Analysis of the roughness regimes for micropolar fluids via homogenization

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.

💡 Research Summary

The paper investigates the asymptotic behavior of incompressible micropolar fluid flows confined in a thin domain whose thickness ηₑ and the wavelength ε of a periodically oscillating upper boundary both tend to zero. The authors adopt a homogenization framework based on two‑scale convergence and the unfolding method, extending the classical analysis of Newtonian thin‑film lubrication to the more complex micropolar model introduced by Eringen.

The thin domain Ωₑ is defined over a smooth planar region ω⊂ℝ² with a height function hₑ(x′)=ηₑ h(x′/ε), where h is a smooth Y‑periodic function on the unit cell Y′. The governing equations are the linearized, nondimensional micropolar system: a Stokes‑type momentum balance coupled with an angular momentum balance for the microrotation w, together with the incompressibility condition. The dimensionless coupling parameter N²∈(0,1) and the microrotation length parameter R_M are assumed to satisfy R_M=ηₑ² R_c (R_c=O(1)), which ensures a strong coupling in the limit. Homogeneous Dirichlet conditions (no slip, no rotation) are imposed on the whole boundary.

A vertical scaling x₃=ηₑ y₃ maps Ωₑ onto a fixed‑height reference domain 𝔈Ωₑ=ω×(0,h(x′/ε)). The rescaled differential operators D_{ηₑ}, ∇{ηₑ}, and rot{ηₑ} are introduced, and the system is rewritten in terms of the rescaled unknowns (ũ_ε, ŵ_ε, p̃_ε). Standard a priori estimates (Poincaré inequality adapted to thin domains, Korn‑type inequalities for the symmetric gradient, and bounds for the rotation) yield uniform H¹‑bounds for the velocity and microrotation, and an L²‑bound for the pressure.

The key parameter is the limit λ=lim_{ε→0} ηₑ/ε, which determines three distinct regimes:

1. Stokes‑roughness regime (0<λ<∞).
   The ratio of thickness to wavelength remains finite. The unfolding method leads to a three‑dimensional cell problem on the reference cell Y, consisting of a coupled micropolar Stokes system. The cell solutions (U,W,P) depend parametrically on λ and are used to define the macroscopic flow factors A_λ (a symmetric positive‑definite tensor) and b_λ (a vector). The homogenized pressure p₀ satisfies the generalized Reynolds equation
   \