On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion

The linearized initial boundary value problem describing the motion of the viscous compressible fluid is studied under Dirichlet zero condition in bounded and unbounded domains. The resolvent estimate for the corresponding operator is proved in the L_p framework and the sharp inner estimate of the resolvent set is obtained. Copyright © 2004 John Wiley Sons, Ltd.     

Exact solution of a boundary value problem describing the uniform cylindrical or spherical piston motion

In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill [1], we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill [1]. We find that although the approximate solution of Lighthill [1] gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s [1] solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions.     

Solvability of a certain problem on the plane motion of two viscous incompressible fluids with free noncompact boundaries

The solvability of a certain two-dimensional boundary-value problem for the system of the Navier-Stokes equations, describing the steady (partially common) motion of two heavy viscous incompressible capillary fluids with free noncompact boundaries, is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 146–153, 1987.     

Axially symmetric wave motion of a viscous liquid in a linearly viscoelastic tube

The problem of the propagation of progressive waves in a tube made of a linearly viscoelastic material which encloses a viscous Newtonian liquid is examined. For numerical calculations, it is proposed that the behavior of the tube wall material be described by the Voigt model. Dispersion curves are constructed for this case.S. M. Kirov Azerbaidzhan State University, Baku. Translated from Mekhanika Polimerov, No. 2, pp. 317–321, March–April, 1976.     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).     

A numerical method for the motion of a mixture of two viscous incompressible fluids

A finite difference technique for the simulation of the motion of a mixture of two viscous incompressible fluids in a closed basin is presented. The mathematical model which has been discretized is the closed system deduced from the general equations, governing the motion of the mixture. The numerical scheme is based on the marker and cell method [4] extended to consider the molecular diffusion process. Computational examples are described and discussed at the end of the paper.     

Solvability of a stationary boundary value problem for a model system of the equations of barotropic motion of a mixture of compressible viscous fluids

Under consideration is some boundary value problem for a model system of equations that describes the steady barotropic motion of a homogeneous mixture of compressible viscous fluids in a bounded three-dimensional domain. We prove the existence theorem for weak solutions of the problem, imposing no restrictions on the structure of total viscosity matrix except the standard requirements of positive definiteness.     

Problem of the motion of two viscous incompressible fluids separated by a closed free interface

A local existence theorem for the problem of unsteady motion of a drop in a viscous incompressible capillary fluid is proved in Sobolev spaces. A linearized problem with known closed interface is also studied in Holder spaces of functions.     

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.     

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.     

Solvability of the initial-boundary-value problem for the equations of motion of a viscous compressible fluid

It is proved that the initial-boundary-value problem for the system of equations describing the motion of a compressible fluid with a constant viscosity is locally solvable with respect to time. The heat conductivity is not taken into account. The solution is found in the class W _q ^2.1 , q>3.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 128–142, 1976.     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain ${\Omega = \mathbb{R}^3_+}$ with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ${\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}$ .     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.