On the effect of a penetrating recirculation region on the bifurcations of the flow past a permeable sphere

We study the flow past a permeable sphere modeled using homogenization theory. The flow through the porous medium is described by the Darcy law, in which the permeability quantifies the resistance for the fluid to pass through the microstructure. A slip condition on the tangential velocity at the interface between the fluid and porous region is employed to account for the viscous effects in the proximity of the interface. The steady and axisymmetric flow is first characterized under the assumption of a homogenous and isotropic porous medium. In a certain range of permeability, the recirculation region penetrates inside the sphere, resulting in a strong modification of the linear stability properties of the flow and in a decrease in the critical Reynolds numbers for the flow instability. However, for very large permeabilities, a critical permeability value is identified, beyond which the steady and axisymmetric flow remains always linearly stable. The hypothesis of a homogenous porous medium is then relaxed, and the effect of polynomial distributions of permeability inside the body is studied. Interestingly, some macroscopic flow properties do not significantly vary with the permeability distributions, provided that their average is maintained constant. The analysis is concluded by outlining a simplified procedure to retrieve the full-scale structure corresponding to a considered distribution of permeability.