Geometric variational approach to the dynamics of porous media filled with incompressible fluid

TL;DR

This paper develops a variational framework to derive the equations governing the dynamics of incompressible fluid-filled porous media, with applications to biological systems and wave propagation analysis.

Contribution

It introduces a novel variational approach to model porous media with incompressible fluids, deriving equations in the spatial frame and connecting to Biot's classical wave equations.

Findings

Derived equations of motion for elastic matrix and fluid

Analyzed wave propagation including S-waves and P-waves

Established stability criteria linked to physical properties

Abstract

We derive the equations of motion for the dynamics of a porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an illustration of the method, the equations of motion for both the elastic matrix and the fluid are derived in the spatial (Eulerian) frame. Such an approach is of relevance e.g. for biological problems, such as sponges in water, where the elastic porous media is highly flexible and the motion of the fluid has a 'primary' role in the motion of the whole system. We then analyze the linearized equations of motion describing the propagation of waves through the media. In particular, we derive the propagation of S-waves and P-waves in an isotropic media. We also analyze the stability criteria for the wave equations and show that they are equivalent to the physicality conditions of the elastic matrix. Finally, we show that the celebrated Biot's equations for waves in porous media are obtained for certain values of parameters in our models.