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Navier slip effects in micropolar thin-film flow: a rigorous derivation of Reynolds-type models

María Anguiano, Igor Pažanin, Francisco J. Suárez-Grau

TL;DR

The paper derives rigorous Reynolds-type equations for stationary incompressible micropolar fluids in thin domains under Navier slip boundary conditions, with a domain-thickness dependent friction parameter. By rescaling to a fixed domain and performing a careful three-regime asymptotic analysis based on the slip scaling $\\lambda_\\epsilon=\\lambda\\epsilon^\\gamma$, the authors obtain reduced coupled 1D and 2D problems: explicit velocity and microrotation fields in the transverse direction and a generalized Reynolds equation for pressure, whose form depends on the slip regime. The analysis rests on uniform a priori estimates, compactness arguments, and pressure decomposition, yielding a rigorous limit system that generalizes classical no-slip thin-film results to Navier-slip with boundary-friction effects. The results clarify how slip alters the effective permeability and the structure of the reduced model, with implications for lubrication theory and microstructured boundary interactions in micropolar fluids. Overall, the work extends sharp thin-domain techniques to micropolar flows under Navier slip, providing explicit limit solutions across multiple boundary regimes.

Abstract

We study the stationary flow of incompressible micropolar fluid in a thin three-dimensional domain under Navier slip boundary condition for the velocity and no-spin condition for microrotation. After rescaling the governing equations, we perform a rigorous asymptotic analysis as the film thickness tends to zero, considering a friction coefficient dependent on the small parameter. According to the scaling of the slip coefficient, we identify three distinct regimes: perfect slip, partial slip, and no-slip. For each regime, we derive the corresponding reduced micropolar system and obtain explicit expressions for the velocity and microrotation fields. This leads to a generalized Reynolds-type equation for the pressure, highlighting the impact of slip effects on the micropolar thin-film flow.

Navier slip effects in micropolar thin-film flow: a rigorous derivation of Reynolds-type models

TL;DR

The paper derives rigorous Reynolds-type equations for stationary incompressible micropolar fluids in thin domains under Navier slip boundary conditions, with a domain-thickness dependent friction parameter. By rescaling to a fixed domain and performing a careful three-regime asymptotic analysis based on the slip scaling , the authors obtain reduced coupled 1D and 2D problems: explicit velocity and microrotation fields in the transverse direction and a generalized Reynolds equation for pressure, whose form depends on the slip regime. The analysis rests on uniform a priori estimates, compactness arguments, and pressure decomposition, yielding a rigorous limit system that generalizes classical no-slip thin-film results to Navier-slip with boundary-friction effects. The results clarify how slip alters the effective permeability and the structure of the reduced model, with implications for lubrication theory and microstructured boundary interactions in micropolar fluids. Overall, the work extends sharp thin-domain techniques to micropolar flows under Navier slip, providing explicit limit solutions across multiple boundary regimes.

Abstract

We study the stationary flow of incompressible micropolar fluid in a thin three-dimensional domain under Navier slip boundary condition for the velocity and no-spin condition for microrotation. After rescaling the governing equations, we perform a rigorous asymptotic analysis as the film thickness tends to zero, considering a friction coefficient dependent on the small parameter. According to the scaling of the slip coefficient, we identify three distinct regimes: perfect slip, partial slip, and no-slip. For each regime, we derive the corresponding reduced micropolar system and obtain explicit expressions for the velocity and microrotation fields. This leads to a generalized Reynolds-type equation for the pressure, highlighting the impact of slip effects on the micropolar thin-film flow.
Paper Structure (15 sections, 15 theorems, 145 equations)

This paper contains 15 sections, 15 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Problem setting and the well-posedness result
  3. Description of the domain and some notation
  4. Micropolar equations with Navier slip boundary conditions
  5. Rescaled problem
  6. A priori estimates and convergences
  7. Auxiliary results
  8. Estimates for velocity and microrotation
  9. Estimates for pressure
  10. Convergences
  11. Limit problem
  12. Expressions of velocity and microrotation
  13. Reynolds equation
  14. Conflict of interest
  15. Data availability statement

Key Result

Theorem 2.1

Under the previous assumptions, for every $\epsilon>0$, there exists a unique weak solution $({\bf u}_\epsilon, {\bf w}_\epsilon, p_\epsilon)\in V^\epsilon_0\times H^1_0(\Omega^\epsilon)^3\times L^2_0(\Omega^\epsilon)$ of (Form_Var_vel)--(Form_Var_micro).

Theorems & Definitions (30)

  • Theorem 2.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4: Estimates for velocity and microrotation
  • proof
  • Proposition 4.5
  • ...and 20 more