Asymptotic analysis of a viscous fluid layer separated by a thin stiff stratified elastic plate

Abstract

A three-dimensional model for a viscous fluid layer separated in two parts by a thin stratified stiff plate is considered. This problem depends on a small parameter ϵ, which is the ratio of the thickness of the plate and that of each of the two parts of the fluid layer. The right-hand side functions are 1-periodic with respect to the tangential variables of the plate. The plate's Young modulus is of order ${\epsilon }^{-3}$, i.e. it is great, while its density is of order 1. At the solid–fluid interfaces, the velocity and the normal stress are continuous. The variational analysis of this model (including the existence, uniqueness of the solution and its regularity) is provided. An asymptotic expansion of the solution is constructed and justified. The error estimate is established for the partial sums of the asymptotic expansion. The limit problem contains a non-standard interface condition for the Stokes equations. The existence, uniqueness and regularity of its solution are proved.

COMMUNICATED BY:

• R. Gilbert

Keywords:

• Linear elasticity
• Stokes equations
• interface
• asymptotic expansion
• dimension reduction
• homogenization

AMS classifications:

• 35C20
• 35B27
• 74K10
• 74F10
• 76M30
• 76M45

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The present work is partially supported by LABEX MILYON (ANR-10-LABX-0070) of University of Lyon, within the program ‘Investissements d'Avenir’ (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR); the second author was supported by Russian Science Foundation [19-11-00033].