Some geometric acoustic problems of one-dimensional unsteady and two-dimensional steady flows

The behavior of discontinuities (weak shocks) of the parameters of a disturbed flow and their interaction with the discontinuities of the basic flow in the geometric acoustics approximation, when the variation of the intensity of such shocks along the characteristics or the bicharacteristics is described by ordinary differential equations, has been investigated by many authors. Thus, Keller [1] considered the case when the undisturbed flow is three-dimensional and steady, and the external inputs do not depend on the flow parameters. An analogous study was made by Bazer and Fleischman for the MGD isentropic flow of an ideal conducting medium [2], while Lugovtsov [3] studied the three-dimensional steady flow of a gas of finite conductivity for small magnetic Reynolds numbers and no electric field. Several studies (for example, [4]) have considered the behavior of discontinuities of the solutions from the general positions of the theory of hyperbolic systems of quasilinear equations. Finally, the interaction of weak shocks (or the equivalent continuous disturbances) with shock waves was studied in [5–11].In what follows we consider one-dimensional (with plane, cylindrical, and spherical waves) and quasi-one-dimensional unsteady flows, and also plane and axisymmetric steady flows. Two problems are investigated: the variation of the intensity of weak shocks in the presence of inputs which depend on the stream parameters, and the interaction of weak shocks with strong discontinuities which differ from contact (tangential) discontinuities.The thermodynamic properties of the gas are considered arbitrary. We note that the resulting formulas for the interaction coefficients of the weak and strong discontinuities are also valid for nonequilibrium flow.     

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.     

Surfaces of discontinuity in a magnetized medium

Features of surfaces of discontinuity in a magnetized medium are considered. It is shown that the polarizing force at interfaces may be directed towards the medium with higher permittivity. It is proved that there exists in an absolutely conducting magnetized medium plane-polarizing discontinuities at which the density of the medium remains invariant.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 23–27, September–October, 1974.     

Decay of a plane shock wave in a two-parameter medium with arbitrary equation of state

Special curves, called shock polars, are frequently used to determine the state of the gas behind an oblique shock wave from known parameters of the oncoming flow. For a perfect gas, these curves have been constructed and investigated in detail [1]. However, for the solution of problems associated with gas flow at high velocities and high temperatures it is necessary to use models of gases with complicated equations of state. It is therefore of interest to study the properties of oblique shocks in such media. In the present paper, a study is made of the form of the shock polars for two-parameter media with arbitrary equation of state, these satisfying the conditions of Cemplen's theorem. Some properties of oblique shocks in such media that are new compared with a perfect gas are established. On the basis of the obtained results, the existence of triple configurations in steady supersonic flows obtained by the decay of plane shock waves is considered. It is shown that D'yakov-unstable discontinuities decompose into an oblique shock and a centered rarefaction wave, while spontaneously radiating discontinuities decompose into two shocks or into a shock and a rarefaction wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 147–153, November–December, 1982.     

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.     

Steady magnetic waves

Steady simple waves are investigated in an incompressible conducting ideal inhomogeneously and isotropically magnetizable fluid moving along the lines of force of a magnetic field. The integration of the system of equations describing such waves is reduced to the calculation of quadrature expressions in the case of an arbitrary magnetization law. It is shown that, depending on the magnetic properties of the medium, different types of steady waves are possible: magnetizing waves in a diamagnetic fluid and demagnetizing waves in a paramagnetic fluid. The results are given of calculations of demagnetizing waves in a conducting ferromagnetic fluid. An analysis is made of the various possible flow regimes of a conducting magnetizable fluid at the point of a perfectly conducting corner.     

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.     

Properties of shock adiabats in the neighborhood of jouguet points

Two well-known properties of shock adiabats in a gas [1] are proved for shock adiabats corresponding to discontinuous solutions of hyperbolic systems of equations expressing conservation laws. If the state on one side of a discontinuity is fixed, then at the point of extremum of the discontinuity velocity on the shock adiabat the velocity of the discontinuity is equal to one of the velocities of the characteristics on the other side of the discontinuity and vice versa. If for the systems there is defined an entropy flux or mass density of entropy, then at the points of extremum of the velocity there is an extremum of the entropy production at the discontinuity and the entropy mass density. If the system is a symmetric hyperbolic system [2, 3], then the extrema of the entropy production at the discontinuity correspond to extrema of the velocity. These properties may be helpful in the study of discontinuities in complex media, since the sections of a shock adiabat whose points can correspond to actually existing discontinuities are frequently bounded by points corresponding to discontinuities whose velocity is equal to the velocity of a characteristic on one of the sides of the discontinuity (see, for example, [1, 4, 5]).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 184–186, March–April, 1979.     

Propagation of solitary waves over a semi-infinte underwater trough

Self-similar solutions describing the incidence of a uniform solitary wave on a semi-infinite linear trough are obtained on the basis of the nonlinear ray method [1]. Previously, in investigating the incidence of a wave on a trough [2] the conditions at the discontinuities present in the solutions were derived on the assumption that they are of low intensity. In the present study the use of the conditions at the discontinuities obtained by investigating soliton interaction [3–5] has made it possible to construct a series of new solutions and take into account wave reflection effects and the formation of a shadow zone beyond the trough. The types of solutions that occur are established in terms of the relations between the wave parameters and the relative depth of the trough. To ensure that self-similar solutions exist for all values of the parameters it was necessary to introduce a type of discontinuity not previously encountered [5–7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–107, July–August, 1987.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for discussing the work.     

Stationary simple waves in an electrically conducting magnetizable fluid

A study is made of the analog of Prandt1—Meyer flow in an incompressible electrically conducting ideal fluid that is magnetizable in accordance with an arbitrary isotropic law. It is shown that inhomogeneity of the magnetization in a conducting fluid makes possible the existence of stationary simple waves with varying magnetic permeability. For a paramagnetic fluid magnetized to saturation, the equations of these waves are integrated completely in the case of a magnetic field parallel to the velocity. Some regions of such flows of magnetizable fluids are discussed in the present paper for the example of the problem of flow around a perfectly conducting profile.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 136–143, September–October, 1980.I thank I.E. Tarapov for his interest in the work and valuable comments made in a discussion.     

Charging of disperse particles in two-phase media with unipolar charge

Charging of disperse particles with good conduction in two-phase media with unipolar charge is considered in the case when the volume concentration of the particles is low. For this, in the framework of electrohydro-dynamics [1, 2], a study is made of the charge of one perfectly conducting liquid particle in a gas (or liquid) with unipolar charge in a fairly strong electric field. The influence of the inertial and electric forces on the motion of the gas is ignored, and the velocities are found by solving the Hadamard—Rybczynski problem. We consider the axisymmetric case when the gas velocity and electric field intensity far from the particle are parallel to a straight line. The analogous problem for a solid spherical particle was solved in [3–6] (in [3], the relative motion of the gas was ignored, while in [4–6] Stokes flow around the particle was considered). The two-dimensional problem of the charge of a solid circular, perfectly conducting cylinder in an irrotational flow of gas with unipolar charge was studied in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 108–115, November–December, 1980.We thank L. I. Sedov and V. V. Gogosov for a helpful discussion of the present work.     

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.     

Steady plane-parallel flow of a magnetic fluid

In the hydrodynamics of a Newtonian fluid, nonlinear effects are connected only with the presence of convective derivatives in the equations and therefore disappear when plane-parallel flows are considered. Non-Newtonian effects are usually taken into account either phenomenologically in the expression for the stress tensor or by explicitly considering additional degrees of freedom. A theory of the effective viscosity of a magnetic fluid is constructed in [1] by regarding a magnetic fluid as a medium with internal rotation. It was shown that the flow of fluid in a magnetic field is non-Newtonian. Later, many authors (see, for example, [2, 3]) studied one- and two-dimensional flows under the influence of a pressure difference. However, the study was usually limited to continuous and smooth solutions. In the present work, we study the plane-parallel flow of a magnetic fluid in a homogeneous magnetic field under the influence of a longitudinal pressure gradient. We also consider discontinuous solutions. It is shown that for large longitudinal pressure gradients and sufficiently great intensities of the magnetic field, the problem has an infinite number of steady solutions which differ in the number and position of discontinuities of the magnetization and the associated abrupt changes in the velocity profile. Steady regimes and their stability are studied numerically with allowance for weak diffusion of magnetization and internal angular momentum. It is shown that the degeneracy is then lifted; however, in a certain region of parameters several stable steady regimes still exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 57–64, November–December, 1984.The authors would like to thank M. I. Shliomis for his constant interest in this work.     

Equations of hydromechanics and compression shock in two-velocity and two-temperature continuum with phase transformations

The equations of motion of multiphase mixtures have been considered in [1–10] and several other studies. In [1] it is proposed that the mixture motion be considered as an interpenetrating motion of several continua when velocity, pressure, mean density, concentration, etc., fields for each phase are introduced in the flowfield. The equations of motion are written separately for each phase, and the force effect of the other components is considered by introducing the interaction forces, which for the entire system are internal. The assumption of component barotropy is used to close the system.The energy equations are used in [2, 3] in place of the component barotropy assumption. Moreover, mixtures without phase transformations are considered. In [4] an analysis is made of the equations of turbulent motion with account for viscous forces for a two-velocity, but single-temperature medium in which equilibrium phase transformations are assumed, i. e., a two-phase medium is considered in which the phase temperatures are the same, the composition is equilibrium, but the phase velocities are different. In [5] the equations are written on the interface in a multicomponent medium consisting of barotropic fluids. A discontinuity classification is also presented here. In the aforementioned work [3] the equations on the shock are written for a continuum with particles without the use of the property of barotropy of the carrier fluid. Various different aspects of the motion of multiphase mixtures are considered in [6–11], for example, the effect of particle collisions with one another, the effect of the volume occupied by the particles on the parameters stream, shock waves, etc. In [7] a study is made of the force effect of an agitated medium on a particle on the basis of the Basset-Boussinesq-Oseen equation.In the following we derive the equations of motion of a two-velocity and two-temperature continuum with drops or particles with nonequilibrium phase transformations, i. e., a medium in which the phase velocities and temperatures are different and the composition may be nonequilibrium. In addition, we study the effect of the presence of particles or drops on the gas parameters behind a shock. Further, the equations obtained here are used to study compression waves, and in particular shock waves.The author wishes to thank Kh. A. Rakhmatulin, S. S. Grigoryan, and Yu. A. Buevich for helpful discussions and valuable comments.     

MHD simulation of the distribution of the gasdynamic parameters and magnetic field behind the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The distributions of the gasdynamic parameters (density, pressure, and velocity) and the magnetic field behind the Earth’s bow shock (on the outer boundary of the magnetosheath) generated under sharp variations in the solar wind dynamic pressure are found in the three-dimensional non-planepolarized formulation with allowance for the interplanetary magnetic field within the framework of the ideal magnetohydrodynamic model using the solution to the MHD Riemann problem of breakdown of an arbitrary discontinuity. Such a discontinuity which depends on the inclination of an element of the bow shock surface arises when a contact discontinuity traveling together with the solar wind and on which the solar wind density and, consequently, the dynamic pressure, increases or decreases suddenly impinges on the Earth’s bow shock and propagates along its surface initiating the development of to six waves or discontinuities (shocks). The general interaction pattern is constructed for the entire bow shock surface as a mosaic of exact solutions to the MHD Riemann problem obtained on computer using an original software (MHD Riemann solver) so that the flow pattern is a function of the angular surface coordinates (latitude and longitude). The calculations are carried out for various jumps in density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that there exist horseshoe zones on the bow shock in which the increase in the density and the magnetic field strength in the fast shock waves or their reduced decrease in the fast rarefaction waves penetrating into the magnetosheath and arising as a result of sharp variation in the solar wind dynamic pressure is superposed on significant drop in the density and growth in the magnetic field strength in slow rarefaction waves. The distributions of the hydrodynamic parameters and the magnetic field can be used to interpret measurements carried out on spacecraft in the solar wind at the libration point and orbiters in the neighborhood of the Earth’s magnetosphere.     

Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

The processes of wave disturbance propagation in a supersonic boundary layer with self-induced pressure [1–4] are analyzed. The application of a new mathematical apparatus, namely, the theory of characteristics for systems of differential equations with operator coefficients [5–8], makes it possible to obtain generalized characteristics of the discrete and continuous spectra of the governing system of equations. It is shown that the discontinuities in the derivatives of the solution of the boundary layer equations are concentrated on the generalized characteristics. It is established that in the process of flow evolution the amplitude of the weak discontinuity in the derivatives may increase without bound, which indicates the possibility of breaking of nonlinear waves traveling in the boundary layer.     

Heating and flow of a conducting gas

A model is proposed for the laminar flow of a conducting gas in the channel of a conical plasmatron (plasma source) and in the surrounding motionless medium. The boundary-layer approximation equations are written taking the intrinsic magnetic fields into account. The flow of an argon plasma in argon and air is analyzed numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 190–193, September–October, 1975.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Magnetohydrodynamic lubrication theory for a cylindrical bearing

Liquid metal, which is a conductor of electric current, may be used as a lubricant at high temperatures. In recent years considerable attention has been devoted to various problems on the motion of an electrically conducting liquid lubricant in magnetic and electric fields (magnetohydrodynamic theory of lubrication), Thus, for example, references [1–3] study the flow of a conducting lubricating fluid between two plane walls located in a magnetic field. An electrically conducting lubricating layer in a magnetohydrodynamic bearing with cylindrical surfaces is considered in [4–8] and elsewhere.The present work is concerned with the solution of the plane magnetohydrodynamic problem on the pressure distribution of a viscous eletrically conducting liquid in the lubricating layer of a cylindrical bearing along whose axis there is directed a constant magnetic field, while a potential difference from an external source is applied between the journal and the bearing. The radial gap in the bearing is not assumed small, and the problem reduces to two-dimensional system of magnetohydrodynamic equations.An expression is obtained for the additional pressure in the lubricating layer resulting from the electromagnetic forces. In the particular case of a very thin layer the result reported in [4–8] is obtained. SI units are used.     

Allowance for longitudinal diffusion in the neighborhood of a discontinuity of catalytic surface properties

In the investigation of flow near surfaces with discontinuous changes in the catalytic properties the question arises of the applicability of parabolic boundary and viscous shock layer equations in the neighborhood of the discontinuity. In the present paper, three types of problem are solved in which longitudinal diffusion is taken into account. In the first an insertion with different catalytic properties is placed in the neighborhood of the stagnation point, in the second the discontinuity lines of the catalytic properties are perpendicular to the oncoming flow, while in the third they are parallel. On the main surface and on the insertion surface the heterogeneous catalytic reactions are assumed firstorder reactions with various rate constants whose values vary in a wide range. The data of the solution are compared with the solution obtained using the boundary layer approximation and the regions of influence of the longitudinal diffusion are estimated. In [1–4] a problem similar to the second one was solved by the numerical method of [1] and the Wiener-Hopf method for the case of transition from a noncatalytic to a perfectly catalytic surface and the region of applicability of the boundary layer was estimated [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 99–105, July–August, 1986.