Question

Consider the Navier - Stokes equations in a bounded region $\Omega_t \subset \mathbb{R}^2$ with a Lipschitz boundary $\partial \Omega_t $ and the domain is time dependent. Can one introduce the notion of the Reynolds number if this problem is posed in a periodic media with scaling parameter $\varepsilon << 1 $. To be precise I consider $\Omega_t = (0, \infty)\times (-H - \eta(x,t), H + \eta(x,t))$ where $\eta \in L^1 (I; L^{\infty}(\mathbb{R}_+ ))$, where $I = (0,T)$ and let $\varepsilon = H/L$ where $L$ is an appropriately chosen characteristic length. A non-dimensionalization of the Navier - Stokes is performed and possibly two Reynolds numbers are obtained, one corresponding to the longitudinal characteristic length $L$ and one corresponding to the transversal characteristic length $H$. Is such a characterization physically reasonable?