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This paper examined the hydromagnetic boundary layer flow of viscoelastic fluid with heat and mass transfer in a vertical channel with rotation and Hall current. A constant suction and injection is applied to the plates. A strong magnetic field is applied in the direction normal to the plates. The entire system rotates with uniform angular velocity (Ω), about the axis perpendicular to the plates. The governing equations are solved by perturbation technique to obtain an analytical result for velocity, temperature, concentration distributions and results are presented graphically for various values of viscoelastic parameter (K2), Prandtl number (Pr), Schmidt number (Sc), radiation parameter (R), heat generation parameter (Qh) and Hall parameter (m).

Hydromagnetic convection with heat transfer in a rotating medium has important applications in geophysics, nuclear power reactors and in underground water and energy storage system. When the strength of the magnetic field is strong, one cannot neglect the effects of Hall currents. A comprehensive discussion of Hall current is given by Cowling [

The constitutive equations for the rheological equation of state for an elastico-viscous fluid (Walter’s liquid B') are

in which

N(t) is the distribution function of relaxation times. In the above equations p_{ik} is the stress tensor, p an arbitrary isotropic pressure, g_{ik} is the metric tensor of a fixed co-ordinate system x_{i} and

where ^{i} at time “t”. The fluid with equation of state (1) and (4) has been designated as liquid B'. In the case of liquids with short memories, i.e. short relaxation times, the above equation of state can be written in the following simplified form

In which

notes the convected time derivative. We consider Oscillatory free convective flow of a viscous incompressible and electrically conducting fluid between two insulating infinite vertical permeable plates. A strong transverse magnetic field of uniform strength

The equations governing the flow of fluid together with Maxwell’s electromagnetic equations are as follows.

Equation of Continuity

Momentum Equation

Energy Equation

Concentration Equation

The generalized Ohm’s law, in the absence of the electric field [

where _{e} are velocity, the electrical conductivity, the magnetic permeability, the cyclotron frequency, the electron collision time, the electric charge, the number density of the electron and the electron pressure, respectively. Under the usual assumption, the electron pressure (for a weakly ionized gas), the thermoelectric pressure, and ion slip are negligible, so we have from the Ohm’s law.

From which we obtain that

The solenoidal relation for the magnetic field

stant) everywhere in the flow, which gives

Since the plates are infinite in extent, all the physical quantities except the pressure depend only on

where _{p} is the specific heat at constant pressure, _{m} is the molecular diffusivity, K_{1} is the chemical reaction rate constant .The radiative heat flux is

given by

tion coefficient, respectively.

The initial and boundary conditions as suggested by the physics of the problem are

where e is a small constant.

We now introduce the dimensionless variables and parameter as follows:

After combining (14) and (15) and taking

where

number,

The boundary conditions (9) can be expressed in complex form as:

The set of partial differential Equations (20) cannot be solved in closed form. So it is solved analytically after these equations are reduced to a set of ordinary differential equations in dimensionless form. We assume that

where R stands for q or

Substituting (22) into (20) and comparing the harmonic and non harmonic terms, we obtain the following ordinary differential equations:

where

The Transformed boundary conditions are

The solutions of (23) under the boundary conditions are

Equation (25) corresponds to the steady part, which gives _{0} as the primary and V_{0} as secondary velocity components. The amplitude (resultant velocity) and phase difference due to these primary and secondary velocities for the steady flow are given by

where

Equation (26) and (27) together give the unsteady part of the flow. Thus unsteady primary and secondary velocity components

The amplitude (resultant velocity) and the phase difference of the unsteady flow are given by

where

The amplitude (resultant velocity) and the phase difference are given by

where u = Real part of q and v = Imaginary part of q.

The amplitude and phase difference of shear stresses at the stationary plate (η = 0), the steady flow can be obtained as

For the unsteady part of flow, the amplitude and phase difference of shear stresses at the stationary plate (

where

where

The amplitude and phase difference of shear stresses at the stationary plate (η = 0) can be as

where

The Nusselt number

The rate of heat transfer (i.e. heat flux) at the plate in terms of amplitude and phase difference is given by

The Sherwood number

The rate of mass transfer (i.e. mass flux) at the plate in terms of amplitude and phase difference is given by.

The system of ordinary differential Equation (23) with boundary condition (24) is solved analytically using perturbation technique. The solutions are obtained for the steady and unsteady velocity fields from (25)-(27), temperature fields from (28)-(30) and concentration fields are given by (31)-(33). The influences of each of the parameters on the thermal mass and hydrodynamic behaviors of buoyancy-induced flow in a rotating vertical channel are studied. The results are presented graphically. Temperature of the heated wall (left wall) _{2}, and e respectively.

resultant velocity increases with increasing values of rotation parameter W. This is due to the fact that the rotation effects being more dominant near the walls, so when W reaches high values secondary velocity component v decreases with increases in W as shown in _{2}.

The phase difference a for the flow is shown graphically in Figures 5-8.

The concentration profile f for the flow is shown graphically in

The temperature profiles q are shown graphically in _{H} by fixing other physical parameters. From this figure, we observe that temperature q decreases with increase in the heat absorption parameter Q_{H} because when heat is absorbed, the buoyancy force decreases the temperature profile.

_{H}, and_{H} which is shown in

Pradip KumarGaur,Abhay KumarJha, (2016) Heat and Mass Transfer in Visco-Elastic Fluid through Rotating Porous Channel with Hall Effect. Open Journal of Fluid Dynamics,06,11-29. doi: 10.4236/ojfd.2016.61002

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