Micropolar fluid flow in a thick domain with multiscale oscillating roughness and friction boundary conditions
type: Item
instance of: scholarly article
title: Micropolar fluid flow in a thick domain with multiscale oscillating roughness and friction boundary conditions
zbMATH Open document ID: 1489.76004
author: Mahdi Boukrouche
author: Laetitia Paoli
author: Fatima Ziane
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-02-25T00:00:00Z
review text: \textbf{Statement of the problem:} Motivated by applications in lubrication, this paper deals with the periodic homogenisation of a complex PDE-system describing micropolar fluid flow in a thin two-dimensional layer with highly oscillating boundary \[ \Omega^{\epsilon} = \Big\{z = (z_1, \, z_2) \in \mathbb{R}^2, \, 0 < z_1 < L, \, 0 < z_2 < \epsilon^m \, h^{\epsilon}(z_1)\Big\} \, , \] with a parameter \(\epsilon \ll 1\), a natural number \(m \geq 2\), and a function \(h^{\epsilon}(z_1) := h(z_1, \, \frac{z_1}{\epsilon}, \, \frac{z_1}{\epsilon^2}, \ldots, \frac{z_1}{\epsilon^m})\) where \(h \in C^{\infty}(\mathbb{R}^{m+1})\) is given and periodic in all variables. This function is moreover strictly positive and subject to uniform bounds from below and above. In the cylindrical domain \((0, \, T) \times \Omega^{\epsilon}\) with \(T> 0\), the following PDEs for the unknowns \(u^{\epsilon}\) (velocity field), \(p^{\epsilon}\) (pressure) and \(\omega^{\epsilon}\) (angular micro-rotation field) are studied: \begin{align*} \frac{\partial u^{\epsilon}}{\partial t} - (\nu+\nu_r) \, \Delta u^{\epsilon} + (u^{\epsilon} \cdot \nabla)u^{\epsilon} + \nabla p^{\epsilon} &= 2 \, \nu_r \, \mathrm{rot} \omega^{\epsilon} + f^{\epsilon}\, ,\\ \operatorname{div}v u^{\epsilon} &= 0\, ,\\ \frac{\partial \omega^{\epsilon}}{\partial t} - \alpha \, \triangle \omega^{\epsilon} + u^{\epsilon} \cdot \nabla \omega^{\epsilon} + 4 \, \nu_r \, \omega^{\epsilon} &= 2 \, \nu_r \, \mathrm{rot} u^{\epsilon} + g^{\epsilon} \, . \end{align*} Here \(f^{\epsilon}\) and \(g^{\epsilon}\) are external force/momentum fields, while \(\nu\), \(\nu_r\) and \(\alpha\) are positive viscosity parameters. The \(L\)-periodicity in the \(z_1\)-direction is assumed for all data and variables. On the upper boundary \(\Gamma^{\epsilon}_1 = \{0 < z_1 < L,\, z_2 = \epsilon^m \, h^{\epsilon}(z_1)\}\) of the layer, the variables are prescribed in the form of \(u^{\epsilon} = (U_0, \, 0)\) and \(\omega^{\epsilon} = W_0\) with…
Mathematics Subject Classification ID: 76A05
Mathematics Subject Classification ID: 76M50
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B27
zbMATH DE Number: 7315356
zbMATH Keywords: reiterated periodic homogenisation
zbMATH Keywords: limit effective problem
zbMATH Keywords: multiscale convergence
zbMATH Keywords: oscillating periodic layer
zbMATH Keywords: Tresca friction law
reviewed by: Pierre-Etienne Druet
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.124688
OpenAlex ID: W3093009989
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DOI: 10.1016/J.JMAA.2020.124688
description: scientific article; zbMATH DE number 7315356