Abstract
In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.
Highlights
Introduction and Main ResultWe consider the Stokes system in which the viscosity is a periodically varying function of the space variable with small period ǫ > 0
The first goal of this paper is to study homogenization of the above systems via Bloch Wave Method which is based on the fact that the homogenized operator can be defined using differential properties of the bottom of the so-called Bloch spectrum
The second goal of this paper is to explore this issue which is especially delicate in the case of Stokes equations
Summary
We consider the Stokes system in which the viscosity is a periodically varying function of the space variable with small period ǫ > 0. The Bloch wave method for scalar equations and systems without differential constraints (like the incompressibility condition) was studied in [12, 13, 14, 21] In such cases, this approach gives a spectral representation of the homogenized tensor A∗ = (A∗)kαlβ in terms of the lowest energy Bloch waves and their behaviour for small momenta (what we call the bottom of the spectrum). We see the effect of differential constraints (the incompressibility condition in the case of Stokes equations) on the homogenization process via Bloch wave method. Extracting macro constitutive relation and macro balance equation from the localized homogenized equation in the Fourier space turns out to be not very straight forward because of differential constraints Let us end this discussion with two general remarks on Bloch wave method. We prove Theorem 1.1 in Section 5 following the Bloch wave homogenization method
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