Title: A new continuum theory for incompressible swelling materials,
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.