High Order Homogenization of the Stokes System in a Periodic Porous Medium

Abstract

We derive high order homogenized models for the incompressible \n Stokes system in a cubic domain filled with periodic obstacles. These models have \n the potential to unify the three classical limit problems (namely the \n ``unchanged Stokes system, the Brinkman model, and the Darcy s law) \n corresponding to various asymptotic regimes of the ratio $\\eta\\equiv \n a_{\\epsilon}/\\epsilon$ between the radius $a_{\\epsilon}$ of the holes and the \n size $\\epsilon$ of the periodic cell. What is more, a novel, rather surprising \n feature of our higher order effective equations is the occurrence of odd order \n differential operators when the obstacles are not symmetric. Our derivation \n relies on the method of two-scale power series expansions and on the existence of \n a ``criminal ansatz, which allows to reconstruct the oscillating velocity and \n pressure $(\\u_{\\epsilon},p_{\\epsilon})$ as a linear combination of the \n derivatives of their formal average $(\\u_{\\epsilon}^{*},p_{\\epsilon}^{*})$ \n weighted by suitable corrector tensors. The formal average \n $(\\u_\\epsilon^{*},p_{\\epsilon}^{*})$ is itself the solution to a formal, infinite \n order homogenized equation, whose truncation at any finite order is in general \n ill-posed. Inspired by the variational truncation method of \n \\cite{smyshlyaev2000rigorous,cherednichenko2016full}, we derive, for any \n $K\\in\\N$, a well-posed model of order $2K+2$ which yields approximations of the \n original solutions with an error of order $O(\\epsilon^{K+3})$ in the $L^{2}$ \n norm. Furthermore, the error improves up to the order $O(\\epsilon^{2K+4})$ if a \n slight modification of this model remains well-posed. Finally, we find \n asymptotics of all homogenized tensors in the low volume fraction limit \n $\\eta\\rightarrow 0$ and in dimension $d\\>3$. This allows us to obtain that our \n effective equations converge coefficient-wise to either of the Brinkman or Darcy \n regimes which arise when $\\eta$ is respectively equivalent, or greater than \n the critical scaling $\\eta_{\\mathrm{crit}}\\sim\\epsilon^{2/(d-2)}$