Nonlinear internal waves in a fluid of infinite depth in the presence of a horizontal magnetic field

An investigation is made into the propagation of long nonlinear weakly nonone-dimensional internal waves in an incompressible stratified fluid of infinite depth in the presence of a horizontal magnetic field. It is shown that such waves are described by an equation representing the extension of the Benjamin-Ono equation to the weakly nonone-dimensional case. The equation obtained differs from that obtained in [4], which is attributable to the anisotropy of the medium resulting from the presence of a magnetic field. The stability of a soliton with respect to flexural perturbations is investigated. A particular case of the variation of the density with height at constant Alfvén velocity is examined in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–72, November–December, 1987.     

Solitons in plasma with a hall dispersion

A system of equations of perfect magnetohydrodynamics is considered with allowance for Hall currents. The study of one-dimensional steady solutions which are damped at infinity can be reduced to the investigation of a Hamiltonian dynamic system with right-hand sides that are not single valued. A qualitative investigation of the system is carried out, with the determination of the region of existence of the given solutions. The solutions have the form of solitary waves — solitons. An exact solution in quadratures is obtained, which describes the structure of the solitons. The existence of two solitons of the Alfvén type is indicated. The existence domain of the corresponding solutions is analyzed. In the limiting cases of magnetosonic and Alfvén solitons, the solutions are expressed in explicit form in elementary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–168, September–October, 1985.     

Transition from regular reflection to mach reflection when a shock wave interacts with a wall in a two-phase gas—liquid medium

A study is made of the transition from regular reflection to Mach reflection when a plane moderately strong or weak shock wave interacts with a wall in a two-phase gas—liquid medium. An equilibrium model that differs from the model of Parkin et al. [1] by the introduction of the adiabatic velocity of sound is used to investigate shock wave reflection in the complete range of gas concentrations. For the reflection of weak shock waves, nonlinear asymptotic expansions [2] are used. In the limiting cases, the results agree with those already known for single-phase media [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 190–192, September–October, 1983.     

Motion of an incompressible conducting liquid in a strong self-consistent magnetic field

One of the common regimes of operation of many laboratory and industrial magnetohydrodynamic (MHD) devices using liquid metals as working medium is the regime for which the Alfvén number A, the ratio of the magnetic and kinetic energy densities, appreciably exceeds unity. For example, for a typical MHD device [1] with characteristic length 0.1 m of the working region, velocity 1 m/sec of the medium, and magnetic induction 1 T (the medium is molten sodium at temperature 330°C) the Alfvén number is A - 900. To simplify the investigation of the processes in such devices, one can use the approximation of a strong magnetic field proposed by Somov and Syrovatskii [2] to describe certain types of hydrodynamic flows of a dissipationless plasma in a magnetic field. In the present paper, the approach to the analysis of the self-consistent magnetohydrodynamic problem in this asymptotic approximation is extended to the case of an incompressible liquid with finite conductivity. A study is made of the closed reduced system of MHD equations obtained from the complete model in the zeroth order in the small parameter A^–1, in which the magnetic field is a force-free field. An investigation is made of the free diffusion of force-free magnetic field with constant coefficient a of proportionality between the current density and the magnetic induction in a spatially unbounded liquid, and the kinematic properties of a velocity field of the liquid in which the force-free nature of the magnetic field is maintained during the damping process are determined. It is shown that the complete class of such velocity fields is represented by the group of rigid-body motions of the liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1991.     

Internal streamwise action of saw-tooth sound waves on turbulent jets

The results of an experimental investigation of the acoustic field produced by turbulent subsonic jets under internal acoustic excitation are presented. It is shown that under the action of saw-tooth finite-amplitude waves the turbulent jets can radiate Mach waves into the ambient medium due to compact acoustic disturbances traveling along the jet at a velocity greater than the speed of sound in the surrounding space.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 153–158. Original Russian Text Copyright © 2004 by Pimshtein.     

Oblique interaction of alfvén and contact discontinuities in magnetohydrodynamics

A general method of solving problems of the interaction of stationary discontinuities is proposed. The problem of the oblique incidence of an Alfvén plane-polarized discontinuity on a contact discontinuity is examined in the general formulation. A solution is constructed numerically over the entire range of variation of the governing parameters. A number of effects associated with the magnetohydrodynamic nature of the interaction are explored. For example, the formation in space of sectors in which the density falls by several orders (almost to a vacuum) is detected. The solutions obtained are of interest, for example, for investigating the interaction between Alfvén discontinuities in the solar wind and the magnetopause, plasmopause and other inhomogeneities whose boundary can be approximated by a contact discontinuity [13–15].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 131–142, January–February, 1990.     

Speed of sound in two-phase vapor-liquid systems

The principal theorems of thermodynamics of irreversible processes are applied to the process of propagation of acoustic waves in a two-phase medium. Expressions are derived which determine the dependence of the sound speed in a vapor-liquid medium on the degree of dryness and the degree of nonequilibrium of different relaxation processes accompanying the propagation of acoustic waves. In the limiting case of equilibrium these expressions reduce to the well-known formulas obtained in equilibrium thermodynamics.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 78–85, September–October, 1970.     

Sound propagation and self-focusing in an inhomogeneous gas-liquid medium

The evolution of sound waves in a gas—liquid medium with an inhomogeneous distribution of the sound speed is considered in this paper on the basis of a nonlinear parabolic equation for the amplitude of the sound wave envelope. It is assumed that the nonlinearity due to the gas inclusions is much greater than the customary hydrodynamic nonlinearity. The influence of the inhomogeneity on the self-focusing of the sound is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1974.The author is grateful to I. R. Shreiber for discussion.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Lift-to-drag ratio at supersonic speeds

Wave lift-to-drag ratio is analyzed ignoring friction and using flows behind oblique shock waves and rarefaction waves. It is shown that the lift-to-drag ratio of an infinite oblique plate can surpass considerably that of triangular plates with subsonic, sonic, or supersonic edges. The simplest finite-span oblique wing is a wing with characteristic edges. However, when the normal-to-the-edge flow velocity component behind a shock reaches the speed of sound, the wing contracts into an edge, and other means must be used to exclude the end effect. Several possible variants are indicated. A straight wedge with side plates is the optimal shape for a lifting body with fixed volume, lift, length, and width. Under the same conditions, the cross-section of a pyramidal body formed by stream planes behind one or two plane shocks has practically no effect on the lift-to-drag ratio, while the region of high lift-to-drag ratio is much narrower than for a wedge. If a pyramid fails to provide the required lift-to-drag ratio, it is necessary to turn to forms that better fill the given area. Redistribution of lift between body and wing permits an improvement in the lift-to-drag ratio.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–141, September–October, 1993.     

Anomalous sound and low-frequency instability in an incompressible viscous liquid with bubbles of soluble gas

A form of instability in a liquid containing bubbles of soluble gas caused by the unbalanced supply of energy to the two-phase medium as a result of the redissolving of the gas atoms in the liquid is investigated. The presence of a phase transition [6] due to the gas atoms being redissolved in the carrier medium leads to the appearance of essentially new effects in the viscous liquid: undamped sound waves, anomalies in the phase velocity of the sound and low-frequency instabilities. These effects are investigated on the basis of the equation of state of a two-phase medium (liquid with bubbles of soluble gas) obtained in [7], which takes into account the kinetics of mass and energy transfer between the phases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–137, September–October, 1991.     

Impact on a plate on the surface of a fluid

Questions of the interaction between solid and elastic structures with an ideal fluid which are associated with the initial stage of the impact and penetration of bodies in the fluid were considered in [1–4]. Results are presented below of an analysis of a central impact on a solid weightless plate which is on the surface of a compressible fluid. The impact velocity is much less than the speed of sound in the medium. Computations are performed by a finite-difference Lagrange method according to a program for plane motions of a continuous medium [5] by using a volume artificial viscosity of Neumann-Richtmayer type [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–145, May–June, 1978.     

Hypersonic flow past a wing at large angles of attack with detached shock wave

The thin shock layer method [1–3] has been used to solve the problem of hypersonic flow past the windward surface of a delta wing at large angles of attack, when the shock wave is detached from the leading edge (but attached to the apex of the wing) and the velocity of the gas in the shock layer is of the same order as the speed of sound. A classification of the regimes of flow past a delta wing at large angles of attack has been made. A general solution has been obtained for the problem of three-dimensional hypersonic flow past the wing allowing for nonequilibrium physicochemical processes of thermal radiation of the gas at high temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–157, May–June, 1985.     

Asymptotic theory of the flow around an obstacle by a sonic flow

The first investigation of the problem of the flow around an obstacle by a gas flow whose velocity is equal to the speed of sound at infinity was carried out in [1, 2], where it is shown in particular that the principal term of the appropriate asymptotic expansion is a self-similar solution of Tricomi's equation, to which the problem reduces in the first approximation upon a hodographic investigation. The requirement that the stream function be analytic as a function of the hodographic variables on the limiting characteristic was an important condition determining the selection of the self-similarity exponent n (xy^–n is an invariant of the self-similar solution). The analytic nature of the velocity field everywhere in the flow above the shock waves, which arise from necessity upon flow around an obstacle, follows from this condition. The latter was found in [3], where one of the branches of the solution obtained in [1] was used in the region behind the shock waves. The principal and subsequent terms of the asymptotic expansion describing a sonic flow far from an obstacle were discussed in [4], where the author restricted himself to Tricomi's equation. Each term of the series constructed in [4] contains an arbitrary coefficient (we will call it a shape parameter) which is not determined within the framework of a local investigation, and consideration of the problem of flow around a given obstacle as a whole is necessary in order to determine these shape parameters. It follows from the results of [4] that the problem of higher approximations to the solution of [1] coincides with the problem, of constructing a flow in the neighborhood of the center of a Laval nozzle with an analytic velocity distribution along the longitudinal axis (a Meyer-type flow). Along with the Meyer-type flow in the vicinity of the nozzle center, which corresponds to a self-similarity exponent n=2, two other types of flow are asymptotically possible with n=3 and 11, given in [5]. The appropriate solutions are written out in algebraic functions in [6]. The results of [5] show that the condition that the velocity vector be analytic on the limiting characteristic in the flow plane is broader than the condition that the stream function be analytic as a function of the hodographic variables, which is employed in [1, 2, 4]. Therefore, the necessity has arisen of reconsidering the problem of higher approximations for the obstacle solution of F. I. Frankl'. It has proved possible for the region in front of the shock waves to use a series which is more general than in [4], which implies the inclusion of an additional set of shape parameters. The solution is given in the hodograph plane in the form of the sum of two terms; the series discussed in [4] corresponds to the first one, and the series generated by the self-similar solution with n=3 or with n=11 corresponds to the second one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 99–107, May–June, 1979.The authors thank S. V. Fal'kovich for a useful discussion.     

Sound absorption in a shock wave

It was shown in [1–4] that the reflection of a sound wave or its transmission through a shock front should be accompanied by attenuation or intensification of the wave is regarded as a discontinuity. In accordance with current representations [5, 6], a shock wave includes a viscous shock and a lengthy relaxation zone. Equilibrium is established with respect to translational and rotational degrees of freedom in the viscous shock and with respect to internal degrees of freedom in the relaxation zone. The result of the interaction of the shock and sound waves is determined by the relationship between the length of the sound wave and the width of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 90–94, May–June, 1986.     

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Transverse waves and discontinuities in an ideal magnetized fluid

Suppose that a constant and uniform field B₀ exists within an ideal fluid medium, that is, a medium in which dissipative processes are absent. If this medium is a conducting one and its magnetization and polarization can be neglected, then finite perturbations of the transverse (perpendicular to B₀) components of the velocity and the magnetic field propagate a-long b₀ with a constant velocity without change of form [1], These plane transverse waves, called Alfven waves, are linear and cannot lead to discontinuities if there are no discontinuities in the initial conditions. In this paper we shall consider plane transverse waves in an ideal fluid medium which is not only electrically conducting, but which can also be magnetized by the magnetic field. In such a medium transverse waves are no longer linear and they can develop into jumps in the magnitudes of the field and the velocity. Like Alfven waves, these waves leave the density of the medium unchanged, so that they can also exist in an incompressible fluid. This circumstance is reflected in the nature of the discontinuity, which is analyzed in Section 5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 116–124, November–December, 1973.The author thanks A. G. Kulikovskii and A. A. Barmin for their valuable remarks contributed during discussions of the work.     

Formation and propagation of cylindrical shock waves in a pulsed induction discharge

An experimental investigation was made into the formation and propagation of cylindrical collapsing shock waves in a pulsed induction gas discharge. The shock fronts were made visible and their propagation velocity measured by means of the schlieren method in conjunction with high-speed cine photography in the photoscanning regime and frame-by-frame detection, which made it possible, using the experimental data, to analyze the formation and propagation of the shock waves. A physical picture of the individual stages in the development of the discharge is given on the basis of the experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–133, January–February, 1981.We are thankful to Yu. P. Raizer for helpful discussions of the work.     

Theory of isothermal multicomponent sorption dynamics for high concentrations of mixture components

An analysis is made of the invariant solutions of the system of quasilinear equations of material balance which describe the motion of sorption shock and dispersing waves of concentration through a porous medium, when the flow velocity is variable (depending on the concentration of the components of a mixture of liquids or gases). It is shown that for linear sorption isotherms the problem formally reduces to one previously solved for a multicomponent system at constant flow velocity and Langmuir isotherms of the mixture. In the presence of dispersion factors and for linear sorption isotherms, solutions are obtained which describe the distributions of the concentrations in a traveling sorption wave regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 91–95, March–April, 1985.