Magnetic hydrodynamics of two-dimensional flows in a plasma with an anisotropic pressure

The hydrodynamic equations of Chew, Goldberger, and Low [1] are used to analyze certain types of two-dimensional flows of a plasma with an anisotropic pressure (the pressure along the magnetic field p differs from the pressure across it p). In Sec. 1 the relationships derived in [2] for the transition of plasma state across surfaces of strong discontinuity are invoked to investigate the variation of the hydrodynamic parameters in weak shock waves in the linear approximation. The flow around bodies which only slightly perturb the main flow is investigated in Sec. 2 in the linear approximation. Similar problems for the case of an isotropic pressure are studied in detail in [3–5], for example.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–10, November–December, 1970.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

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.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Effect of ion heat fluxes on the stability of an anisotropic collisionless plasma

Small perturbations of an unbounded volume of anisotropic collisionless plasma in a strong magnetic field are studied on the basis of MHD equations. It is assumed that there are present in the plasma ion heat fluxes connected with the third-order moments of the ion distribution function. The dispersion equation obtained, determining the velocity of five types of waves, is analyzed. In the space of the undisturbed plasma parameters the regions of values in which small perturbations are damped exponentially with time are found.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 153–157, March–April, 1993.     

Propagation of long nonlinear waves in a ponderable fluid beneath an ice sheet

In Sec. 1 the stability of small-amplitude steady-state periodic solutions of Eq. (0.1) in the neighborhood of k=k_n are investigated. The results of the investigations are consistent with those of [1]. In Sec. 2 the stability of periodic waves not lying in the neighborhood of resonance is considered. It is shown that in the region of instability when =1 steady-state solutions of the soliton type with oscillatory structure may exist. In Sec. 3 the properties of certain exact solutions — periodic waves and solitons — are studied in relation to the nature of the singular points of the dynamical system derived from (0.1). In Sec. 4 the evolution of rapidly decreasing Cauchy data is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1989.     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.     

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.     

Investigation of the properties of a shock wave arising in a supersonic flow of plasma,passing through a transverse magnetic field

In a flow of plasma, set up by an ionizing shock wave and moving through a transverse magnetic field, under definite conditions there arises a gasdynamic shock wave. The appearance of such shock waves has been observed in experimental [1–4] and theoretical [5–7] work, where an investigation was made of the interaction between a plasma and electrical and magnetic fields. The aim of the present work was a determination of the effect of the intensity of the interaction between the plasma and the magnetic field on the velocity of the motion of this shock wave. The investigation was carried out in a magnetohydrogasdynamic unit, described in [8]. The process was recorded by the Töpler method (IAB-451 instrument) through a slit along the axis of the channel, on a film moving in a direction perpendicular to the slit. The calculation of the flow is based on the one-dimensional unsteady-state equations of magnetic gasdynamics. Using a model of the process described in [9], calculations were made for conditions close to those realized experimentally. In addition, a simplified calculation is made of the velocity of the motion of the above shock wave, under the assumption that its front moves at a constant velocity ahead of the region of interaction, while in the region of interaction itself the flow is steady-state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–91, January–February, 1975.     

Propagation of a soliton along a fluid-filled pipe

In [1, 2] a mathematical model of the motion of a fluid in a pipe whose axis is a curve in space was discussed and certain simplifications of the problem were studied. The propagation of linear and nonlinear waves in the framework of the model was studied. In the present paper we consider a simple wave flow in a pipe with elastic walls suing one of the models introduced in [1], which, unlike [2], takes into account axial displacements of the pipe. The basic equations describing the propagation of waves in the pipe are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 3, pp. 58–63, May–June, 1986.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

Turbulent flow of a fluid in the neighborhood of the trailing edge of a plate at zero angle of attack

The laminar flow regime of an incompressible fluid at the trailing edge of a plate was studied by Stewartson and Messiter [1, 2] by means of the method of matched asymptotic expansions. In. the present paper, this method is used to analyze the same problem, but in the case of turbulent flow in the boundary layer and the wake. A system of linear equations of elliptic type with variable coefficients is obtained for the averaged values of the flow parameters in the main part of the boundary layer and the wake that is responsible for the change in the displacement thickness. A solution of this system is constructed by the Fourier method in the case of a power law of the velocity in front of the interaction region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–23, November–December, 1983.     

Generation of Nonlinear Waves on a Viscoelastic Coating in a Turbulent Boundary Layer

Self–induced excitation of periodic nonlinear waves on a viscoelastic coating interacting with a turbulent boundary layer of an incompressible flow is studied. The response of the flow to multiwave excitation of the coating surface is determined in the approximation of small slopes. A system of equations is obtained for complex amplitudes of multiple harmonics of a slow (divergent) wave resulting from the development of hydroelastic instability on a coating with large losses. It is shown that three–wave resonant relations between the harmonics lead to the development of explosive instability, which is stabilized due to the deformation of the mean (Sover the wave period) shear flow in the boundary layer. Conditions of soft and hard excitation of divergent waves are determined. Based on the calculations performed, qualitative features of excitation of divergent waves in known experiments are explained.     

Use of flows with straight sonic line in nozzles with corners

The optimal scheme of a Laval nozzle is discussed. In the case of a profiled nozzle with a corner it is possible to use in the region of mixed flow both flows of general form with curvilinear sonic line as well as the special case when the sonic line is straight. It is shown that the latter alternative is preferable: when the supersonic part of the profile is determined by the simple wave method, the velocity at the wall increases monotonically and the flow does not contain shock waves. In contrast, in nozzles with curvilinear sonic line, a section in which the velocity decreases is formed immediately behind the corner, which can lead to boundary layer separation. In addition, for values of the supersonic velocity at the nozzle exit near the velocity of sound it is proved that the characteristics of the simple wave intersect in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 168–170, January–February, 1981.     

Modeling of long nonlinear waves on the interface in a horizontal two-layer viscous channel flow

The dynamics of two-dimensional waves of small but finite amplitude are theoretically studied for the case of a two-layer system bounded by a horizontal top and bottom. It is shown that for relatively large steady-state flow velocities and at certain fluid depth ratios the vertical velocity profile is nonlinear. An evolutionary equation governing the fluid interface disturbances and allowing for the long-wave contributions of the layer inertia and surface tension, the weak nonlinearity of the waves, and the unsteady friction on all the boundaries of the system is derived. Steady-state solutions of the cnoidal and solitary wave type for the disturbed flow are determined without regard for dissipation losses. It is found that the magnitude and the direction of the flow can alter not only the lengths of the waves but also their polarity.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 143–158. Original Russian Text Copyright © 2005 by Arkhipov and Khabakhpashev.     

On the expansion into a vacuum of a rarefied plasma with two sorts of ions

The one-dimensional expansion of a plasma with different temperatures and two sorts of ions into a vacuum is examined. When the ion velocity distribution in the plasma is Maxwellian, propagation of a rarefaction wave is observed, the boundary of which is a weak discontinuity moving with the velocity of ionic sound in the plasma. The value of this velocity is found for the plasma in question. Attention is mainly focused on finding the first two moments of the distribution functions, i.e., the mean velocities and the densities of the heavy particles. An approximate asymptotic solution is obtained for the system of transport equations in the case when the two kinds of ions have similar masses, and the system is solved numerically by computer. Some features of the solutions, typifying a plasma in which the different sorts of ions have different masses, are analyzed in detail.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 1, pp. 32–38, January–February, 1970.In conclusion the author thanks N. T. Pashchenko for suggesting the problem and showing unfailing interest.     

Pulsating regime corresponding to an obstacle in a stationary inhomogeneous flow

The pulsating regime produced by the presence of a cylindrical cavity in a stationary inhomogeneous supersonic flow is simulated mathematically. The system of equations for an inviscid thermally nonconducting gas is solved by a numerical method based on a two-step difference scheme of second order of approximation. This method makes it possible to calculate in each time step the complete flow field at once, which makes it possible to follow the development of the nonstationary flow, which in the present case is a pulsating flow. The flow pattern in the pulsating regime is studied in detail. The pressure pulsations in the cavity are due to the alternating passage through it of shock waves and rarefaction waves, and the pulsations are nonlinear. The influence of the basic parameters on the characteristics of the pulsating flow is studied and some estimates are made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 64–71, September–October, 1979.     

Friction and heat transfer in an incompressible fluid near a plane stagnation point

In the neighborhood of a plane stagnation point, the flow and heat transfer of an incompressible fluid are studied. In the inner flow region, the velocity and pressure fields are described by the complete Navier-Stokes equations, and the temperature field is described by the complete energy equation. In the outer flow region, a two-term asymptotic solution of the corresponding equations is obtained. The problem is reduced to the numerical solution of ordinary differential equations. Numerical results are discussed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–65, July–August, 1996.     

Shock wave intersection in magnetohydrodynamics

The flow formed in the neighborhood of the discontinuity intersection point when shock waves collide at a nonzero angle is studied. The investigation can be directly applied to problems of shock wave interaction in the interplanetary plasma [9–12]. In magnetohydrodynamics the nature of the flow and its investigation are much more complex than in gas dynamics because of the greater number of possible waves and governing parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 132–143, May–June, 1991.