem are:

t = 0: u = 0 = v,T = T_{0}

t > 0: u = 0 = v, ,

for y = –d u = 0 = v, ,

for y = d(5)

We introduce the following non-dimensional quantities,

Introducing these non-dimensional quantities, Equations (1), (2), (3) & (4) reduce to

Now initial and boundary condition (5) according to new system becomeT = 0: u = 0 = v, T = 0

t > 0: u = 0 = v, T = 1 – e^{-at}, C = 1 – e^{-at}, for y = –1 u = 0 = v, T = 1 – e^{-at}, C = 1 – e^{-at}, for y = 1 (10)

where, (mass concentration of dust particle)

M (Hartmann number)

W (relaxation time parameter for particles)

G_{r} (grashof number)

G_{m} (Modified grashof number)

S_{c} (Schmidt number)

S (heat source/sink parameter)

K (Chemical reaction parameter)

(Visco-elastic parameter), , P_{r} (Prandtl number)

To solve the Equations (6) to (9) subject to the boundary conditions (10), according to I. Pop [

Substituting the equations like (11) into the Equations (6) to (9) and equating harmonic and non-harmonic terms, we get the following set of equations.

u_{0}_{ }= v_{0}_{ }& u_{1} = v_{1 }(1-aw)(14)

Where dashes represents differentiation w. r. to y.

Boundary conditions are reduced to:

Solutions of the Equations (12) to (18) under the boundary conditions (19) after substituting in (11), we have:

Let and be the skin friction for dusty fluid and dust particles respectively then we have:

, , , , , , ,

, , , , ,

Numerical solutions for velocity profile, skin friction for dusty fluid as well as dust particles and also temperature field, concentration profile for dusty fluid have been calculated. The values of different parameters and their effects on velocity, Temperature, concentration and skin friction have been displayed through graphs.

A temperature field has been represented in

It is observed that increasing values of heat source/sink parameter and Prandtl number decreases the temperature. Also we see that the temperature is minimum at the centre of the channel (y = 0) and increasing towards the plates.

Observation of

From Figures 3 and 4 we observe that increasing value of thermal diffusion parameter (Soret number) increases the velocity of dusty fluid and dust particles while chemical reaction parameter decreases the same. Also Figures 3 and 4 bears that the velocity is maximum at the centre of the channel and decreasing towards the plates.

Skin friction for different values of S_{T} (M = 3, K = 0.4,

S = 0.3):

Skin friction for different values of S, (M = 3, K = 1, S_{T} = 1.4):

Skin friction for different values of K (M = 3, S_{T} = 2.7.4, S = 0.3): Skin friction for different values of M (S_{T} = 1.4, K = 1, S = 0.4):

The results displayed in Figures 5.1-5.4 are as, the increasing value of thermal diffusion parameter and heat source/sink parameter decreases the skin friction of dusty fluid and dust particles. Increasing value of chemical re-

action parameter and magnetic field parameter increases the skin friction of dusty fluid and dust particles.

The theoretical and numerical solutions are obtained for different profiles. From graphical representations, we have the following observations:

1) Velocity and skin friction of the dust particles behaves same as dusty fluid.

2) Increasing value of y increases the temperature, concentration while decreases the velocity of dusty fluid and dust particles.

3) Velocity of dust particles is less than velocity of dusty fluid and skin friction of dust particles is greater than that of dusty fluid.

4) Increasing values of thermal diffusion parameter (Soret number) decreases the concentration, skin friction while increases the velocity of dust particles and dusty fluid.

5) Increasing values of magnetic field parameter increases the skin friction for both the dusty fluid and dust particles.

6) Increasing values of heat source/sink parameter decreases the skin friction for the dusty fluid as well as dust particles.

The author is sincerely thankful to the reviewers for the critical comments and suggestions to improve the quality of the manuscript.