Micropolar fluid flow in a thick domain with multiscale oscillating roughness and friction boundary conditions
\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 functions \(U_0\) and \(W_0\) depending only on time. On the lower part \(\Gamma_0 = \{0< z_1 < L, \, z_2 = 0\}\), a non-penetration condition \(u^{\epsilon}_2 = 0\) for the velocity is supplemented by vanishing angular momentum \(\omega^{\epsilon} = 0\), and by the Tresca-type friction condition for the tangential part of the stress \(((\sigma_{ij})) = \big(\big(- p^{\epsilon} \, \delta_{ij} + (\nu+\nu_r) \, (\partial_{z_j} u_i^{\epsilon} + \partial_{z_i} u_j^{\epsilon})\big)\big)\) in the form of \begin{align*} |\sigma^{\epsilon}_{\tau}| < k^{\epsilon} \, \, \Rightarrow \, \, & u_1^{\epsilon} = s_0(t)\, , \\ |\sigma^{\epsilon}_{\tau}| = k^{\epsilon} \, \, \Rightarrow \, \, & \exists \lambda \geq 0 \, : \, u_1^{\epsilon} = s_0(t) - \lambda \, \sigma_{\tau}^{\epsilon} \, , \end{align*} where \(k^{\epsilon}\) is a given function on \((0,T) \times \Gamma_0\) and \(s_0(t)\) is the sliding velocity of the wall.\\[1ex] \textbf{The main results:} Several aspects of the problem, as for instance the existence analysis for weak solutions in Hilbert spaces, or also the two-scale homogenisation, have been considered by the same team of authors in previous publications. The present paper focusses on the derivation of the limit effective problem for \(\epsilon \rightarrow 0\) and \(m\geq 2\) independent scales. By means of the rescaling \(y_1 = z_1\) and \(y_2 = z_2/\epsilon^m \, h^{\epsilon}(z_1)\) of the position-variables, the problem is first reformulated on the fixed domain \(\Omega = (0, \, L) \times (0,1)\). Then, \(\epsilon\)-independent estimates are established, and the multiscale (weak) compactness techniques developed by \textit{G. Allaire} [SIAM J. Math. Anal. 23, No. 6, 1482--1518 (1992; Zbl 0770.35005)], and \textit{G. Allaire} and \textit{M. Briane} [Proc. R. Soc. Edinb., Sect. A, Math. 126, No. 2, 297--342 (1996; Zbl 0866.35017)], help extracting converging subsequences. If the data are subject to certain natural regularity assumptions, and they satisfy the following scaling conditions: \begin{gather*} \epsilon^{2m} \, f^{\epsilon}(t, \, y) = f(t, \, y, \, \frac{y_1}{\epsilon}, \ldots, \frac{y_1}{\epsilon^m}), \quad \epsilon^{2m} \, g^{\epsilon}(t, \, y) = g(t, \, y, \, \frac{y_1}{\epsilon}, \ldots, \frac{y_1}{\epsilon^m})\, ,\\ \epsilon \, k(t, \, y_1) = k(t, \, y_1, \, \frac{y_1}{\epsilon}, \ldots, \frac{y_1}{\epsilon^m}) \, , \end{gather*} where \(f\), \(g\) and \(k\) are fixed functions, then the solutions converge to limits \[u^{\epsilon} \rightarrow \rightarrow^{m+1} u^0, \quad p^{\epsilon} \rightarrow \rightarrow^{m+1} p^0, \quad \omega^{\epsilon}\rightarrow \rightarrow^{m+1} \omega^0 \, ,\] in the sense of \(m+1\)-scale convergence. The convergence analysis for the pressure, which is introduced only as a distribution, requires a more delicate treatment. For the limits \(u^0\), \(p^0\) and \(\omega^0\) several structural properties are established. An interesting feature is for instance the asymptotic condition affecting the divergence of the velocity field, in which the geometry of the problem is encoded via occurrence of the function \(h\). Finally, it is shown how the limits can be computed by means of solving a few purely linear second-order elliptic problems with the time as a parameter. An elliptic variational inequality accounts for the Tresca-type friction condition.
scientific article; zbMATH DE number 7315356
scientific article