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

• Published in: Soft matter 2020, Vol.16 (4), p.939-944
• Main Authors: Rosti, Marco Edoardo, Pramanik, Satyajit, Brandt, Luca, Mitra, Dhrubaditya
• Format: Article
• Language: English
• Subjects: Channels; Computational fluid dynamics; Computer simulation; Constitutive relationships; Cylinders; Darcys law; Deformation; Fluid flow; Formability; Hexagonal lattice; Incompressible flow; Liquid phases; Mathematical models; Mechanical properties; Modulus of elasticity; Nonlinear response; Permeability; Porous materials; Porous media; Shear modulus; Solid phases; Two dimensional models; Viscoelasticity; Viscous fluids
• ISSN: 1744-683X; 1744-6848; 1744-6848
• DOI: 10.1039/c9sm01678c

We perform direct numerical simulations of the flow through a model of 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 (Δ 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 material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is an universal function of Δ P / G , which is confirmed by the present simulations. We show that the flux through a poroelastic material is a super-linear function of the pressure-difference. The permeability is a universal function of the ratio of the pressure-difference over the shear modulus, proportional to the cube of porosity.