Two‐scale convergence for thin domains and its applications to some lower‐dimensional models in fluid mechanics

Abstract

Inspired by the similar ideas from the homogenization theory, in this paper we introduce the notion of two‐scale convergence for thin domains that allow lower‐dimensional approximations. We prove the compactness theorem, analogous to the one in homogenization theory. Using those results we derive the lower‐dimensional models for potential flow in thin (possibly degenerated) pipe, the degenerated Reynold’s equation for viscous flow in degenerated thin domain and the 1‐dimensional approximation for the non‐Newtonian (power‐law) flow in a thin pipe.