Oseen and Stokes asymptotics for the problem on stationary motion of two immiscible liquids
type: Item
instance of: scholarly article
title: Oseen and Stokes asymptotics for the problem on stationary motion of two immiscible liquids
zbMATH Open document ID: 0965.35116
DOI: 10.1007/BF02433443
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-08-24T00:00:00Z
review text: Let \(\Omega_1\) be a bounded domain in \(\mathbb R^3\), \(0\in\Omega_1\), \(\Omega_2 = \mathbb R^3\setminus\overline\Omega_1\), \(\Gamma\) is the boundary between \(\Omega_1\) and \(\Omega_2\). The motion is considered of two viscous incompressible fluids with no phase transitions. The following boundary value problem can be posed: Find a solution \(\mathbf v^{(i)}(x) = (v_1^{(i)},v_2^{(i)},v_3^{(i)})\), \(p^{(i)}(x)\) (the velocity field and pressure) in \(\Omega_i\), \(i = 1,2\), to the equations \[ \begin{aligned} &{-}\mu_i\Delta\mathbf v^{(i)} + \rho_i\lambda(\mathbf v^{(i)}\cdot\nabla)\mathbf v^{(i)} + \nabla p^{(i)} = \mathbf f^{(i)},\\ &\nabla\cdot \mathbf v^{(i)} = 0,\quad x\in\Omega_i, \end{aligned} \] satisfying the conditions \[ \begin{aligned} &[\mathbf v]|_{\Gamma} = \mathbf a,\quad \mathbf v^{(1)}\cdot\mathbf n|_{\Gamma} = b, \quad [\mu(S(\mathbf v)\mathbf n)_{\tau}]|_{\Gamma} = \mathbf d, \\ & \mathbf v^{(2)}(x)\to \mathbf e_3,\quad p^{(2)}\to 0\quad (|x|\to\infty), \end{aligned} \] where \([\mathbf v]|_{\Gamma} = \mathbf v^{(1)}|_{\Gamma} - \mathbf v^{(2)}|_{\Gamma}\), \(S(\mathbf v) = \|S_{ij}(\mathbf v)\|\), \(S_{ij}(\mathbf v) = \frac{\partial v_i} {\partial x_j} + \frac{\partial v_j}{\partial x_i}\), \(\mathbf e_3 = (0,0,1)\), \(\mathbf n\) is the normal to \(\Gamma\), the symbol \(\tau\) denotes the tangent component of a vector field. The aim of the article is to study the Oseen and Stokes asymptotics for the above-mentioned problem. The author obtains a priori estimates for ``residuals'' in the asymptotic representation of the solution. Then the result obtained is employed for proving solvability of a free boundary problem of a uniform falling viscous drop in an infinite fluid.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 76D27
zbMATH DE Number: 1495304
zbMATH Keywords: uniform drop in infinite liquid
zbMATH Keywords: Stokes flow
zbMATH Keywords: Oseen flow
zbMATH Keywords: a priori estimates
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: slow motion of two immiscible liquids
zbMATH Keywords: free-boundary problem
author: V. A. Solonnikov
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
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cites work: Q3707632
cites work: Estimates of Oseen's Potentials in Weighted Hölder Spaces
full work available at URL: https://doi.org/10.1007/bf02433443
OpenAlex ID: W1999028788
description: scientific article; zbMATH DE number 1495304