Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall

International audienceAbstract. In this paper we continue the study of a fluid-structure interaction problem with the non periodic case. We consider the non stationary flow of a viscous fluid in a thin rectangle with an elastic membrane as the upper part of the boundary. The physical problem which corresponds to non homogeneous boundary conditions is stated. By using a boundary layer method, an asymp- totic solution is proposed. The properties of the boundary layer functions are established and an error estimate is obtained

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

International audienceThe non-steady viscous ow 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: epsilon, that is the ratio of the thickness of the channel and its length, and delta = epsilon^gamma, gamma>=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 di erence 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

Asymptotic expansion of the solution to the Stokes flow problem in a thin cylindrical elastic tube

International audienceThe purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions

Asymptotic analysis for the Kelvin–Voigt model for a thin laminate

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Asymptotic analysis of a viscous fluid-thin plate interaction: periodic flow

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

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Asymptotic analysis of a thin rigid stratified elastic plate – viscous fluid interaction problem

International audienceA three-dimensional model for the interaction of a thin stratified rigid plate and a viscous fluid layer is considered. This problem depends on a small parameter $\varepsilon$ which is the ratio of the thickness of the plate and that 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’s modulus is of order $\varepsilon^{-3}$, i.e. it is great, while its density is of order 1. At the solid–fluid interface, 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 proved for the partial sums of the asymptotic expansion. The limit problem contains a non-standard boundary condition for the Stokes equations. The existence, uniqueness, and regularity of its solution are proved. The asymptotic analysis is applied to the partial asymptotic dimension reduction of the solid phase and the derivation of the asymptotically exact junction conditions between two-dimensional and three-dimensional models of the plate