A Note on Two-Fluid Starting Flow in a Porous Spaced Channel with a Magnetic Field

Abstract

This study is concerned with starting flow of immiscible fluids in porous space owing to a sudden pressure gradient in the presence of a transverse magnetic field. The flow is divided into two regions, Region1(upper layer) and Region2(lower layer), and they are of variable widths. The required eigenvalues and eigenfunctions, along with the orthogonality, are developed. The analytic solution took on an infinite series structure due to the time-dependent initial transient component of the velocity. Analytical expressions for fluid velocity, volumetric flow rate, and shear stress are evaluated for pertinent parameters. We take cases when channels are filled with air over water and oil over water for analyzing the results. Channel with air over water illustrates that the upper layer is filled with air and the lower layer is filled with water. Similarly, a channel with oil over water illustrates that the upper layer is filled with oil, and the lower layer is filled with water. The effect of Hartmann number and time on velocity profiles has been seen in this study for variable fluid widths in both cases. It is observed that the starting flow velocity slows down with the increase of Hartmann number and porous medium parameter. The effect of the Hartmann number and porous medium parameter on the volumetric flow rate for oil over water case is also shown graphically. For a better understanding of the physical characteristics, the results of shear stress on the lower and upper walls of the channel also have been presented in tabular form.