Viscous flow past a porous sphere with an impermeable core : effect of stress jump condition

An arbitrary flow of a viscous, incompressible fluid past a porous sphere of radius `a' with an impermeable core of radius `b', using Brinkman's equation in the porous region is discussed. At the interface of the clear fluid and porous region, stress jump boundary condition for the tangential stresses along with the continuity of normal stresses and the velocity components are used. On the surface of the impermeable core no slip condition is used. The corresponding Faxen's laws are derived to compute the drag and torque acting on the surface r=a. It is found that the drag and torque not only change with the change of the permeability, but also a significant effect of the stress jump co-efficient is observed. The variation of drag and torque with permeability for different thickness (a-b) of the porous region as well as for different values of stress jump coefficient is discussed when the basic flow is due to uniform flow, two dimensional irrotational flow, doublet in a uniform flow, stokeslet, rotlet. In case of uniform flow the flow field has been plotted. In all the cases, a significant effect of the stress jump coefficient has been realized.