Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel

Abstract

The non-steady viscous flow in a thin channel with elastic wall is considered. The viscosity is constant everywhere except for some small neighborhood of the origin of the coordinate system, where it is a variable function. The problem contains two small parameters: $\varepsilon$, that is the ratio of the thickness of the channel and its length, and $ \delta = \varepsilon^\gamma, $ $ \gamma \geq 3 ,$ that is the "softness of the wall", i.e. its inverse (rigidity) is great. An asymptotic expansion of the solution is constructed and, in particular, the leading term is described. An important new element of this paper is the procedure of construction of the boundary layer in the neighborhood of the origin of the coordinate system, generated by the variable viscosity. The error estimates for the difference of a truncated asymptotic ansatz and the exact solution are obtained. To this end, the existence and uniqueness of the solution are studied and some a priori estimates are proved.