The paper deals with a fluid-structure interaction problem. A non steady-state viscous flow in a thin channel with an elastic wall is considered. The problem contains two small parameters: One of them is the ratio of the thickness of the channel to its length (i.e., to the period in the case of periodic solution); the second is the ratio of the linear density to the stiffness of the wall. For various ratios of these two small parameters, an asymptotic expansion of a periodic solution is constructed and justified by a theorem on the error estimates. To this end we prove the auxiliary results on existence, uniqueness, regularity of solution and some a priori estimates. The leading terms of the asymptotic solution are compared to the Poiseuille flow in a channel with absolutely rigid walls. In critical case a non-standard sixth order equation for the wall displacement is obtained. © 2005 Elsevier SAS. All rights reserved.
Panasenko, G. P., & Stavre, R. (2006). Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall. Journal Des Mathematiques Pures et Appliquees, 85(4), 558–579. https://doi.org/10.1016/j.matpur.2005.10.011