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

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.

Variational geometric approach to the thermodynamics of porous media

“…In Arnold's theory the Lagrangian is simply the kinetic energy, as the potential energy of the fluid is absent, and the fluid pressure enters the equations from the incompressibility condition. This paper extends the initial derivation of [3,54], to include thermodynamic processes via the variational approach to thermodynamics initially developed in [1,2]. In particular, we achieve the following novel results:…”

“…This origin of this difficulty in the interpretation of fluid content was explained in [3], in terms of the fluid content being a constraint in the fluid's incompressibility, and the fluid pressure being the Lagrange multiplier related to the incompressibility. That description in [3] was based on the classical Arnold description of incompressible fluid as geodesic motion, hence Euler-Lagrange equations, on the group of volumepreserving diffeomorphisms of the fluid domain [53]. In Arnold's theory the Lagrangian is simply the kinetic energy, as the potential energy of the fluid is absent, and the fluid pressure enters the equations from the incompressibility condition.…”

“…In this paper, we take an alternative view, namely, we choose the same coordinate system of a stationary observer (Eulerian frame) for the description of both the fluid and the elastic media. This is the approach taken in [3,54]. This frame is more frequently used in the classical fluid description, but is less common in the description of elastic media.…”

“…Ironically, the Eulerian frame is also frequently used in the description of wave propagation in the media, in particular, classical Biot's theory [14][15][16][17]. Physically, our description is more relevant for the case of a porous media consisting of a dense network of elastic 'threads' positioned inside the fluid, which is a case that has not been considered before apart from [3]. In our formulation, we choose the Eulerian description for both the fluid and the elastic matrix.…”

“…In this paper we derive the equations of motion for the dynamics of a deformable porous media which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics [1,2]. This theory extends the recently developed variational derivation of the mechanics of deformable porous media [3] to include thermodynamic processes and can easily include incompressibility constraints. The model for the combined fluid-matrix system, written in the spatial frame, is developed by introducing mechanical and additional variables describing the thermal energy part of the system, writing the action principle for the system, and using a nonlinear, nonholonomic constraint on the system deduced from the second law of thermodynamics.…”