Compaction in a deformable porous cylinder with elastic boundaries
Perfusion of soft materials such as biological tissue or hydrogels is essential for the functioning of organ and laboratory systems such as chromatographic columns and bioreactors. Inspired by these applications, we model fluid-driven compaction in a long, thin cylindrical porous medium bounded by an impermeable elastic membrane and study how flow regimes relate to elastic parameters. Using a Lagrangian formulation of Darcy flow coupled to small-strain linear elasticity with porosity dependent permeability and elastic moduli, we perform an asymptotic reduction in the small aspect ratio limit and obtain a leading-order nonlinear diffusion equation for the porosity, which we solve numerically. Whereas rigid boundaries produce a compaction plateau, compliant walls exhibit, at most, an intermediate plateau beyond which the flow increases once the imposed pressure becomes comparable to the product of membrane stiffness and initial porosity. When the membrane is less stiff than the porous medium, flow rate can exceed that expected for a rigid medium. A parameter space map distinguishes regimes where plateau and breakthrough occur, where the steady flow rate is below (sub-Darcy) or above (super-Darcy) the undeformable-medium prediction, and delineates the small-strain domain in which the theory applies. An asymptotic solution for negligible gravity captures the departure from the plateau and yields compact expressions for effective permeability and flow rate.