The breakdown of Darcy's law in a soft porous material

Abstract

We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus $G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux ($q$) is a nonlinear function -- steeper than linear -- of the pressure-difference ($\Delta P$) across the medium. Furthermore, the flux is larger for a softer medium (smaller $G$). We construct a theory of this super-linear behavior by modelling the channels between the solid cylinders as elastic channels whose walls are made of a material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is a universal function of $\Delta P/G$, which is confirmed by the present simulations.

6 pages, 3 figures, Some minor changes (including the title) from the previous submission

References in corpus (5)

• Flow rate--pressure drop relation for deformable shallow microfluidic channels
• Passive control of viscous flow via elastic snap-through
• Numerical simulation of turbulent channel flow over a viscous hyper-elastic wall
• Rheology of suspensions of viscoelastic spheres: deformability as an effective volume fraction
• From arteries to boreholes: Steady-state response of a poroelastic cylinder to fluid injection

Cited by in corpus (4)

• Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits
• Flow rate--pressure drop relations for new configurations of slender compliant tubes arising in microfluidics experiments
• A homogenised model for flow, transport and sorption in a heterogeneous porous medium
• Growth morphology and symmetry selection of interfacial instabilities in anisotropic environments