Stability of an ionizing shock wave in an electromagnetic field

The stability of a plane shock wave, which is ionizing a gas, against small two-dimensional perturbations in the presence of a uniform electromagnetic field is analyzed. The applied electric field is assumed to be normal to the wave front while the magnetic field is parallel to the front and perpendicular to the plane in which the perturbations are propagating. The medium is assumed to be a two-parameter gas. The case of an ideal gas is analyzed in detail. The presence of an electric field component normal to the front leads to the formation of a surface charge at the front whose interaction with the electromagnetic field considerably affects the stability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 134–142, November–December, 1978.In conclusion, the authors thank A. G. Kulikovskli for useful discussions of the present work.     

Method of narrow strips for nonlinear problems of a stratified fluid

Plane steady flow is considered for an ideal incompressible stratified fluid in a gravitational field of force. It is a characteristic feature of these flows that the density is constant and Bernoulli's constant remains the same along a streamline. Internal waves arise because of ponderability in the stratified fluid; they are not due to the presence of a free surface. These wave motions are studied in detail in the linear formulation, but flows of the solitary wave type can be described only by nonlinear equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–178, March–April, 1986.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Influence of soluble surface-active substances on emulsification

A study is made of the influence of a film of soluble surface-active substance on the emulsification of drops of a viscous liquid in the field of an acoustic wave. The drops are in an ideal incompressible fluid. It is assumed that the drop radius is much less than the acoustic wavelength but much greater than the capillary radius. The motion of the drops is not taken into account.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 160–164, September–October, 1981.I thank V. A. Briskman for constant interest in the work.     

Internal waves in a two-layer electrically conducting fluid in the presence of a magnetic field

The propagation of linear and nonlinear internal waves along the interface between two weakly conducting media differing in density and electrical conductivity is investigated and the influence of MHD interaction effects on their characteristics is analyzed. It is shown that in this system the waves propagate with dispersion and dissipation, and for harmonic waves of infinitesimal amplitude there exists a range of wave numbers on which propagating modes do not exist. For waves of finite amplitude a nonlinear Schrödinger equation with a dissipative perturbation is obtained and its asymptotic solution is found. It is established that the presence of electrical conductivity and an applied magnetic field leads to a decrease in the amplitude and the frequency of the envelope of the wave train.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–108, September–October, 1990.     

Absolute instability of an interface between two magnetic fluids in the presence of a tangential velocity discontinuity and an external magnetic field

The nature of the instability of the surface of a tangential velocity discontinuity between two incompressible, inviscid, capillary, nonconducting, linearly magnetizable fluids in an external magnetic field is considered. An absolute instability criterion is obtained in analytic form. When the gravity force is negligible, this criterion does not depend on the value of the surface tension. When the surface tension is negligible, the absolute instability criterion is obtained for the region of the flow parameters in which the causality condition for the system in question is satisfied. If the magnetic field is tangential to the interface, all the criteria obtained are also applicable to the case of nonmagnetic, perfectly conducting fluids, i.e., to the case of ideal magnetohydrodynamics.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 21–26, September–October, 1995.     

Stabilization of the interface instability of moving magnetizable fluids

A study is made of the stabilization of the interface between two moving magnetizable fluids by means of an external magnetic field H={H₀ cos t, H₀ sin t, 0} of circular polarization. The fluids are assumed to be ideal, incompressible, nonconducting, and electrically neutral. An equation of motion is derived for the perturbed interface. The Borg criterion is used to obtain sufficient conditions of stability of small perturbations of the interface; these conditions relate the amplitude H₀ and the frequency of the external magnetic field to the characteristic parameters of the problem and the wave vector k. The dependence of H₀ and on the modulus of the wave vector is investigated. The obtained results are compared with the results of [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 118–122, July–August, 1980.     

Regular oblique interaction of shock waves coming from the same direction in a conducting medium with a magnetic field

The regular interaction of two plane MHD shock wave fronts overtaking one another at an arbitrary angle is investigated within the framework of the ideal magnetohydrodynamic model. The self-similar solution in which all the discontinuities are explicitly distinguished is obtained by means of a computer. The solution is analyzed over a wide range of the key parameters.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 127–143, November–December, 1995.     

Waves in a Bed under Periodic Tangential Loading

The far-zone structure of the wave field in an elastic bed on a rigid foundation is considered. The wave field is generated by a tangential periodic force applied to the bed surface. The amplitude- frequency characteristics of surface vibrations for propagating modes are found. The partition of energy between different modes is considered.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 109–115, July– August, 2005.     

Control of the motion of a vessel with a heavy inhomogeneous liquid

The problem of controlling the internal waves inside a closed rectangular cavity filled with a heavy two-layer fluid is considered in the linear approximation. The fluid is assumed to be stably stratified, ideal, and incompressible. The controlling horizontal force is applied to the body containing the cavity. It is assumed that at the initial moment there are no oscillations of the fluid and the interface is horizontal. The problem is to bring the vessel as a whole into a prescribed state of linear motion without relative wave motion of the fluid. The Cauchy-Poisson problem and the self-consistent integro-differential equation of the vessel motion are solved using the Fourier method and taking into account the reaction of the internal waves. On the basis of an analysis of the corresponding generalized momentum problem, approaches are proposed for solving the problem of control. It is shown that a control action with a sufficiently high order of Steklov smoothness ensures the approximate solution of the control problem with the required accuracy for all the characteristics of the motion of the hybrid oscillating system under consideration.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–22, May–June, 1995.The study was performed with financial support from the Russian Foundation for Fundamental Research (project No. 94-01-01368).     

Influence of porosity of the base on a plane standing wave of a heavy liquid

A study is made of the simple problem of the contact of two plane-parallel potential flows of incompressible fluid when one takes place in a layer of finite thickness and the other in a semiinfinite space of a porous medium. At the interface, which is taken to be a plane, the same conditions are used as earlier in problems of the contact of two wave flows of fluids with different densities and the contact of a wave motion in a layer of compressible fluid and wave motions in an elastic semi-infinite space. These conditions reduce to equalities of the pressures and projections of the velocity vectors onto the normal to the interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 160–163, July–August, 1984.     

Development of taylor instability in compressible conducting media in a magnetic field

Article [1] is devoted to the investigation of the interface of viscous incompressible conducting media in the presence of a current and with a magnetic field with a small magnetic Reynolds number. Article [2] discusses the stability of a contact discontinuity in compressible media. In this article, in an analysis of the case of long-wave vibrations in the region z<0, the boundary condition is unsatisfactorily met. Therefore, in this part of it the author actually considered a problem with a mass force f=(0, 0, –g sign z) instead of f=(0, 0, –g). The present, article considers the stability of the interface of compressible conducting media in a magnetic field. It is postulated that the magnetic intensity can undergo a discontinuity at this boundary. The article gives the dependence of the maximal increment of the rise in the instability on the determining parameters. An analysis is made of the stability of a contact discontinuity as a function of the angles formed by the wave vector and the intensity of the magnetic field. The stabilizing effect of the walls on the stability is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, 15–19, September–October, 1975.The author thanks A. G. Kulikovskii for his aid in stating the problem and for his continuing interest in the work.     

Solitary stokes waves in heavy-fluid slit flow

Flow with a solitary Stokes wave is obtained as a result of solving the plane nonlinear problem of the steady flow of an ideal heavy incompressible fluid into a slit at the bottom of a vessel for a narrow range of Froude numbers on which there exists a solution with a jet surface descending monotonically to the slit [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 173–176, November–December, 1991.The authors are grateful to A. N. Ivanov and G. Yu. Stepanov for their assistance and useful discussions.     

Interaction between an acoustic wave and a disc

A numerical solution is obtained to the problem concerning a pressure measurement at the boundary between an ideal compressible fluid and a solid wall. It is assumedthat the fluid occupies a semiinfinite cylinder with a rigid bottom into which an elastic disc is inserted and heldfirmly around its edges. Motion is produced by a pressure wave originating at infinity. A finite-difference grid for this application is described and the results of actual calculations are shown.Deceased.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 84–91, January–February, 1972.The authors thank L. M. Flitman for reviewing the work.     

Arrival of a shock wave at the expanding part of a channel

In this paper the flow of an ideal gas, observed with the arrival of a shock wave at the expanding part of a channel is discussed. It proposes a scheme of the flow approximately modeling the complex of discontinuities arising in this case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 191–193, November–December, 1978.     

Stability of a radiation-absorbing ionizing shock wave in an electromagnetic field

The linear stability of a radiation-absorbing ionizing shock wave (light detonation waves) in the presence of a uniform electromagnetic field is investigated. The applied electric field is considered to be normal to the wave front and the magnetic field to be parallel to the front and perpendicular to the plane in which perturbations propagate. The medium satisfies a two-parameter equation of state. Analytic stability criteria are obtained. For a perfect gas the effect of the electromagnetic field and radiation on shock wave stability is determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–30, January–February, 1996.     

Development of the Richtmyer-Meshkov instability on an interface with continuous density variation

The development of the Richtmyer-Meshkov instability driven by a shock wave reflected from a rigid wall is investigated. It is shown that the perturbation amplitude growth rate depends on the nature of the shock wave refraction on the interface between gases of different density. In the case of regular-soft refraction the reduction in the growth rate of the Richtmyer-Meshkov instability associated with the continuous change in density on the interface is measured.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 52–58, September–October, 1996.     

Parameters determining the shape of the subsonic section of a shock wave in the case of axisymmetric flow

Supersonic flow around a blunt body by an ideal gas with a constant specific heat ratio is considered. The dependence of the geometry of the subsonic section of the shock wave on the shape of the body and the freestream Mach number is studied. Analysis of the large quantity of numerical data has permitted simple approximate relations to be formulated for the principal geometrical parameters of the wave, which can be used for solving the problem of flow around a quite broad class of bodies. The question of the characteristic dimensions in such problems is also discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 77–83, July–August, 1976.In conclusion, the author expresses his thanks to V. N. Ivanova for carrying out the calculations and to é. é. Shnol' for for several useful comments.     

Taylor instability in an electromagnetic field

A study is made of the gravitational instability of the interface between two incompressible liquids in an electromagnetic field parallel to the interface when one of the liquids has a finite conductivity and the other is nonconducting. The magnetic Reynolds number R_m is assumed to be finite. It is shown that for all R_m, the electromagnetic field cannot stabilize the interface (if both liquids conduct, there is also no stabilization), although there may exist stable directions of propagation of perturbations. The greatest growth rate of perturbations corresponds to waves propagating at right angles to the vector of the initial magnetic field, and the electromagnetic field and conductivity of the walls do not affect these perturbations. The small-parameter method is used to obtain a dispersion relation for the small induced magnetic fields. It is shown that the range of angles between the wave vector and the vector of the initial magnetic field that correspond to unstable perturbations is greater than in the case when P_m 1 [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 118–121, November–December, 1979.I thank A. A. Barmin and A. G. Kulikovskii for constant interest in the work.