Memory effects in the homogenization of a viscous fluid containing elastic particles

Abstract

This work deals with a fluid-structure interaction problem involving an incompressible viscous fluid and a periodic array of tiny elastic inclusions in a three-dimensional domain. We focus on a critical regime where the volume fraction of the inclusions tends to zero, but their density and stiffness are scaled so that they still influence the macroscopic behavior. The asymptotic regime depends on the parameter $γ=\lim_{\varepsilon\to 0}r_\varepsilon/\varepsilon^3\in [0,+\infty]$, where $\varepsilon$ is the period and $r_\varepsilon$ the inclusion size. For $γ=0$ the solid and fluid equations uncouple in the limit. For $γ\in (0,+\infty]$, we show that the macroscopic fluid velocity satisfies a Stokes system with memory: the standard Stokes equations are augmented by convolution-in-time terms arising from the elastic vibrations of the inclusions. The existence of the convolution kernels is established for both $γ\in(0,+\infty)$ and $γ=\infty$ using semigroup theory. For $γ=\infty$ the microscopic problem reduces to a purely conservative elastodynamic equation and the kernels can be computed explicitly via spectral decomposition. They are oscillatory and almost-periodic in time, yielding a persistent, non-decaying memory. For $γ\in(0,+\infty)$ the microscopic elastodynamic equation is coupled to a local Stokes problem, whose viscous dissipation yields memory kernels that decay to zero at infinity.