Minimum principle for the flow of inelastic non-Newtonian fluids in macroscopic heterogeneous porous media

Abstract

The minimization of dissipation is a general principle in physics. It stipulates that a nonequilibrium system converges toward a state minimizing the energy dissipation. In fluid mechanics, this principle is well known for Newtonian fluids governed by Stokes equation. It can be formulated as follows: among all admissible velocity fields, the solution of the Stokes equation is the one that minimizes the total viscous dissipation. In this paper, we extend these approaches to non-Newtonian fluids in macroscopic heterogeneous porous media, or fractures. The flow is then governed by a nonlinear Darcy equation that can vary in space. In this case, a minimization principle can still be written depending on the boundary conditions. Moreover, such minimization principle can be derived either for the velocity or pressure field.