Analysis of the Roughness Regimes for Micropolar Fluids via Homogenization
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Analysis of the Roughness Regimes for Micropolar Fluids via Homogenization Francisco J. Suárez-Grau1 Received: 1 July 2019 / Revised: 27 July 2020 / Accepted: 11 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We study the asymptotic behavior of micropolar fluid flows in a thin domain of thickness ηε with a periodic oscillating boundary with wavelength ε. We consider the limit when ε tends to zero and, depending on the limit of the ratio of ηε /ε, 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. Keywords Homogenization · Micropolar fluid flow · Reynolds equation · Thin-film fluid Mathematics Subject Classification 76D08 · 76A20 · 76A05 · 76M50 · 35B27 · 35Q35
1 Introduction We study in this paper the effect of small domain irregularities on thin film flows governed by the linearized 3D micropolar equations. In the case of Newtonian fluids governed by the Stokes or Navier–Stokes equations, this problem has been widely studied since Bayada and Chambat [4] provided, by means of homogenization techniques, a rigorous derivation of the classical 2D Reynolds equation h3 ∇ p + b = 0, div − 12ν
(1)
Communicated by Syakila Ahmad.
B 1
Francisco J. Suárez-Grau [email protected] Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, 41012 Seville, Spain
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F. J. Suárez-Grau
where h represents the film thickness, p is the pressure, ν is the fluid viscosity and b is a vectorial function that usually appears from the exterior forces or from the imposed velocity on a part of the boundary. In this sense, various asymptotic Reynolds-like models, in special regimes, have been obtained depending on the ratio between the size of the roughness and the thickness of the domain and the boundary conditions considered on a part of the boundary, see for example Bayada et al. [8], Benhaboucha et al. [10], Benterki et al. [11], Bresch et al. [14], Boukrouche and Ciuperca [15], Chupin and Martin [18], Letoufa et al. [24], Suárez-Grau [33], and references therein. More precisely, a very general result was obtained in Bayada and Chambat [5,6], see also Mikelic [29]. Assuming that the thickness of the domain is rapidly oscillating, i.e., the thickness is given by a small parameter ηε and one of the boundary is rough with small roughness of wavelength ε, it was proved that depending on the limit of the ratio ηε /ε, denoted as λ, there exist three characteristic regimes: Stokes roughness (0 < λ < +∞), Reynolds roughness (λ = 0) and high-frequency roughness (λ = + ∞). In particular, it was obtained that the flow is governed by a generalized 2D Reynolds equation of the form div (−Aλ ∇ p + bλ ) = 0,
(2)
for 0 ≤ λ ≤ +∞, where Aλ and bλ are macroscopic quantities known as flow factors, which take into account the microstructure of the roughness. Moreover, it holds that in the Stokes roughness regime the flow fact
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