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Modeling of a micropolar thin film flow with rapidly varying thickness and non-standard boundary conditions

María Anguiano, Francisco J. Suárez-Grau

TL;DR

This work analyzes the asymptotic behavior of stationary micropolar fluid flow in a thin film with a rapidly varying thickness and non-standard boundary conditions under a Reynolds roughness regime. Employing rescaling and an adapted unfolding approach, it derives a generalized Reynolds equation with effective coefficients that encode the rough boundary and boundary-slip effects, and provides explicit expressions for the averaged velocity and microrotation via local cell problems. The study establishes uniform a priori estimates and convergence to a two-pressure homogenized limit, enabling a rigorous dimensional reduction to a 2D model applicable to lubrication and microfluidic contexts. The resulting framework supports numerical computation of the effective coefficients and offers a principled approach to predict micropolar lubrication phenomena in rough micro-geometries.

Abstract

In this paper, we study the asymptotic behavior of the micropolar fluid flow through a thin domain assuming zero Dirichlet boundary condition on the top boundary, which is rapidly oscillating, and non-standard boundary conditions on the flat bottom. Assuming ``Reynolds roughness regime", in which the thickness of the domain is very small compared to the wavelenth of the roughness (i.e. a very slight roughness), we rigorously derive a generalized Reynolds equation for pressure clearly showing the roughness-induced effects. Moreover, we give expressions for the average velocity and microrotation.

Modeling of a micropolar thin film flow with rapidly varying thickness and non-standard boundary conditions

TL;DR

This work analyzes the asymptotic behavior of stationary micropolar fluid flow in a thin film with a rapidly varying thickness and non-standard boundary conditions under a Reynolds roughness regime. Employing rescaling and an adapted unfolding approach, it derives a generalized Reynolds equation with effective coefficients that encode the rough boundary and boundary-slip effects, and provides explicit expressions for the averaged velocity and microrotation via local cell problems. The study establishes uniform a priori estimates and convergence to a two-pressure homogenized limit, enabling a rigorous dimensional reduction to a 2D model applicable to lubrication and microfluidic contexts. The resulting framework supports numerical computation of the effective coefficients and offers a principled approach to predict micropolar lubrication phenomena in rough micro-geometries.

Abstract

In this paper, we study the asymptotic behavior of the micropolar fluid flow through a thin domain assuming zero Dirichlet boundary condition on the top boundary, which is rapidly oscillating, and non-standard boundary conditions on the flat bottom. Assuming ``Reynolds roughness regime", in which the thickness of the domain is very small compared to the wavelenth of the roughness (i.e. a very slight roughness), we rigorously derive a generalized Reynolds equation for pressure clearly showing the roughness-induced effects. Moreover, we give expressions for the average velocity and microrotation.

Paper Structure

This paper contains 17 sections, 17 theorems, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Position of the problem
  3. Definition of the domain and some notation.
  4. Micropolar equations with non-standard boundary conditions
  5. Mathematical study
  6. Rescaling
  7. A priori estimates and convergences
  8. Estimates for velocity and microroration
  9. The extension of $({\bf \widetilde{u}}^\epsilon, {\bf \widetilde{w}}^\epsilon)$ to the whole domain.
  10. Estimates for pressure
  11. Convergences
  12. Adaptation of the unfolding method
  13. Definition and properties
  14. Equivalent weak variational formulation
  15. Convergences and limit problem
  16. ...and 2 more sections

Key Result

Proposition 2.2

Sufficiently regular solutions of (system_1_J)--(BCTopBot1_J) satisfy the weak formulation: Find $({\bf v}_\epsilon, w_\epsilon, p_\epsilon)\in V^\epsilon_0\times V^\epsilon\times L^2_0(\Omega^\epsilon)$ such that

Figures (2)

  • Figure 1: Domain $\Omega^\epsilon$, bottom flat boundary $\Gamma_0$ and top oscillating boundary $\Gamma_1^\epsilon$
  • Figure 2: Domain $\Omega^\epsilon$ in 2D and the reference cell Z in 2D

Theorems & Definitions (37)

  • Remark 2.1
  • Proposition 2.2: Theorem 2.1 in Bayada_NewModel
  • proof
  • Theorem 2.3
  • Lemma 4.1: Poincaré's inequality
  • Lemma 4.2
  • proof
  • Lemma 4.3: Trace estimates
  • proof
  • Lemma 4.4: A priori estimates
  • ...and 27 more