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

We derive the equations of motion for the dynamics of porous medium, filled with 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 medium, 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 medium,. In particular, we derive the propagation of S-waves and P-waves in an isotropic medium,. 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 medium, are obtained for certain values of parameters in our models.