The steady motions of a heavy ellipsoid filled with liquid on a plane with friction

The motion of a heavy uniform thin-walled ellipsoid of revolution, completely filled with an ideal incompressible liquid, performing uniform vortex motion is investigated. It is assumed that the ellipsoid is situated on a horizontal plane, from the side of which a normal reaction and a force of viscous sliding friction act on it. The equations of motion of the system, suitable both in the general case and in limiting cases of zero ellipsoid mass or zero liquid mass, are set up. Steady and periodic motions of the ellipsoid with the liquid are obtained. The conditions for uniform rotations of the ellipsoid about a vertically situated axis of symmetry to be stable are obtained.     

The plane isothermal motions of an ideal gas without expansions

A general solution of the equations of plane isothermal motions of an ideal gas without expansions is obtained. A method of reducing the overdetermined system of differential equations to an involution is proposed, which consists of obtaining integrable relations. All representations of the solution in time are obtained: polynomial, harmonic and biharmonic, and representations of linearly and exponentially increasing harmonics. All the representations give solutions of the overdetermined system, among which there are all the solutions with linear velocity fields, obtained by Ovsyannikov.     

A submodel of the rotational motions of a gas in a uniform force field

An invariant submodel, constructed using a subalgebra of the sum of the rotation, time transfer and Galilean transfer is considered within the framework of the Podmodeli program [1]. A group classification is constructed and simple solutions are obtained. The submodel is reduced to symmetrical form. Assertions are made on the hyperbolicity, characteristics and force discontinuities. The necessary conditions far solutions to exist without discontinuities on the axis of symmetry are derived and their asymptotic submodel is investigated.     

Continuous conjugation of special nonisentropic one-dimensional gas motions

The exact partially invariant solution of equations of motion of a compressible fluid describing the collapse of particles to a point and an instantaneous source from the point in a one-dimensional nonisentropic motion is cut off by the characteristics and glued into a continuous solution of a one-dimensional submodel in a finite domain. The possibility of a continuous periodic nonisentropic motion of a compressible fluid in a bounded domain under the action of a piston is shown.     

Goursat problem of the continuous conjugation of radial rectilinear motions of a gas

Gas dynamics equations have an isentropic solution describing the radial rectilinear motion of particles to the center and from the center with constant velocities. Two such solutions can be continuously conjugated if the Goursat problem is solved in a spatially similar domain with matched data on the characteristics. We prove the existence and uniqueness of a smooth solution of the Goursat problem in a small ball for a polytropic gas with exponent 5/3.     

On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities

We consider the Cauchy problem for the equations of selfgravitating motions of a barotropic gas with density-dependent viscosities μ(ρ), and λ(ρ) satisfying the Bresch–Desjardins condition, when the pressure P(ρ) is not necessarily a monotone function of the density. We prove that this problem admits a global weak solution provided that the adiabatic exponent γ associated with P(ρ) satisfies ${\gamma > \frac{4}{3}}$ .     

Contact discontinuity with general perturbations for gas motions

The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the Navier-Stokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the Navier-Stokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate , it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of . Thus, this reciprocal order of decay rates for the time evolution of the perturbation is essential to close the a priori estimate containing the uniform bounds of the L^∞ norm on the lower order estimate and then it gives the decay of the solution to the contact wave pattern.     

Small motions and eigenoscillations of a “fluid–barotropic gas” hydrosystem

This paper deals the problem of small motions and eigenoscillations of the hydrodynamical system” ideal fluid–barotropic gas” with regard for gravitational and capillary forces. The spectrum structure and the basis property of eigenfunctions are studied. The variational principles for eigenvalues are presented. The theorems of strong solvability of an initial boundary-value problem and the inverse Lagrange’s theorem of stability of the hydrosystem are proved.     

On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities

We consider the Cauchy problem for the equations of selfgravitating motions of a barotropic gas with density-dependent viscosities μ(ρ), and λ(ρ) satisfying the Bresch–Desjardins condition, when the pressure P(ρ) is not necessarily a monotone function of the density. We prove that this problem admits a global weak solution provided that the adiabatic exponent γ associated with P(ρ) satisfies ${\gamma > \frac{4}{3}}${\gamma > \frac{4}{3}}.     

Numerical investigation of a liquid displacing a gas in thin porous layers

This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which occurs e.g. during the filling process of a lithium-ion battery with an electrolyte. The investigation is based upon the local volume-averaged Navier-Stokes equations, using a Volume-of-Fluid method to treat the interface. For the flow the wall effect and capillary forces have to be considered. Capillary rise experiments are used to determine the permeability. Since the layers are thin and the characteristic size of the particles is comparatively large, friction with the electrode is taken into account with respect to the mobility of the contact line. The implemented models are validated against analytical results, showing a good agreement. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The steady motions of a three-body system in weightlessness conditions

The stability of the steady motions by inertia of a combination of three point masses, forming an open chain, is investigated using the Routh–Lyapunov theorem. The problem is investigated in two different formulations: in the first formulation the mean mass is fixed with respect to a thread, and in the second it can move along that thread without friction, while constant tension of the thread is ensured by additional devices, consuming external energy. In steady motions, the configurations of the arrangement of the masses in both systems are similar, but the stability conditions are found to be different.     

The conditions for the symmetric instability of the vortex motions of an ideal stratified liquid

The problem of the symmetric instability of the steady-state motions of an incompressible ideal liquid which is stratified with respect to its density is investigated in the case of two types of motion, axially symmetric and with translational symmetry. It is shown that the sufficient condition for stability obtained in [1] using a variational method (the direct Lyapunov method) for the motions under consideration is closely related to the extremal nature of their energy; stable motions are characterized by a conditional minimum of the energy. A minimum of the energy holds in the class of states for which a potential vortex, expressed in terms of the Lagrangian invariants, angular momentum and density, is represented by the same function of these invariants. Conditions for instability are formulated and estimates of the increase in the kinetic energy of perturbations are given.     

The Hamiltonian dynamics of self-gravitating liquid and gas ellipsoids

The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).      

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.     

The quasistatic motions of a three-body system on a plane

A controlled three-body system on a horizontal plane with dry friction is considered. The interaction forces between each pair of bodies are controls that are not subject to prior constraints but must be chosen in such a way that the motions of the system generated by them are quasistatic, that is, the total force acting on each of the bodies must be close to zero. All motions in which one body moves and the other two are fixed are found in the class of quasistatic motions. The problem of the optimal displacement of a moving body between two specified positions on a plane such that the absolute magnitude of the work of the friction forces along the trajectory is a minimum is solved. The quasistatic controllability of a three-body system is demonstrated and algorithms for bringing it into a specified position are discussed. The system considered simulates a mobile robot consisting of three bodies between which control forces act that can be realized by linear motors. The sizes of the bodies are assumed to be negligibly small compared with the distances between them so that the bodies are treated as particles.     

The optimal periodic motions of a two-mass system in a resistant medium

The rectilinear motions of a two-mass system, consisting of a container and an internal mass, in a medium with resistance, are considered. The displacement of the system as a whole occurs due to periodic motion of the internal mass with respect to the container. The optimal periodic motions of the system, corresponding to the greatest velocity of displacement of the system as a whole, averaged over a period, are constructed and investigated using a simple mechanical model. Different laws of resistance of the medium, including linear and quadratic resistance, isotropic and anisotropic, and also a resistance in the form of dry-friction forces obeying Coulomb's law, are considered.     

Numerical investigation into a liquid displacing a gas in thin porous layers

To examine the filling process in a lithium-ion battery, a numerical model to characterize the displacing flow of a liquid in air-filled pores of thin heterogeneous porous materials is elaborated. The investigation is based on the volume-averaged Navier-Stokes equations for small Reynolds numbers, using a volume-of-fluid method to cover the multiphase flow. The flow is investigated with respect to the wall effect and to capillary action within the porous matrix. On the one hand, model experiments with similar particle-size distributions as in the battery layers are conducted to extract the porosity as function of the wall distance. On the other hand, experiments with the three different porous layers of the battery are performed to receive mean values for the most important properties related to the two-phase flow. Results for the displacement flow in parts of the battery are presented and discussed, showing a considerable influence of the modeled effects onto the flow characteristics. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)