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Asymptotic analysis of a viscous fluid layer separated by a thin stiff stratified elastic plate. (English) Zbl 1458.35348

Summary: 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 \(\varepsilon \), 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 \(\varepsilon^{-3}\), 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35C20 Asymptotic expansions of solutions to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74K20 Plates
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74B10 Linear elasticity with initial stresses
76D07 Stokes and related (Oseen, etc.) flows
76M30 Variational methods applied to problems in fluid mechanics
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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