A viscoelastic fluid flow through mixing grids

Abstract

We study the asymtotic behaviour of a viscoelastic fluid in a porous medium $ω_\varepsilon (\varepsilon>0)$ obtained by removing from an open set $ω$ small obstacles $(T^\varepsilon_v)_{1≤{v}≤{n(\varepsilon)}}$ of size $a_\varepsilon$ periodically distributed on a hyperplane H which intersects $ω$. We establish that the fluid behave differently depending on whether the size $a_\varepsilon$ is greater than or smaller than a critical size $c_\varepsilon$. If $a_\varepsilon = c_\varepsilon$ , a convolution term appears in the limit problem. This corresponds to a long memory effect. If $a_\varepsilon$ is smaller than $c_\varepsilon$, the fluid behaves as if there where no obstacles. If $a_\varepsilon$ is greater than $c_\varepsilon$ or is of the order of the period, the fluid adheres on the hyperplane H which plays a thin solid plate role and the fluid behaves separately on each side of this plate.