Hydrodynamic stability of the spherical front of a reaction accompanied by a strong increase in viscosity

The hydrodynamic stability of the steady spherical front of a reaction accompanied by a strong increase in viscosity taking place in a radial, centrally symmetric flow of condensed reaction medium is considered. The flow is assumed to be bounded by two permeable concentric spherical surfaces. The stability is investigated in the linear approximation with respect to small perturbations. A relation is obtained for the dependence of the perturbation growth rate on the parameters of the problem: the viscosity ratio of the medium at the front, the ratio of the boundary surface radii to the radius of the front, and the dimensionless hydrodynamic resistances of the boundary surfaces. It is shown that the front is stable over almost the entire physical region of variation of the parameters and that instability occurs only in the case of a low hydrodynamic resistance of the outer boundary surface as the front approaches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 46–53, May–June, 1988.     

Vortex motion of a disperse mixture

A study is made of the motion of fine spherical particles in a given steady vortex flow of an incompressible fluid. The results are given of an investigation into the Lyapunov stability of a particle trajectory coincident with the vortex axis, and of trajectories from which the distance to the vortex axis is determined by the condition of equality of the radial components of the force of the phase interaction and of the centrifugal force which acts on the particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–169, November–December, 1984.     

Convective motion of a conducting liquid carrying a current in a transverse magnetic field

The convective instability of a layer of conducting liquid carrying a current and lying in a magnetic field perpendicular to the current is considered. The problem of the nonconductive approximation in a linear setting is solved. The relationships between the Rayleigh number and the Hartmann number (determining the neutral stability) are derived.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza. No. 2, pp. 20–26, March–April, 1971.     

Instability of a spherical gas bubble in a variable-pressure field

Only a few studies, of which we mention [1–5], have been addressed to the problem of the stability of the accelerated motion of a spherical interface of two fluids. In the present paper we consider the problem of the stability of radial motion of the spherical boundary of a gas bubble in an incompressible inviscid liquid under the action of variable external pressure. Surface tension is not taken into account. We study the possibility of the existence of stable motions for broad classes of time dependence of the external pressure, namely for monotonic and periodic dependences. It is shown that stability is possible only for infinitely large bubble radii or for very specific assumptions concerning the initial conditions and the pressure-time dependence law.     

Evolution of a Small Distortion of the Spherical Shape of a Gas Bubble under Strong Expansion-Compression

The evolution of a small distortion of the spherical shape of a gas bubble which undergoes strong radial expansion-compression upon a single oscillation of the ambient liquid pressure under a harmonic law are analyzed by numerical experiments. It is assumed that the distortions of the spherical bubble shape are axisymmetric and have the form of individual spherical surface harmonics with numbers of 2–5. Bubble-shape oscillations prior to the beginning of expansion are taken into account. Generally, the distortion value during bubble expansion-compression depends on the phase of bubble-shape oscillation at the beginning of the expansion (initial phase). Emphasis is placed on the dependence of the maximum distortions in the initial phase at certain characteristic times of bubble expansion-compression on the amplitude of the external excitation, liquid viscosity, and distortion mode (harmonic number). The parameters of the problem are typical of the stable periodic sonolumiescence of an individual air bubble in water at room temperature. An exception is the liquid pressure oscillation amplitude, which is varied up to values that are five times the static pressure. That large excitation amplitudes are beyond the stability threshold of periodic oscillations of spherical bubbles. Their consideration is of interest from the point of view of increasing the compression ratio of the bubble gas, i. e., increasing the maximum temperature and density achievable in the final compression stage.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 17–28, July–August, 2005.     

Parametric Excitation of a Secondary Flow in a Vertical Layer of a Fluid in the Presence of Small Solid Particles

Thermal convection is studied in an inhomogeneous medium consisting of a fluid and a solid admixture under conditions of finite–frequency vibrations. Convection equations are derived within the framework of the generalized Boussinesq approximation, and the problem of flow stability in a vertical layer of a viscous fluid with horizontal oscillations along the layer to infinitely small perturbations is considered. A comparison with experimental data is made.     

Numerical analysis of oscillations of a gaseous cavity in a compressible liquid

The radial oscillations of a gas-filled spherical cavity in an infinite compressible ideal liquid are investigated. The numerical solution of the equation of motion of a gaseous sphere is used for the comparison of acoustic and energy characteristics of underwater trotyl explosion and small-amplitude explosion sound sources (detonation of explosive gaseous mixture, expansion of a compressed inert gas).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 176–182, March–April, 1971.     

Nonlinear interaction of longitudinal—Transverse acoustic waves in a circular cylinder

The limiting amplitudes of acoustic oscillations in a cylindrical volume of a heat releasing medium in which one or several modes are unstable in the linear approximation are determined. One of the mechanisms limiting the amplitudes of unstable acoustic modes is the transfer of energy from them to damped modes by nonlinear interaction. The nonlinear interactions of plane acoustic waves in a long channel have been considered by Artamonov and Vorob'ev [1]; in the present paper, the interaction of mixed longitudinal—transverse acoustic modes in a closed cylindrical volume is considered. The equations describing the interaction of two and three longitudinal—transverse modes are derived and investigated in the quadratic approximation by the method of slowly varying amplitudes and phases of the oscillations [2]. The treatment is applicable to a high-temperature gas, for which general stability conditions in the linear approximation have been formulated by Artamonov [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–9, September–October, 1982.I should like to express my thanks to K. I. Artamonov (deceased) for suggesting the problem and for scientific supervision and A. P. Vorob'ev for constant interest in the work and helpful advice.     

Stability of viscous flow between rotating and stationary disks

The linear problem of the stability of viscous flow between rotating and stationary parallel disks is solved in the locally homogeneous formulation using the method of normal modes. The main flow is assumed to be selfsimilar with respect to the radial coordinate. The system of sixth-order equations, derived for the amplitude functions of the disturbances, is integrated by a finite difference method. The stability characteristics with respect to disturbances of four types are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 79–87, November–December, 1991.     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

Dynamics of a cavitational cavity in a non-newtonian fluid

The dynamics of a spherical cavity in a non-Newtonian fluid, described by the Reiner-Rivlin rheological equation [1], is investigated. The equation of radial cavity motion is obtained, where the gas in the cavity is subject to a polytropic law and surface tension is taken into account. The equation of cavity motion is solved numerically for a number of values of the transverse viscosity coefficient. The influence of the transverse viscosity on the collapse process of vapor and gas-filled cavities is shown. Numerical computations are also carried out for the rate of energy dissipation and the pressure distribution in the fluid.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 170–173, July–August, 1973.The authors are grateful to A. T. Listrove for attention to the research.     

Mass transfer between particles and a flow in the presence of a volume chemical reaction

The exterior problem of the mass transfer between a spherical drop and a linear shear flow in the presence of a first-order volume reaction is solved in the diffusion boundary layer approximation. A simple approximate expression for calculating the average Sherwood number for a drop or solid particle of arbitrary shape is proposed. At large Péclet numbers this expression is applicable to any type of flow over the entire range of variation of the reaction rate constant. The problem of diffusion to a spherical drop in a translational Stokesian flow in the presence of a first-order volume reaction was investigated in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 109–113, November–December, 1987.     

A spherical wave in a viscoelastic half-space

The propagation of spherical waves in an isotropie elastic medium has been studied sufficiently completely (see, e.g., [1–4]). it is proved [5, 6] that in imperfect solid media, the formation and propagation of waves similar to waves in elastic media are possible. With the use of asymptotic transform inversion methods in [7] a problem of an internal point source in a viscoelastic medium was investigated. The problem of an explosion in rocks in a half-space was considered in [8]. A numerical Laplace transform inversion, proposed by Bellman, is presented in [9] for the study of the action of an explosive pulse on the surface of a spherical cavity in a viscoelastic medium of Voigt type. In the present study we investigate the propagation of a spherical wave formed from the action of a pulsed load on the internal surface of a spherical cavity in a viscoelastic half-space. The potentials of the waves propagating in the medium are constructed in the form of series in special functions. In order to realize viscoelasticity we use a correspondence method [10]. The transform inversion is carried out by means of a representation of the potentials in integral form and subsequent use of asymptotic methods for their calculation. Thus, it becomes possible to investigate the behavior of a medium near the wave fronts. The radial stress is calculated on the surface of the cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 139–146, March–April, 1976.     

Conditions of toroidal equilibrium of a moving plasma

Necessary conditions for stationary plasma flow in a magnetic field of toroidal geometry are derived in a magnetohydrodynamic approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 31–35, January–February, 1976.     

Numerical study of the dynamics of a vapor envelope round a heated solid particle in a liquid

A study is made of the radial motion of a vapor envelope surrounding an isolated spherical particle in an unbounded mass of liquid. It is assumed that the liquid is viscous and incompressible and that the temperature is distributed uniformly in the solid particle. A model of a calorifically perfect gas is used for the vapor phase. The same assumptions are made as in Rayleigh's formulation for the problem of the dynamics of a single bubble: that the process is spherically symmetric and that the pressure P₂ (t) in the vapor phase is homogeneous. The justification for making these assumptions in problems of the dynamics of gas, vapor, and vaporgas bubbles is discussed in [1–5]. In this paper, the collapse of the vapor layer and the boiling of the liquid on the surface of the heated particle are not considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 154–157, July–August, 1985.     

Convective instability of a system of horizontal layers of immiscible liquids with a deformable interface

The convective stability of equilibrium is considered for a system of two immiscible fluids which differ little in density. A generalized Boussinesq approximation is developed, making it possible to take the interface deformations properly into account. The stability of the equilibrium state of two fluids in a horizontal layer with a vertical temperature gradient is investigated. Several instability mechanisms are identified: long-wave and cellular monotonic disturbances and oscillatory disturbances. Increasing the deformability is shown to cause switching between instability mechanisms.Perm. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 32–39, March–April, 1996.     

“Bunched” model of corona discharge in a gas-dynamical flow

In the present paper, a theoretical model of corona discharge is proposed for the case when electric charge transport is implemented by means of the motion of discrete charged bunches of finite dimensions. A system of equations and boundary conditions is formulated for the study of unsteady cyclic processes in a corona discharge. The electric field induced by the space charge of bunches and the presence of an external electric circuit are taken into account. A solution of the formulated system of equations for corona discharge with spherical geometry is obtained. The integrated (current-voltage) characteristics and the amplitude-frequency characteristics of the corona discharge are found. The proposed theory is generalized to the case of a corona discharge in a moving gas. The unsteady characteristics of corona discharge with spherical geometry for gas motion in a radial direction are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 153–160, January–February, 1986.The authors wish to express their gratitude to V. A. Likhter and V. I. Shul'gin for their useful discussions and valuable observations.     

Nonlinear stabilization of unstable acoustical vibrations in a bounded heat-evolving medium

The working process in many power units, taking place with the evolution of energy, under determined conditions is acoustically unstable in a linear approximation. The problem of determination of the amplitudes of the unstable waves, set up as a result of the pumping of energy from the unstable mode to the damping mode with their linear interaction is of practical interest. In this paper equations are derived for the fully established amplitudes of plane acoustical vibrations in a three-wave approximation, taking account of boundary impedances, breaking down the internal resonance of the acoustical overtones excited. The discussion regards a high-temperature heat-evolving gas, whose general stability conditions were formulated in [1] and discussed also in [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 34–41, November–December, 1978.     

Electroconvective stability of a weakly conducting liquid

In inhomogeneous electric fields, at sufficiently high field strengths, a weakly conducting liquid becomes unstable and is set in motion [1–4]. The cause of the loss of stability and the motion is the Coulomb force acting on the space charge formed by virtue of the inhomogeneity of the electrical conductivity of the liquid [4–13]. This inhomogeneity may be due to external heating [4–6], a local raising of the temperature by Joule heating [2, 7, 8], and nonlinearity of Ohm's law [9–13]. In the present paper, in the absence of a temperature gradient produced by an external source, a condition is found whose fulfillment ensures that the influence of Joule heating on the stability can be ignored. Under the assumption that this condition is satisfied, a criterion for stability of a weakly conducting liquid between spherical electrodes is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1979.     

Numerical study of unsteady evaporation and heat transfer from a spherical surface

An unsteady radial problem of evaporation and heat transfer from a spherical surface is considered for a model kinetic equation. The problem is solved numerically using a second-order implicit conservative method.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 181–192. Original Russian Text Copyright © 2005 by Titarev and Shakhov.