Strong expansion–contraction of a cavity in a fluid under acoustic action

The simplest model of the strong expansion–contraction of a spherical evacuated cavity in an unbounded ideal incompressible fluid under acoustic action is constructed. It is assumed that the cavity is formed at the instant of maximum dilatation of the fluid as the result of the breakdown in its continuity by elementary particles, and the amplitude of the induced acoustic oscillations exceed the mean pressure in the fluid by an order of magnitude or more. The trajectory of the motion of the interface is divided into three stages, in each of which it is assumed that there is a certain constant value of the pressure. This enables an approximate analytical solution of the problem to be obtained. It is shown that there is an agreement with the numerical solution of the initial problem that is sufficient for qualitative estimates. The solution obtained enables a parametric analysis of the expansion–contraction processes of a cavity to be carried out and enables the volume of calculations in the numerical modelling to be reduced when taking account of additional factors.     

Investigation of the problem of the flow of a compressible viscous fluid past a bounded body in the hölder classes of functions

It is shown that the stationary exterior problem for the equations of motion of a viscous compressible fluid is uniquely solvable in weighted Hölder spaces if the exterior forces and the value of the velocity at infinity are sufficiently small. As a weight function, a power function (1+|x|)^?m, m>0, is taken. The proof, which is carried out by the method of decomposition, relies on the estimates of singular integrals and solutions of the transport equations in weighted spaces. Bibliography: 5 titles.     

Solvability “in the small” of a nonstationary flow problem for an ideal incompressible fluid,I

For the system of Euler equations and the incompressibility equation one considers the following initial-boundary value problem: the field of velocities is prescribed at the initial moment and for all to one gives the following boundary conditions: on the entire boundary of the normal component of the velocity is prescribed and on that part S₁ of the boundary of where inflow occurs one prescribes the value of the velocity =rot |S₂, whose components satisfy a certain necessary equality, derived in the paper. For such a problem one proves its unique solvability on a small interval of time.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 39–56, 1980.     

Mathematical analysis of a fluid—solid interaction problem

This paper analyzes a fluid—solid interaction model which describes the interaction between an inviscid fluid and an elastic solid In the model, the linear elastodynamic equations complemented with appropriate interface and boundary conditions are used to describe the wave propagation in the fluid and solid regions, and absorbing boundary conditions are used to minimize unphysical wave reflections. It is shown that the initial boundary value problem of the mathematical model posses a unique global (in time) quasi-strong solution. Regularity of the quasi-strong solution is also obtained under some reasonable assumptions on the data and on the domain.     

Energy criterion of nonlinear stability of a flow of the Rivlin–Ericksen fluid in porous medium

Nonlinear stability of the motionless state of the thermosolutal Rivlin–Ericksen fluid in porous medium for the case of stress-free boundaries is studied by generalized energy method. By means of introducing an energy functional we will prove a sufficient condition for unconditional nonlinear stability of the motionless state. The nonlinearly stabilizing effect of concentration on the system is proved. Furthermore, for certain range of system parameters our sufficient condition for stability is also necessary.     

On a linearized problem arising in the Navier–Stokes flow of a free liquid jet

In this work, we analyze a Stokes problem arising in the study of the Navier–Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.     

On the existence and the uniqueness of the solution of a fluid–structure interaction scattering problem

The existence and uniqueness of the solution of a fluid–structure interaction problem is investigated. The proposed analysis distinguishes itself from previous studies by employing a weighted Sobolev space framework, the DtN operator properties, and the Fredholm theory. The proposed approach allows to extend the range of validity of the standard existence and uniqueness results to the case where the elastic scatterer is assumed to be only Lipschitz continuous, which is of more practical interest.     

On the practical possibility of a ‘no show paradox’ under the single transferable vote

In a recent paper Brams and Fishburn (1983) have demonstrated that the ‘single transferable vote’ system for a single member electoral district admits a ‘no show paradox’. The present work indicates that under the single transferable vote the probability of occurence of a no show paradox is very high in situations where the outcome under the single transferable vote differs from that under plurality. This considerably weakens the case for replacing the plurality system by the single transferable vote system. It is also shown that under the single transferable vote the occurrence of a ‘no show paradox’ is consistent with one-dimensional left-right voting behaviour.     

On flow of a Navier–Stokes fluid in curved pipes. Part I: Steady flow

We consider the steady, fully developed motion of a Navier–Stokes fluid in a curved pipe of cross-section $D$ under a given axial pressure gradient $G$. We show that, if $G$ is constant, this problem has a smooth steady solution, for arbitrary values of the Dean’s number $\kappa$, for $D$ of arbitrary shape and for any curvature ratio $\delta$ of the pipe. This solution is also unique for $\kappa$ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if $\kappa =0$. Finally, we show the same properties for the approximations to the Navier–Stokes equations called “Dean’s equations” and provide a rigorous way in which solutions to the full Navier–Stokes equations approach those to this approximation in the limit of $\delta \to 0$.     

Sensitivity of the slip rate coefficient in fluid flow – poroelastic coupling conditions

To model flow-induced structural vibrations, an interface to couple fluid flow and poroelastic material in a finite element formulation has been developed. One parameter of this interface condition is the slip rate coefficient, resulting from the so-called Beavers-Joseph-Saffman condition. This condition states that the jump in tangential velocity at a fluid flow – porous interface is proportional to the shear stress. Up to now no a priori determination of this parameter exists, and the known parameter range has been deducted from measurements, i. e., in our case from the results of the pore-resolving simulations. When modeling realistic problems assuming incompressible fluids, vectorial flow velocity and scalar pressure interact with the poroelastic material. As the slip rate coefficient only influences the tangential contributions, its overall influence is not clear. In this work, the sensitivity of the slip rate coefficient regarding the interface's coupling conditions is evaluated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the reconstruction of the potential in the inverse problem for a power of the Sturm—Liouville operator

In this paper, we prove the existence and uniqueness theorem for the inverse problem of spectral analysis for a power of the Sturm—Liouville operator on an interval with power > 3/4 and a quadratically summable, complex-valued, nonsymmetric potential. In this case, the potential is reconstructed by the mix of four spectra or by two spectra.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 8 , Suzdal Conference—2, 2003.This revised version was published online in April 2005 with a corrected cover date.     

A quantum particle under the action of “white noise” type forces

We obtain the Schrödinger equation for the wave function of a particle interacting with the external field having a potential containing a component of the white noise type and the Kolmogorov equations for the distribution of a wave function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 907–914, July, 1993.     

On the geometric analysis of a quartic–quadratic optimization problem under a spherical constraint

Mathematical Programming - This paper considers the problem of solving a special quartic–quadratic optimization problem with a single sphere constraint, namely, finding a global and local...     

Steady flow of a viscous incompressible fluid in an unbounded «funnel-shaped» domain

In this paper we consider a domain which is tube-like at one exit to infinity and the halfspace at the other side. We prove existence of steady motions for the Navier-Stokes problem and for the case in which the fluid is moving through a porous medium at rest filling . In both cases the proof holds for arbitrary fluxes. We describe the asymptotic behaviour of the solutions in the halfspace for both problems.