An analysis has been performed to study magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid. We investigate the combined effect of inertia, electromagnetic forces, and suction or injection. With the introduction of a similarity transformation, the continuity and momentum equations governing the squeeze flow are reduced to a single, nonlinear, ordinary differential equation. An approximate solution of the equation subject to the appropriate boundary conditions is derived using the homotopy perturbation method (HPM) and compared with the direct numerical solution (NS). Results showing the effect of squeeze Reynolds number, Hartmann number and the suction/injection parameter on the axial and radial velocity distributions are presented and discussed. The approximate solution is found to be highly accurate for the ranges of parameters investigated. Because of its simplicity, versatility and high accuracy, the method can be applied to study linear and nonlinear boundary value problems arising in other engineering applications.
This paper deals with the study of magneto-hydrodynamic (MHD) squeeze flow of an electrically conducting fluid between two infinite, parallel disks. The lower disk is stationary and permeable with (suction or injection). The upper disk is impermeable and moves toward the lower disk with a specified time dependent velocity. The use of a MHD fluid in lubrication prevents the adverse impact of temperature on the fluid viscosity when the system operates under extreme conditions. The problem considered is of general interest in the theory of magneto-hydrodynamic lubrication and other related applications. In particular, the results of the present investigation are directly applicable to the hydrodynamics of high temperature bearings lubricated with liquid metals. A number of theoretical and experimental investigations into magneto-hydrodynamic effects in lubrication have been reported. These include among other works of Hughes and Elco [
In the present work we investigate the combined effect of inertia, electromagnetic forces, and surface suction or injection in a squeeze film between two parallel disks. This combination of effects in squeezing flow has not been studied previously. The plates are made of a nonconducting material. There is no externally applied electric field and the induced electric field is negligible. Since the problem defies an exact analytical solution, special techniques must be used to derive approximate analytical solution.
One such technique is the homotopy perturbation method (HPM) proposed and applied by He [
The basic idea embodied in the HPM and a brief summary of the method can be found in [
We consider axisymmetric incompressible flow between two parallel infinite disks, which at time
The boundary conditions are given by
According to the HPM, we can construct a homotopy of (2) as follows:
We consider a three term-solution for
Assuming
Once
The axial and radial velocities are each functions of parameters
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Variation of axial velocity for
Figures
The results for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Variation of radial velocity for
Figure
Variation of radial velocity with
Variation of radial velocity with
For every case investigated (Figures
He’s homotopy perturbation method (HPM) has been utilized to derive approximate analytical solutions for the radial and axial velocity distributions in magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid. The approximate solutions have been compared with the direct numerical solutions generated by the symbolic algebra package Maple 11 which uses a Fehlberg fourth-fifth order Runge-Kutta finite-difference method for solving nonlinear boundary value problems. The comparison showed that the HPM solutions are highly accurate and provide a rapid means of computing the flow velocities between the plates.
For both the cases of no injection and injection, the axial component of the velocity increases monotonically as the similarity variable increases. The velocity profiles are not significantly affected by the increase in the fluid electrical conductivity and/or the magnetic field. Similarly, the effect of squeeze Reynolds number
The authors express their gratitude to the reviewers for their valuable suggestions which were incorporated to enhance the value of the paper.