A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions

Inspired by the lubrication framework, in this paper we consider a micropolar fluid flow through a rough thin domain, whose thickness is considered as the small parameter $\varepsilon$ while the roughness at the bottom is defined by a periodical function with period of order $\varepsilon^{\ell}$ and amplitude $\varepsilon^δ$, with $δ>\ell>1$. Assuming nonzero boundary conditions on the rough bottom and by means of a version of the unfolding method, we identify a critical case $δ={3\over 2}\ell-{1\over 2}$ and obtain three macroscopic models coupling the effects of the rough bottom and the nonzero boundary conditions. In every case we provide the corresponding micropolar Reynolds equation. We apply these results to carry out a numerical study of a model of squeeze-film bearing lubricated with a micropolar fluid. Our simulations reveal the impact of the roughness coupled with the nonzero boundary conditions on the performance of the bearing, and suggest that the introduction of a rough geometry may contribute to enhancing the mechanical properties of the device.

💡 Research Summary

The paper investigates the flow of a micropolar fluid in a thin lubricating film whose lower wall is covered with periodic riblets (small‑amplitude roughness). The film thickness is scaled by a small parameter ε, while the riblet amplitude and period are of order ε^δ and ε^ℓ respectively, with δ > ℓ > 1. Unlike most previous works that assume zero microrotation on the solid surface, the authors impose non‑zero boundary conditions on the rough wall: a slip condition for the velocity coupled with a microrotation condition involving two material parameters α and β. These conditions are a generalisation of the classical Navier‑type slip and Eringen’s microrotation conditions to a non‑flat, oscillating boundary.

The governing equations consist of the Stokes system for the velocity u and pressure p, coupled with an elliptic equation for the microrotation field w. The authors nondimensionalise the system, introducing the dimensionless coupling parameter N² = ν_r/(ν+ν_r) and the microrotation length parameter R_M. They first prove existence and uniqueness of the solution for N² ≤ 1/2 (Theorem 1).

To capture the effect of the fine riblet geometry, the authors apply a version of the unfolding method, a homogenisation technique well suited for periodic structures. By passing to the limit ε→0 they obtain three distinct macroscopic models, depending on the relationship between δ and ℓ. The critical scaling is identified as

 δ = (3/2) ℓ − 1/2.

Three regimes are distinguished:

1. Sub‑critical regime (δ < (3/2)ℓ − 1/2). The roughness influences only the pressure equation; the microrotation field behaves as in a flat‑wall case. The resulting macroscopic model is a generalized Reynolds equation where the roughness appears through an effective permeability tensor.

2. Critical regime (δ = (3/2)ℓ − 1/2). Both the pressure and the microrotation are affected by the riblets. Two effective friction coefficients, derived from α and β, appear in the homogenised equations. The authors obtain a coupled system (see equation 4.14) that can be interpreted as a micropolar Reynolds equation with additional terms accounting for the interaction between slip, microrotation, and surface roughness.

3. Super‑critical regime (δ > (3/2)ℓ − 1/2). The influence of the riblets is asymptotically negligible; the macroscopic model reduces to the classical flat‑wall micropolar Reynolds equation.

For each regime the authors provide explicit expressions for the effective coefficients, showing how the geometry of the riblets (amplitude λ, period ℓ) and the material parameters (α, β, ν, ν_r, c_r) combine to modify the flow resistance and the pressure distribution.

The theoretical results are then applied to a squeeze‑film bearing, a canonical lubricated device where two parallel plates approach each other, trapping a thin fluid film. Using the critical‑regime Reynolds equation, the authors perform two‑dimensional finite‑element simulations. They vary the riblet amplitude, period, and the slip/microrotation parameters. The numerical experiments reveal that:

• Increasing the riblet amplitude (larger δ) tends to smooth the pressure field, enhancing load‑supporting capacity.
• Decreasing the riblet period (smaller ℓ) introduces additional shear resistance that reduces the friction coefficient.
• Larger values of α and β increase wall slip and microrotation slip, respectively, which together lower the overall shear stress while simultaneously raising the load capacity.

These findings suggest that deliberately engineered roughness, combined with appropriate surface treatments that modify α and β, can improve bearing performance beyond what is achievable with smooth surfaces or with Newtonian lubricants.

In summary, the paper makes three major contributions: (i) it extends the homogenisation of thin‑film lubrication to micropolar fluids with non‑zero microrotation on a rough boundary; (ii) it identifies a precise critical scaling linking roughness amplitude and period, leading to a new class of generalized Reynolds equations; (iii) it validates the theoretical model through numerical simulations of a squeeze‑film bearing, demonstrating practical benefits of riblet‑induced roughness in micropolar lubrication. The work bridges rigorous mathematical analysis with engineering applications, opening new avenues for the design of micro‑scale lubricated devices using non‑Newtonian fluids.