A new continuum theory for incompressible swelling materials

TL;DR

This paper develops a novel continuum model for incompressible swelling materials based on particle packing heuristics, capturing complex geometrical effects beyond traditional Darcy laws.

Contribution

It introduces a new continuum theory derived from packing heuristics, accounting for geometrical features not captured by existing models.

Findings

Derived a formula for equilibrium density under confinement.

Formulated evolution equations based on particle non-overlap and energy minimization.

Identified complex geometrical effects in swelling media beyond Darcy law.

Abstract

Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.