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Analysis of the roughness regimes for micropolar fluids via homogenization

Francisco J. Suárez-Grau

Abstract

We study the asymptotic behavior of micropolar fluid flows in a thin domain of thickness $η_\varepsilon$ with a periodic oscillating boundary with wavelength $\varepsilon$. We consider the limit when $\varepsilon$ tends to zero and, depending on the limit of the ratio of $η_\varepsilon/\varepsilon$, we prove the existence of three different regimes. In each regime, we derive a generalized Reynolds equation taking into account the microstructure of the roughness.

Analysis of the roughness regimes for micropolar fluids via homogenization

Abstract

We study the asymptotic behavior of micropolar fluid flows in a thin domain of thickness with a periodic oscillating boundary with wavelength . We consider the limit when tends to zero and, depending on the limit of the ratio of , we prove the existence of three different regimes. In each regime, we derive a generalized Reynolds equation taking into account the microstructure of the roughness.

Paper Structure

This paper contains 10 sections, 6 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. The domain.
  4. The problem.
  5. Weak variational formulations.
  6. A priori estimates
  7. The extension of $(\tilde{u}_\varepsilon,\tilde{w}_\varepsilon,\tilde{p}_\varepsilon)$ to the whole domain $\Omega$
  8. Adaptation of the unfolding method
  9. Weak variational formulation.
  10. Stokes roughness regime ($0<\lambda<+\infty$)

Key Result

Lemma 3.1

For every $v\in H^1_0(\Omega_\varepsilon)^3$, the following inequality holds where $c_2>0$ is independent of $v$, $\varepsilon$ and $\eta_\varepsilon$.

Theorems & Definitions (12)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Lemma 3.6
  • proof
  • ...and 2 more