Elastohydrodynamics of a deformable porous packing in a channel competing under shear and pressure gradient

Abstract

Elastohydrodynamics of a deformable porous medium sandwiched between two parallel plates is investigated under the influence of an externally applied pressure gradient as well as an induced shear due to the movement of the upper plate. Biphasic mixture theory is used to describe the macroscopic governing equations for the fluid velocity and the solid displacement, assuming the deformable porous medium as a continuum space. The corresponding reduced mathematical model is a coupled system of elliptic partial differential equations. It is assumed that the fluid at the lower plate experiences slip due to the surface roughness of the plate. The exact solution for unidirectional fluid velocity and solid deformation resembling plain Poiseuille–Couette flow are presented for steady and unsteady states. Asymptotic analysis of the biphasic mixture in the case of low and high Darcy numbers is performed to validate the obtained solution using Prandtl’s matching technique. It is observed that the Womersley number dictates whether the fluid is trapped inside the channel or escapes the channel. The competition between the shear and the pressure gradient is analyzed, and a critical criterion is established that dictates the dominant factor. A mathematical analysis of the current problem is invaluable in understanding the mechanical behavior of biomass under pressure-driven flow in applications such as tissue engineering or shear driven flow inside endothelial glycocalyx layers, which are discussed in brief. In this context, our analysis on the extent of tissue deformation in response to frequency variations is expected to give useful insights to identify the right diagnosis.