Homogenization of a non-homogeneous fluid

TL;DR

This paper analyzes the asymptotic behavior of a non-homogeneous, heat-conducting incompressible fluid in a perforated domain, deriving a Brinkman-type limit with temperature-dependent properties as the hole size shrinks.

Contribution

It provides the first homogenization result for an inhomogeneous heat-conducting fluid in the critical perforation regime, accounting for temperature-dependent viscosity and heat conductivity.

Findings

Derives a Brinkman-type limit with a friction term influenced by geometry and viscosity.

Establishes homogenized heat conductivity coefficients for fluid and holes.

First to analyze critical perforation effects in inhomogeneous heat-conducting fluids.

Abstract

We consider a non--homogeneous incompressible and heat conducting fluid confined to a 3D domain perforated by tiny holes. The ratio of the diameter of the holes and their mutual distance is critical, the former being equal to $\epsilon^3$, the latter proportional to $\epsilon$, where $\epsilon$ is a small parameter. We identify the asymptotic limit for $\epsilon \to 0$, in which the momentum equation contains a friction term of Brinkman type determined uniquely by the viscosity and geometric properties of the perforation. Besides the inhomogeneity of the fluid, we allow the viscosity and the heat conductivity coefficient to depend on the temperature, where the latter is determined via the Fourier law with homogenized (oscillatory) heat conductivity coefficient that is different for the fluid and the solid holes. To the best of our knowledge, this is the first result in the critical case for the inhomogenous heat--conducting fluid.