## Reynolds Number in Multiple Scales

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?

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 Maybe try asking this question on theoreticalphysics.stackexchange.com – Piotr Migdal Mar 21 2012 at 10:09