Ray method for flexural vibrations of a shell immersed in a liquid

Equations are obtained by the ray method for the flexural vibrations of a shell accompanied by a surface wave in a liquid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 111–123, 1976.     

Asymptotics of homogeneous oscillations in a compressible viscous fluid

The large amplitude high frequency oscillations of the flow of a compressible viscous fluid have been shown to obey to an integro-differential system. We consider here the case of homogeneous oscillations, with both gas-like pressure law and a van der Waals one. We show that the solution admits a limit as the time increases. This limit is constant in the former case, but may take up to three distinct values in the latter.     

Resonance frequencies of a cavity containing a compressible viscous fluid

The aim of this paper is to study the resonance spectrum of a cavity containing a compressible viscous fluid. This system admits a discrete infinite sequence of eigenvalues whose real parts are negative, which is interpreted as the damping effect introduced by viscosity. Only a finite number of them have non-zero imaginary parts and this number depends on viscosity; a simple criterion is given for their position in the complex plane. The case of a cavity containing an elastic mechanical system immersed in the fluid is also examined; from a qualitative point of view, the nature of the resonance spectrum remains unchanged.     

The motion of a compressible viscous fluid around rotating body

We consider the motion of compressible viscous fluids around a rotating rigid obstacle when the velocity at infinity is non zero and parallel to the axis of rotation. We prove the existence of weak solution.     

Boundary tangency for density interfaces in compressible viscous fluid flows

We prove that, for solutions of the Navier-Stokes equations of two-dimensional, viscous, compressible flow, curves which are initially transverse to the spatial boundary and across which the fluid density is discontinuous become tangent to the boundary instantaneously in time. This effect is seen to result from the strong pressure gradient force, which in this case includes a vector measure supported on the curve, together with the fact that singularities in this system are convected with the fluid velocity.     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

Analyticity of the semigroup generated by the operator of a viscous compressible fluid

The linearized initial‐boundary value problem describing small motions of the viscous, barotropic compressible fluid in a bounded vessel is studied under various boundary conditions (Dirichlet, Neumann and intermediate). It is shown that the corresponding operator generates an analytic semigroup in the space L₂. Copyright © 1999 John Wiley & Sons, Ltd.     

On global motion of a compressible viscous heat-conducting fluid bounded by a free surface

We consider the motion of a viscous compressible heat-conducting fluid in ³ bounded by a free surface which is under constant exterior pressure. We present the global existence theorems in two cases: when the free surface is under the surface tension and without it.     

On the plane couette flow of a viscous compressible fluid with transpiration cooling

The exact solution for the plane couette flow of a viscous compressible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity profiles and the enthalpy are shown graphically.     

Adaptive approximation of a filtration problem for a viscous compressible fluid in a fissured porous medium

A filtration problem for a viscous compressible multiphase fluid mixture in a porous medium is considered in the case of high degree anisotropy caused by fissures. A correct spatial discretization is discussed. A comparison between the subgrid method and other multipoint methods is performed.     

The vibrations of a conical shell filled with a variable volume of fluid

We consider the vibrations of a conical shell filled with a fluid. We obtain a system of differential equations that describes the interaction of a variable volume of fluid with the shell. We solve the problem of determining the natural frequencies for a completely full shell. We obtain a formula for computing the fundamental frequency as a function of the geometric parameters of the shell. One figure. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 85–88, 1991.     

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.     

Fluid dynamics of an underwater electric discharge on the axis of a shell immersed in a fluid

We study the motion of a fluid during an electrical discharge on the axis of an elastic cylindrical shell immersed in the fluid. We estimate the influence of the parameters of the shell on the dynamics of a gasvapor chamber. We point out several properties of the computation of the pulsed flows in an infinite volume of fluid (the choice of the method of constructing the grid, the limitation of the computational region). We obtain the pressure distribution over space at large distances from the shell. We analyze the influence of the shell on the parameters of the flow in the interior and exterior regions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 120–126.