Hydromagnetic effects on couple stress fluid flow through a porous sphere embedded in another porous medium

This work concerns the axisymmetric steady flow of an incompressible couple stress fluid through a porous sphere implanted in another unbounded porous medium. The governing equations of motion for couple stress fluid flow through porous media describe the flow. A uniform magnetic field is applied in the direction perpendicular to the flow. The stream function solution for the flow is obtained analytically. Appropriate boundary conditions are used on the surface of the porous sphere, and uniform velocity conditions at infinity are used. Expressions for flow velocities (normal and tangential), vorticities, couple stresses, tangential stresses, and normal stresses of the fluid are obtained. The variations of couple stress parameter, permeability parameter, and Hartman number on flow velocities, streamlines, and drag coefficient are plotted and discussed.