On the problem of non-stationary motion of two viscous incompressible liquids

The problem of stability of uniformly rotating liquid mass consisting of two viscous incompressible capillary self-gravitating liquids separated by a free interface is studied. Bibliography: 12 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 103–119.     

Uniqueness and nonuniqueness of the Stokes and Oseen flows

The uniqueness class for the solutions to the Cauchy problem for flows modeled by the time-dependent Stokes and Oseen systems of equations is determined as in the growth class C exp(α|x|²). An example of a type considered by Tychonoff [Mat. Sb. 42 (1935), 199–216] is given that establishes the lack of uniqueness for such Stokes and Oseen systems. Even for the incompressible Navier–Stokes system, an example shows that rapidly growing nonphysical mathematical solutions exist.     

An iterative solver for the Oseen problem and numerical solution of incompressible Navier–Stokes equations

Incompressible unsteady Navier–Stokes equations in pressure–velocity variables are considered. By use of the implicit and semi‐implicit schemes presented the resulting system of linear equations can be solved by a robust and efficient iterative method. This iterative solver is constructed for the system of linearized Navier–Stokes equations. The Schur complement technique is used. We present a new approach of building a non‐symmetric preconditioner to solve a non‐symmetric problem of convection–diffusion and saddle‐point type. It is shown that handling the differential equations properly results in constructing efficient solvers for the corresponding finite linear algebra systems. The method has good performance for various ranges of viscosity and can be used both for 2D and 3D problems. The analysis of the method is still partly heuristic, however, the mathematically rigorous results are proved for certain cases. The proof is based on energy estimates and basic properties of the underlying partial differential equations. Numerical results are provided. Additionally, a multigrid method for the auxiliary convection–diffusion problem is briefly discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Weighted extended B-spline method for the approximation of the stationary Stokes problem

A new stabilized, mesh-free method for the approximation of the Stokes problem, using weighted extended B-splines (WEB-splines) as shape functions has been proposed. The web-spline based bilinear velocity–constant pressure element satisfies the so called inf–sup condition or Ladyshenskaya–Babus˘ka–Brezzi (LBB) condition. The main advantage of this method over standard finite element methods is that it uses regular grids instead of irregular partitions of domain, thus eliminating the difficult and time consuming pre-processing step. Convergence and Condition number estimates are derived. Numerical experiments in two space dimensions confirm the theoretical predictions.     

A two immiscible liquids penetration model for surface-driven capillary flows

This is a mathematical and numerical study of liquid dynamics in a horizontal capillary. We present a two-liquids model which takes into account the effects of real phenomena like the outside flow dynamics. Moreover, we report on results obtained by an adaptive numerical method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces

We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.      

Spectral asymptotics of two parameter nonlinear Sturm-Liouville problem

In this paper we establish an asymptotic formula of n ? th variational eigenvalue λ = λ_n(μ,α) of the nonlinear Sturm-Liouville problem with two parameters on the general level set N _α,μ as μ → ∞.     

A maximum modulus theorem for the Oseen problem

Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R ³. It is proved a maximum modulus estimate of the following form: If ${{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}$ and ${p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}$ in Ω, and if ${|{\bf u}| \le M}$ on ${\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}$ as ${|{\bf x}|\to \infty }$ , then ${|{\bf u}({\bf x})|\le c M}$ in Ω. Here the constant c depends only on Ω and λ.     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

An inverse boundary value problem for the Oseen equation

Based on the two‐dimensional stationary Oseen equation we consider the problem to determine the shape of a cylindrical obstacle immersed in a fluid flow from a knowledge of the fluid velocity on some arc outside the obstacle. First, we obtain a uniqueness result for this ill‐posed and non‐linear inverse problem. Then, for the approximate solution we propose a regularized Newton iteration scheme based on a boundary integral equation of the first kind. For a foundation of Newton‐type methods we establish the Fréchet differentiability of the solution to the Dirichlet problem for the Oseen equation with respect to the boundary and investigate the injectivity of the linearized mapping. Some numerical examples for the feasibility of the method are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

On the free boundary problem to the two viscous immiscible fluids

In this paper, we prove the local solvability of the free boundary problem describing the motion of two layers of immiscible, heavy, viscous, incompressible fluid lying above an infinite rigid bottom and with surface tension on the interfaces, and global solvability near the equilibrium state.     

Solutions of the Stokes,Oseen, and Navier-Stokes Equations in Three Dimensions

A constructive analytical procedure is described for generating a class of solutions pertaining to the fluid velocity field satisfying the three-dimensional Navier-Stokes equations. The arbitrary functions occurring in the solution are related to those that arise in the general solution of the Stokes and Oseen equations.     

An inhomogeneous boundary value problem for the stationary motion equations of Jeffreys viscoelastic medium

We consider the inhomogeneous Dirichlet problem for the stationary motion equations of an incompressible viscoelastic medium of Jeffreys type. We prove the existence of weak solutions of this problem. Moreover, we show that the weak solution set is sequentially weakly closed.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Plane couette flow of two immiscible incompressible fluids with uniform suction at the stationary plate

The contact layer in the plane couette flow of two immiscible fluids with uniform suction at the stationary plate has been investigated. The effect of suction is to supply an adverse pressure gradient to the fluid which causes back flow near the stationary plate. The streamlines are drawn to show the effect of uniform suction at the stationary plate on the contact layer.