In this paper, we make an initial value investigation of the unsteady flow of incompressible viscous fluid between two rigid non-conducting rotating parallel plates bounded by a porous medium under the influence of a uniform magnetic field of strength H0 inclined at an angle of inclination α with normal to the boundaries taking hall current into account. The perturbations are created by a constant pressure gradient along the plates in addition to the non-torsional oscillations of the upper plate while the lower plate is at rest. The flow in the porous medium is governed by the Brinkman’s equations. The exact solution of the velocity in the porous medium consists of steady state and transient state. The time required for the transient state to decay is evaluated in detail and the ultimate quasi-steady state solution has been derived analytically. Its behaviour is computationally discussed with reference to the various governing parameters. The shear stresses on the boundaries are also obtained analytically and their behaviour is computationally discussed.

Hall Effects Unsteady Rotating Flows Three-Dimensional Flows Parallel Plate Channels Incompressible Viscous Fluids Brinkman’s Model
1. Introduction

2. Formulation and Solution of the Problem

We consider the unsteady flow of an incompressible electrically conducting viscous fluid bounded by porous medium with two non-conducting rotating parallel plates. A uniform transverse magnetic field is applied to z-axis. In the presence of strong magnetic field a current is inclined in a direction normal to the both electric and magnetic field viz. Magnetic field of strength H0 inclined at angle of inclination to the normal to the boundaries in the transverse xz-plane. The inclined magnetic field gives rise to a secondary flow transverse to the channel. The hydro magnetic flow is generated in a fluid system by non-torsional oscillations of the upper plate. The lower plate is at rest. The origin is taken on the lower plate and the x-axis parallel to the direction of the upper plate. Since the plates are infinite in extent, all the physical quantities except the pressure depend on z and t only. In the equation of motion along x-direction, the x-component current density―and the z- component current density. We choose a Cartesian system 0 (x, y, z) such that the boundary walls are at z = 0 and z = l. The flow through porous medium governed by the Brinkman equations. The unsteady hydro magnetic equations governing flow through porous medium under the influence of a transverse magnetic field with reference to a rotating frame are

where, (u, w) is the velocity components along O (x, z) directions respectively. is the density of the fluid, is the magnetic permeability, is the coefficient of kinematic viscosity, k is the permeability of the medium, is the applied magnetic field. When the strength of the magnetic field is very large, the generalized Ohm’s law is modified to include the Hall current, so that

where, q is the velocity vector, H is the magnetic field intensity vector, E is the electric field, J is the current density vector, is the cyclotron frequency, is the electron collision time, is the fluid conductivity and, is the magnetic permeability. In Equation (3) the electron pressure gradient, the ion-slip and thermo- electric effects are neglected. We also assume that the electric field E = 0 under assumptions reduces to

where is the Hall parameter.

On solving Equations (4) and (5) we obtain

Using the Equations (6) and (7), the equations of the motion with reference to rotating frame are given by

By combining the Equations (8) and (9), we get.

Let

The boundary and initial conditions are

We introduce the following non dimensional variables are

Using non-dimensional variables, the governing equations are (dropping asterisks)

where,

is the Hartmann number;

is the rotation parameter;

is inverse Darcy parameter and;

is the Hall parameter.

We choose is the prescribed of pressure gradient, then the Equation (13) reduces to

Corresponding initial and boundary conditions are

Taking Laplace transform of Equation (14) using initial condition (15) the governing equations in terms of the transformed variable reduces to

The relevant transformed boundary conditions are

Solving the Equation (17) and making use of the boundary conditions (18) and (19), we obtain

where

,

Taking inverse Laplace transform to the Equation (20), we obtain

The shear stresses on the upper plate and the lower plate are given by

3. Results and Discussion

The flow is governed by the non-dimensional parameters M the Hartman number, D−1 the inverse Darcy parameter, K is the rotation parameter and m is the Hall parameter. The velocity field in the porous region is evaluated analytically its behaviour with reference to variations in the governing parameters has been computationally analyzed. The profiles for u and w have been plotted in the entire flow field in the porous medium. The solution for the velocity consists of three kinds of terms 1) steady state, 2) the quasi-steady state terms associated with non-torsional oscillations in the boundary, 3) the transient term involving exponentially varying time dependence. From the expression (21), it follows that the transient component in the velocity in the fluid region

decays in dimensionless time. When the transient terms decay the steady oscillatory

solution in the fluid region is given by

We now discuss the quasi steady solution for the velocity for different sets of governing parameters namely viz. M the Hartman number and D−1 the inverse Darcy parameter, K the rotation parameter, m is the Hall parameter, P0 & P1 the non dimensional pressure gradients, the frequency oscillations ω, a and b the constants related to non torsional oscillations of the boundary, for computational analysis purpose we are fixing the axial pressure gradient as well as a and b, and, , ,. Figures 1-8 corresponding to the velocity components u and w along the prescribed pressure gradient for different sets of governing parameters when the upper boundary plate executes non-torsional oscillations. The magnitude of the velocity u and w increases for the sets of values 0.1 ≤ z ≤ 0.3 as well as which reduces for all values of z with increase in the intensity of the magnetic field (Figure 1 and Figure 5). The resultant velocity q decreases with increasing the Hartmann number M. The magnitude of the velocity u decreases in the upper part of the fluid region 0.1 ≤ z ≤ 0.2 while it experiences enhancement lower part 0.3 ≤ z ≤ 0.9 with increasing the inverse Darcy parameter D−1 (Figure 2). The magnitude of the velocity w increases in the upper part of the fluid region 0.1 ≤ z ≤ 0.3, while it reduces in lower part 0.4 ≤ z ≤ 0.9 with increasing the inverse Darcy parameter D−1 (Figure 6). The resultant velocity q reduces with increasing the inverse Darcy parameter D−1. The magnitude of velocity u decreases in the upper part of the fluid region while it experiences enhancement lower part 0.3 ≤ z ≤ 0.9 and also the magnitude of velocity w increases throughout the fluid region (Figure 3 and Figure 7). However the resultant velocity q enhances with increasing the Hall parameter m. Finally we notice that, from (Figure 4 and Figure 8) the magnitude of the velocity component enhances for 0.1 ≤ z ≤ 0.3 and z = 0.7, and reduces within the region 0.4 ≤ z ≤ 0.6 and 0.8 ≤ z ≤ 0.9 with increase in rotation parameter K. while the velocity component w enhances for 0.3 ≤ z ≤ 0.4 and z = 0.9, and reduces for 0.1 ≤ z ≤ 0.2, with increase in rotation parameter K.

The shear stresses and on the upper plate have been calculated for the different variations in the governing parameters and are tabulated in the Table 1, Table 2. On the upper plate we notice that the magnitudes of enhances the inverse Darcy parameter D−1, the hall parameter m, rotation parameter K decreases with increase in the Hartmann number M (Table 1). The magnitude of decreases with increase in the Hartmann number M, the inverse Darcy parameter D−1 rotation parameter K and the Hall parameter m fixing the other parameters (Table 2). The similar behaviour is observed on the lower plate (Table 3, Table 4). We also notice that the magnitude of the shear stresses on the lower plate is very small compare to its values of the upper plate.

The velocity profile for u with M The velocity profile for u with D<sup>−</sup><sup>1</sup> The velocity profile for u with m The velocity profile for u with K The velocity profile for w with M The velocity profile for w with D<sup>−</sup><sup>1</sup> The velocity profile for w with m The velocity profile for w with K The shear stress <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/2-1100363x69.png" xlink:type="simple"/></inline-formula> on the upper plate
MIIIIIIIVVVIVII
20.0452740.0527980.6688760.0527870.0655250.0844740.144589
50.0329050.0435350.0504870.0434650.0518960.0522480.125547
D−11000200030001000100010001000
m1112311
K2222234
MIIIIIIIVVVIVII
2−0.05356−0.040556−0.03558−0.04955−0.32511−0.041125−0.0044585
5−0.04555−0.034255−0.02622−0.03512−0.02222−0.024451−0.0001254
D−11000200030001000100010001000
m1112311
K2222234
MIIIIIIIVVVIVII
20.0085540.0055420.0025540.0066580.0033250.000144−0.104595
50.0078850.0041020.0010010.0051140.0021140.000025−0.002852
D−11000200030001000100010001000
m1112311
K2222234
MIIIIIIIVVVIVII
2−0.000255−0.000149−0.000025−0.000228−0.000187−0.0000145−0.0000054
5−0.000246−0.000124−0.000012−0.000193−0.000078−0.0000102−0.0000029
D−11000200030001000100010001000
m1112311
K2222234
4. Conclusions

1) The resultant velocity q enhances with increasing hall parameter m and rotation parameter K, and decreases with increasing inverse Darcy parameter D−1 as well as the Hartmann number M.

2) On the upper plate the magnitude of enhances when increasing the hall parameter m; rotation parameter K and the inverse Darcy parameter D−1 decrease with increase in the Hartmann number M.

3) On the upper plate the magnitude of shear stress enhances when increasing the hall parameter M; rotation parameter K and the inverse Darcy parameter D−1 decrease with increase in the Hartmann number M.

4) The similar behaviour is observed on the lower plate.

5) The magnitude of the shear stresses on the lower plate is very small than the values of the upper plate.