A Variational Principle for Steady Frictional Flow In Nonlinear Porous Media
A Fermat-like principle of minimum time is formulated for nonlinear steady paths of fluid flow in isotropic porous media. The principle describes an optimal nature of nonlinear paths in steady Darcy’s flows. An expression for the total path resistance leads to a basic analytical formula for an optimal shape of a steady trajectory in nonlinear flows of fluid. In the physical space an optimal path ensures the maximum flux or shortest transition time of the fluid through the porous medium. A sort of “law of bending” holds for the frictional fluid flux in Lagrange coordinates, which shows that - by minimizing the total resistance - a ray spanned between two given points takes the shape assuring that its relatively large part resides in the region of lower flow resistance (a 'rarer' region of the medium).