Oseen and Stokes asymptotics for the problem on stationary motion of two immiscible liquids
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.
scientific article; zbMATH DE number 1495304