_{1}

^{*}

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.

The rotating flow between parallel plates is a classical problem that has important applications in magneto hydro dynamic (MHD) power generators and pumps, accelerators, aerodynamic heating, electrostatic precipitation polymer technology, petroleum industry, purification of crude oil and fluid droplets, sprays, designing cooling systems with liquid metal, centrifugal separation of matter from fluid and flow meters. The flows of fluids through porous medium are very important particularly in the fields of agricultural engineering for irrigation processes; in petroleum technology to study petroleum transport; in chemical engineering for filtration and purification processes. A series of investigations have been made by (Raptis et al., 1981 [

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 H_{0} inclined at angle of inclination

where, (u, w) is the velocity components along O (x, z) directions respectively.

where, q is the velocity vector, H is the magnetic field intensity vector, E is the electric field, J is the current density vector,

where

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,

We choose

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

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

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, P_{0} & P_{1} 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^{−1} (^{−1} (^{−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 (

The shear stresses ^{−1}, the hall parameter m, rotation parameter K decreases with increase in the Hartmann number M (^{−1} rotation parameter K and the Hall parameter m fixing the other parameters (

M | I | II | III | IV | V | VI | VII |
---|---|---|---|---|---|---|---|

2 | 0.045274 | 0.052798 | 0.668876 | 0.052787 | 0.065525 | 0.084474 | 0.144589 |

5 | 0.032905 | 0.043535 | 0.050487 | 0.043465 | 0.051896 | 0.052248 | 0.125547 |

D^{−1} | 1000 | 2000 | 3000 | 1000 | 1000 | 1000 | 1000 |

m | 1 | 1 | 1 | 2 | 3 | 1 | 1 |

K | 2 | 2 | 2 | 2 | 2 | 3 | 4 |

M | I | II | III | IV | V | VI | VII |
---|---|---|---|---|---|---|---|

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^{−1} | 1000 | 2000 | 3000 | 1000 | 1000 | 1000 | 1000 |

m | 1 | 1 | 1 | 2 | 3 | 1 | 1 |

K | 2 | 2 | 2 | 2 | 2 | 3 | 4 |

M | I | II | III | IV | V | VI | VII |
---|---|---|---|---|---|---|---|

2 | 0.008554 | 0.005542 | 0.002554 | 0.006658 | 0.003325 | 0.000144 | −0.104595 |

5 | 0.007885 | 0.004102 | 0.001001 | 0.005114 | 0.002114 | 0.000025 | −0.002852 |

D^{−1} | 1000 | 2000 | 3000 | 1000 | 1000 | 1000 | 1000 |

m | 1 | 1 | 1 | 2 | 3 | 1 | 1 |

K | 2 | 2 | 2 | 2 | 2 | 3 | 4 |

M | I | II | III | IV | V | VI | VII |
---|---|---|---|---|---|---|---|

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^{−1} | 1000 | 2000 | 3000 | 1000 | 1000 | 1000 | 1000 |

m | 1 | 1 | 1 | 2 | 3 | 1 | 1 |

K | 2 | 2 | 2 | 2 | 2 | 3 | 4 |

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 ^{−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.