An analysis of the swimming problem of a singly flagellated micro-organism in an MHD fluid flowing through a porous medium

The focus of the present work is concerned with the study of the swimming of microscopic organisms that use a single flagellum for propulsion in a magnetohydrodynamic (MHD) fluid flowing through a porous medium. The flow is modelled by appropriate equations and the organism is modelled by an infinite flexible but inextensible transversely waving sheet, which represents approximately the flagellum. The governing equations subject to appropriate boundary conditions are solved analytically. Expressions for the velocity of propulsion of the microscopic organism are obtained. We show that as the MHD character of the fluid is removed the results match those of an earlier analysed problem of propulsion through a fluid in a porous medium. In addition, in the final case of a simple viscous fluid (absence of magnetic field), we show that as the permeability becomes large the results reduce to the swimming of such organisms in a viscous fluid (discounting the pores and the MHD character).