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

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 the 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 Letter, 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 a minimization principle can be derived either for the velocity or the pressure field.