On a Linear Problem Connected with Equilibrium Figures of a Uniformly Rotating Viscous Liquid

We consider the behavior (for large time) of the solution to a linear problem arising in the study of the stability of an equilibrium figure of a viscous incompressible liquid that is not subject to the action of the surface tension forces. Bibliography: 16 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 113–128.     

Estimates for the Resolvent of the Operator Appearing in the Study of Equilibrium Figures of a Rotating Viscous Incompressible Fluid

An estimate in the -norm is obtained for a solution to a linear problem with parameter connected with the evolution problem for an isolated liquid mass. Bibliography: 7 titles.     

On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

两个旋转球之间粘性不可压缩流动的Lorenz系统

两个不同角速度旋转球之间粘性流动问题是地球外部大气流动的简化模型.通过引入球Bessel函数的有理表达式,得到Stokes算子特征值与特征函数的有理表达形式.利用Stokes算子特征函数作为基函数系,对两个旋转球间流动问题进行谱Galerkin逼近.由三模态的Glerkin逼近方程得到—个类Lorenz系统,我们对此系统进行分歧问题和吸引子的讨论,从而得到原问题的稳定性判定.     

Arbitrary Motions of a Viscous Incompressible Liquid in a Finite Region

By using an entirely constructive analysis it is shown that there exist solutions of the viscous flow equations in which two of the Cartesian components of velocity are arbitrary and the third component can be constructed uniquely by the application of integrability conditions. The solutions are in general defined in a finite region of the fluid and can be matched onto a well-defined deterministic solution of the Navier–Stokes equations such that all of the relevant physical quantities are continuous at the interface.     

Influence of Inertia of a Compliant Surface on Viscous Instability of an Incompressible Boundary Layer

Computational Mathematics and Mathematical Physics - An incompressible boundary layer on a compliant plate is considered. The influence exerted by the inertia of the plate on the stability of the...     

Double Potential Method for Modeling the Internal Flow of a Viscous Incompressible Liquid

Doklady Mathematics - The problem of numerical modelling water purification from iron impurities is considered. The cleaning task is relevant for many industrial applications, including the...     

Global Existence of Heat-Conductive Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heat-conducting incompressible viscous fluids in \(\mathbb{R}^{2}\), where the viscosity and heat-conductivity coefficient vary with the temperature. It is shown that the Cauchy problem has a unique global-in-time strong solution \((u, \theta)(x,t)\) on \(\mathbb{R}^{2}\times(0,\infty)\), provided the initial norm \(\|\nabla u_{0}\|_{L^{2}}\) is suitably small, or the lower-bound of the coefficient of heat conductivity (i.e. \(\underline{\kappa}\)) is large enough, or the derivative of viscosity (i.e. \(|\mu'(\theta)|\)) is small enough.     

On the Instability of Certain Motions of an Ideal Incompressible Fluid

Short-wave instabilities in the general three-dimensional time-dependent flows of an ideal incompressible fluid are considered. A new geometrical optics equation for the perturbed vorticity amplitude is derived. It is shown that under certain conditions this equation can be solved explicitly. Using this observation a number of known instability conditions are obtained and some new ones are derived.     

Small Motions and Normal Oscillations of a Double Pendulum with Cavities Containing a Viscous Incompressible Liquid

The linear hydrodynamic problem involving the small motions and normal oscillations of a double pendulum with cavities completely filled with a liquid is examined. The problem is solved using the methods of functional analysis. An existence theorem is formulated for the solutions to the Cauchy problem and the properties of the normal oscillations are described.     

A Class of Solutions for the Equations Governing the Steady Two-Dimensional Motion of a Viscous Incompressible Liquid

A general viscosity dependent solution for the stream function is found satisfying the simplest nondegenerate form of the steady flow Navier-Stokes equations for a viscous incompressible liquid. The solution is two-dimensional and is expressed in terms of arbitrary analytic functions in the fluid domain. This class of flows is generated by complex stream functions, and the region of definition is restricted by an inequality containing these analytic functions. A general potential flow, and degenerate Stokes or creeping flows are recovered as particular solutions in limiting cases.     

非牛顿粘性不可压流方程的近似惯性流形

本文利用对非牛顿粘性不可压缩流方程对时间 t的解析性和长时间渐近性估计 ,具体构造了它的近似惯性流形 ,并得出收敛阶估计 .     

A Constructive Analytical Approach to the Calculation of Flows Representing the Steady Motion of a Viscous Incompressible Liquid

A constructive analytical technique is described for the calculation of the fluid velocity components satisfying the Navier-Stokes equations representing the steady motion of a viscous, incompressible liquid. These components are found in terms of the derivatives of two functions which in turn satisfy linear partial differential equations for which known solutions of general type are readily available.     

On the Lorenz Systems for the Incompressible Flow between Two Concentric Rotating Cylinders

A spectral method is used to derive a series of equations for axisymmetric Couette-Taylor flow. A three-modes system, which is similar to the Lorenz systems, is obtained by a suitable three-modes truncation of the Navier-Stokes equations for the incompressible flow between two concentric rotating cylinders. The stability of the three-modes systems is discussed. Moreover, the existence of its attractor and the estimation of Hausdorff dimension are given.     

Blow-Up Solution of the 3D Viscous Incompressible MHD System

The 3D viscous incompressible magneto-hydrodynamic (MHD) system comprised by the 3D incompressible viscous Navier-Stokes equation couples with Maxwell equation. The global well-posedness of the coupled system is still an open problem. In this paper, we study the Cauchy problem of this coupled system and establish some logarithmical type of blow-up criterion for smooth solution in Lorentz spaces.     

About the Decay of Vorticity in a Two-dimensional Incompressible Viscous Fluid

The main purpose of this article is to investigate the behavior of the vorticity in a two dimensional incompressible viscous medium. We give bounds for the maximal amount of vorticity which can be concentrated in a domain of a given size. In order to obtain bounds we compare the vorticity equation to the heat equation in ²     

Absolute Instability of Incompressible Boundary Layer over a Compliant Plate

An incompressible boundary layer on a compliant plate is considered. The influence exerted by the tensile stress and bending stiffness of the plate on the stability of the boundary layer is investigated in the limit of high Reynolds numbers on the basis of the triple-deck theory. It is shown that upstream-propagating growing waves can be generated in a certain range of parameters characterizing the plate properties. As a result, the flow becomes absolutely unstable in the conventional sense.     

Strong Solutions for Nonhomogeneous Incompressible Viscous Heat-Conductive Fluids with Non-Newtonian Potential

We consider the Navier-Stokes system with non-Newtonian potential for heat-conducting incompressible fluids in a domain Ω R3. The viscosity, heat conduc- tion coefficients and specific heat at constant volume are allowed to depend smoothly on the density and temperature. We prove the existence of unique local strong solu- tions for all initial data satisfying a natural compatibility condition. The difficult of this type model is mainly that the equations are coupled with elliptic, parabolic and hyper- bolic, and the vacuum of density cause also much trouble, that is, the initial density need not be positive and may vanish in an open set.     

The Incompressible Limits of Viscous Polytropic Fluids with Zero Thermal Conductivity Coefficient

ABSTRACT

We study the singular limit of viscous polytropic fluids without thermal conductivity as the Mach number tends to zero. A uniform existence result for the Cauchy problem in R ³ is proved under the assumption that the initial data belongs uniformly to H ^k (R ³) with k = 2, 3 and is well-prepared in H ¹ (R ³).