Two-dimensional flow of a perfect conducting gas in crossed electric and magnetic fields

Reference [1, 2] give a solution of the problem of the two-dimen-, sional flow of an inviscid thermally-nonconducting gas with constant conductivity in a channel of constant cross section for particular forms of the given applied magnetic field. The present paper obtains a solution of the problem of the two-dimensional flow of a gas with variable conductivity in crossed electric and arbitrary magnetic fields by means of the small parameter method. The magnetic Reynolds number R_m and the magnetohydrodynamic interaction parameter S are chosen as parameters. The international system of units is employed.Notation V flow velocity - j electric current density - p pressure in the flow - E electric field strength - gas density - electrical conductivity of the gas - T gas temperature - ratio of specific heats at constant pressure and volume - L channel half-height - ] permeability (magnetic) - B magnetic induction vector - B₀ applied magnetic field     

Solution of the direct problem of the mixed flow of a conducting gas in a laval nozzle with a meridional magnetic field

We consider the direct problem in the theory of the axisymmetric Laval nozzle (including sonic transition) for the steady flow of an inviscid and nonheat-conducting gas of finite electrical conductivity. The problem is solved by numerical integration of the equations of unsteady gas flow using an explicit difference scheme that was proposed by Godunov [1,2], and was used to calculate steady and unsteady flows of a nonconducting gas in nozzles by Ivanov and Kraiko [3]. The subsonic and the supersonic flows of a conducting gas in an axisymmetric channel when there is no external electric field, the magnetic field is meridional, and the magnetic Reynolds numbers are small have previously been completely investigated. Thus, Kheins, Ioller and Élers [4] investigated experimentally and theoretically the flow of a conducting gas in a cylindrical pipe when there is interaction between the flow and the magnetic field of a loop current that is coaxial with the pipe. Two different approaches were used in the theoretical analysis in [4]: linearization with respect to the parameter S of the magnetogasdynamic interaction and numerical calculation by the method of characteristics. The first approach was used for weakly perturbed subsonic and supersonic flows and the solutions obtained in analytic form hold only for small S. This is the approach used by Bam-Zelikovich [5] to investigate subsonic and supersonic jet flows through a current loop. The numerical calculations of supersonic flows in a cylindrical pipe in [4] were restricted to comparatively small values of S since, as S increases, shock waves and subsonic waves appear in the flow. Katskova and Chushkin [6] used the method of characteristics to calculate the flow of the type in the supersonic part of an axisymmetric nozzle with a point of inflection. The flow at the entrance to the section of the nozzle under consideration was supersonic and uniform, while the magnetic field was assumed to be constant and parallel to the axis of symmetry. The plane case was also studied in [6]. The solution of the direct problem is the subject of a paper by Brushlinskii, Gerlakh, and Morozov [7], who considered the flow of an electrically conducting gas between two coaxial electrodes of given shape. There was no applied magnetic field, and the induced magnetic field was in the direction perpendicular to the meridional plane. The problem was solved numerically in [7] using a standard process. However, the boundary conditions adopted, which were chosen largely to simplify the calculations, and the accuracy achieved only allowed the authors [7] to make reliable judgments about the qualitative features of the flow. Recently, in addition to [7], several papers have been published [8–10] in which the authors used a similar approach to solve the direct problem in the theory of the Laval nozzle (in the case of a nonconducting gas).Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 14–20, September–October, 1971.In conclusion the author wishes to thank M. Ya. Ivanov, who kindly made available his program for calculating the flow of a conducting gas, and also A. B. Vatazhin and A. N. Kraiko for useful advice.     

Stability of flow of a thin layer of viscous magnetic liquid

The laminar flow of a thin layer of heavy viscous magnetic liquid down an inclined wall is examined. The stability and control of the flow of an ordinary liquid are affected only by alteration of the angle of inclination of the solid wall and the velocity of the adjacent gas flow. When magnetic liquids are used [1, 2], an effective method of flow control may be control of the magnetic field. By using magnetic fields of various configurations it is possible to control the flow of a thin film of viscous liquid, modify the stability of laminar film flow, and change the shape of the free surface of the laminarly flowing thin film, a factor which plays a role in mass transfer, whose rate depends on the phase contact surface area. The magnetic field significantly affects the shape of the free surface of a magnetic liquid [3, 4]. In this paper the velocity profile of a layer of viscous magnetic liquid adjoining a gas flow and flowing down an inclined solid wall in a uniform magnetic field is found. It is shown that the flow can be controlled by the magnetic field. The problem of stability of the flow is solved in a linear formulation in which perturbations of the magnetic field are taken into account. The stability condition is found. The flow stability is affected by the nonuniform nature of the field and also by its direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–65, September–October, 1977.     

Some conditions for the existence of a “Pinch” structure of the current sheet in plasma

In pulsed plasma accelerators of various configurations it is frequently possible to observe an instability of the current sheet characterized by separation into individual narrow channels (local pinches). In this paper the flow of plasma over these channels is qualitatively investigated. It is shown that a large part of the discharge current can flow through a series of narrow channels only when they completely decelerate the plasma flow. This places upper and lower bounds with respect to pressure on the region of existence of the pinch structure, if it is assumed that deceleration by the magnetic field predominates over viscous deceleration. The upper bound is obtained from the condition of deceleration of the plasma by the magnetic field near the pinches, the lower bound from the condition that the ratio of specific heats of the gas must be small. As the initial gas pressure decreases, so does the plasma deceleration time, but the characteristic times of the excitation and ionization processes increase; therefore the condition 1 ceases to be satisfied. The results are compared with experiment. The induced currents in the plasma near the pinches are also a reason for the stability of the network of current filaments.According to [1] a pulsed high-current discharge in a low-pressure gas develops as follows. After breakdown and the attainment of a certain uniform value of the gas conductivity the current continues to increase only in a surface layer (skin effect), which, if the particle density is sufficient, acts as an impermeable piston on the gas in front of it (snowplow model). Clearly, to judge from the collapse time of the current cylinder [2], this gives a good picture of the situation in the Z pinch. However, in coaxial and rail guns the gas is by no means always completely raked up by the current sheet (see, for example, [3]). This is also indicated by the observed broad spectrum of plasmoid velocities.It has been noted in high-speed photographs of pulsed discharges of very different configurations (Z and pinches, coaxial and rail plasma guns) that the current sheet is often divided into a series of channels [4–12], through which the main current flows. This follows from their high acceleration as compared with the rest of the plasma [4, 5], from the arrangement of the cathode spots along the electrodes, and from the large value of the plasma density in the channels [11]. This effect is well reproduced from discharge to discharge.Experiments with the Z pinch [7] have shown that the division of the current sheet into pinches has almost no effect on its velocity, which is in good agreement with calculations based on the snowplow model, although the diameter of the channels (determined, it is true, from the luminescence distribution and not the current density) is much less than the distance between them. On the other hand, in rail guns the pinches only slightly entrain the gas filling the tube [11]. The surface layer pinch instability is observed in the pressure range from several mm to several Hg [9]. Both pressure limits depend heavily on the nature of the gas in which the discharge takes place, the upper limit being approximately inversely proportional to the molecular weight of the gas [9], In a rail gun at low initial gas pressures (p < <20 Hg) pinches appear at some optimal rate of gas release from the electrodes and the walls [12].A uniform transverse magnetic field has only a slight influence on the pinches, since it is smaller in magnitude than the magnetic field of the pinches themselves [11]. A transverse magnetic field nonuniform along the length of the pinches causes them to decay, if the magnitude of the external field is comparable with the pinch field. Pinch decay was observed by the author in a rail gun when one of the electrodes was cut out so that on a certain section of the rail its transverse dimension was reduced to the diameter of the pinch, which was much less than the distance between electrodes (this setup was described in [11]).The instability of a plane current sheet in a plasma has been demonstrated on several occasions [13, 14]. Under experimental conditions the pinches move through the plasma. Therefore we will consider the case when a considerable part of the discharge current flows through a series of narrow channels (assuming that for this case the conductivity of the pinches is infinite).In conclusion, the author thanks A. K. Musin for his helpful suggestions.     

Dissipative instability of a nonisothermal electrically conducting flow between parallel plates in a transverse magnetic field

Problems of dissipative instability (in particular, overheating) in magnetohydrodynamies has been studied in [1–6]. The Leontovieh mechanism of overheating instability is explained in [I] by the example of a stationary homogeneous plasma in a strong magnetic field along which current flows. The rate of buildup of perttbations is estimated in [2] to explain the effect of overheating instability on the operation of an MHD generator. The effect of inhomogeneity in the temperature field and in the boundaries of the region on the formarion of this instability has been studied by the example of discharge in a stationary medium in the absence of a magnetic field [3], Certain cases of overheating instability in magnetohydrodynamies are considered in [4, 6], where it is shown that it can be aperiodic as well as oseillatery (Alfven and acoustic waves). Finally, the hydro-dynamic and overheating branches of instability in the ease of non-isothermal plasma flow in a plane MHD channel was investigated in [6]. But the overheating instability was examined without allowance for the dependence of the viscosity and thermal-conductivity coefficients on temperature in the limiting case S R_m 1 and only for small perturbation wavelengths. The development of shortwave perturbations is studied below with allowance for viscosity and thermal conductivity and for a wider range of conditions A 1. Overheating instability over the entire range of wavelengths for the ease considered in [6] is also studied.The author thanks Yu. M. Zolotaikin for programming and performing the calculations.     

Magnetic field in a cylindrical conductor moving with a velocity proportional to r−1

Nonstationary magnetic field problems in a moving conductor are of interest in connection with obtaining pulsed magnetic fields by magnetic cumulation [1], The field penetrates into the conductor as a result of the growth of the skin layer and is carried along with the conductor. The first mechanism of the interaction of a field with a conductor is called the diffusion of the field, and the second convection. Five self-similar solutions of magnetic field problems in a conductor which has a velocity v =q/2 r and a conductivity =const are discussed and a numerical solution of the problem of the compression of a field in a cylindrical cavity when the conductor moves toward the axis is presented. One of the self-similar solutions is compared with the numerical solution.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 18–25, September–October, 1973.     

Influence of internal degrees of freedom on flow of a rarefied gas over a flat plate

Flow of a rarefied gas over a flat plate has been investigated numerically by a number of authors, using both the kinetic model equations (e.g., 1, 2]) and the Boltzmann equation [3, 6], In most cases a solution was found for a monatomic gas. The appreciable influence of the molecule structure on local and total aerodynamic characteristics and on the flow field over a flat plate at small angles of attack was noted in [1, 5, 7], where the authors examined various models for the rotational molecular degrees of freedom. In the present paper a two-point repulsion center model with constant collision cross section is used to investigate the influence of internal degrees of freedom of the molecule in flow over a plate, positioned parallel to (angle of attack = 0), and transverse to ( = 90 °) a rarefied gas stream. The data are compared with those calculated for a monatomic gas and from experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 151–156, November–December, 1978.     

Stability of a flow in a circular tube

Many studies, both theoretical and experimental, have been dedicated to the stability of flow in a circular tube (see, for example, review [1]). In every case mathematical investigation has not succeeded in obtaining an expression for hydrodynamic instability of such a flow for disturbances of sufficiently low amplitude. (An exception is [2].) Experiment also indicates the stability of such a flow [3], with a laminar mode being extended to Reynolds numbers of the order of tens of thousands. These facts are the basis for the assumption that the flow of a viscous incompressible liquid in a circular tube is stable for small perturbations. However, there is no analytical or even numerical proof of this hypothesis. Moreover, some studies, for example [2], indicate the instability of such a flow in relation to three-dimensional nonaxiosymmetric perturbations. The analysis of hydrodynamic stability with respect to three-dimensional disturbances of flow within a circular tube conducted in this study showed the stability of the flow over a wide range of wave numbers and Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 20–24, January–February, 1973.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Effect of surface potential and intrinsic magnetic field on resistance of a body in a supersonic flow of rarefied partially ionized gas

The character of flow over a body, structure of the perturbed zone, and flow resistance in a supersonic flow of rarefied partially ionized gas are determined by the intrinsic magnetic field and surface potential of the body. The effects of intrinsic magnetic field and surface potential were studied in [1–4]. There have been practically no experimental studies of the effect of intrinsic magnetic field on flow of a rarefied plasma. Studies of the effect of surface potential have been limited to the case R/_d<50 [1, 3]; this is due to the difficulty of realization of flowover regimes at R/_d>10² (where R is the characteristic dimension of the body and X is the Debye radius). At the same time R/_d>10², the regime of flow over a large body, is of the greatest practical interest. The present study will consider the effect of potential and intrinsic magnetic field on resistance of a large (R/_d>10²) axisymmetric body (disk, sphere) in a supersonic flow of rarefied partially ionized gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 41–47, May–June, 1986.     

Linear theory of unsteady gas flow under the influence of nonpotential external forces

We determine in the linear formulation the velocity and pressure fields excited in a compressible medium by a lifting filament that displaces and deforms arbitrarily. For general unsteady motion of such a filament we give explicit formulas that express the velocity at a given point in terms of the intensity of the free vortices entering the audio signal audibility zone constructed for this point. We examine gas flow caused by an arbitrary external body force field.Studies devoted to the determination of gas velocity fields for flow past slender bodies relate primarily to translational motion of a body with a dominant constant velocity [1–3]. Gas velocities for helical motion of a rectilinear lifting filament within the gas have been examined in [4].     

Nonstationary flow of a conducting gas in crossed electric and magnetic fields

The plasma is inviscid, cool, and not thermally conducting; it flows in a channel of constant cross section. The solution is derived by the small parameter method, for which purpose the magnetic interaction N is used. There have been previous studies of the transient-state flow of an inviscid and thermally nonconducting plasma in crossed electric and magnetic fields [1–3]. A plasma of infinite conductivity has been considered [1], as well as flow involving entropy change in an MHD system with strong electromagnetic fields [2, 3].     

Self-similar magnetogasdynamic flows accompanied by detonation and combustion waves

Magnetogasdynamic (MGD) flows with detonation waves and combustion fronts have attracted more and more attention in recent years. Intensive heat supply assures such a significant increase in the temperature and pressure behind the heat liberation fronts that the gaseous combustion products become conductive so that the flow map in the electric and magnetic fields can vary substantially as compared with ordinary gasdynamics. In the case of finite gas conductivity, when the magnetic Reynolds numbers R_m are low, the asymptotic laws of detonation wave propagation which either go over into the Chapman-Jouguet (CJ) mode (in a number of cases at a finite distance from the initiation source) or remain overcompressed, have been studied [1]. Stationary flow modes behind detonation waves have been investigated in [2] and the problem of the detonation wave originating at the closed end of the tube emerging in the stationary mode in crossed homogeneous magnetic and electric fields has been examined. Results are presented in this paper of an investigation of one-dimensional self-similar flows caused by piston motion in a hot gas mixture in which a detonation wave or combustion front is propagated. The motion is realized in external electric and magnetic fields which exert a substantial effect on the flow of the conductive combustion products. Domains of application of the governing parameters in which the various flow modes are realized are found by using a qualitative and numerical analysis. The results obtained are used to solve problems about the hypersonic gas flow around a thin wedge in an axial magnetic field.     

Investigation of plasma flux interaction with an axisymmetric magnetic field

The motion of the plasma flux in an axisymmetric magnetic field is examined for a magnetic Reynolds number R_m 10, magnetohydrodynamic interaction parameter N 1, and Hall parameter 1. Flux deceleration in a circular channel is studied at the entrance to the magnetic field because of the formation of azimuthal electrical current eddies.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–39, May–June, 1972.     

Flow stability of a conducting liquid flowing down an inclined plane in the presence of a magnetic field

The stability of a steady flow of incompressible, conducting liquid down an inclined plane in the presence of longitudinal and transverse magnetic fields is studied. Solutions of the linearized magnetohydrodynamic equations with corresponding boundary conditions are found on the assumption that the Reynolds number R_g and the wave number are small. It is shown that the longitudinal magnetic field plays a stabilizing role. It is known [1] that the flow of a viscous liquid over a vertical wall is always unstable. In this article it is shown that the instability effect at small wave numbers may be eliminated if the longitudinal magnetic field satisfies the conditions found. The case when the Alfvén number and the wave number are small and the Reynolds number is finite is also examined.     

Turbulent film condensation on a horizontal elliptical tube

An investigation has been made of turbulent film condensation on a horizontal elliptical tube. The present study is based on Colburn analogy [1] and potential flow theory to determine the high tangential velocity of vapor flow at the boundary layer and to define the local interfacial shear owing to high velocity vapor flow across the tube surface. The condensate film flow and local/or mean heat transfer characteristics from a horizontal elliptical tube with variable ellipticities, e, under the influence of Froude number, sub-cooling parameter and system pressure have been performed. The present result for dimensionless mean heat transfer coefficient reduces to the same result obtained by Sarma et al.s [2] e=0 (circular tube). Compared with laminar model by Yang and Hsu [3], the present turbulent model shows in better agreement with Michaels experimental data [4] (for e=0). The dependence of mean Nusselt coefficient on the effect of n (power of Reynolds) [1] is also discussed.     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.     

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.     

Transient minor disturbances in flat streams of a highly conductive plasma in a channel

Transient magnetohydrodynamic disturbances of flat streams of a highly conductive, nonviscous, thermally nonconductive, quasineutral plasma in a channel of slowly varying cross section with sectioned electrodes are analyzed in a linear approximation. The influence of the Hall, effect is taken into account in the analysis. It is shown that the evolution of a disturbance in the isomagnetic parameter B/p is comprised of transport along the channel together with the plasma stream, transport along the undisturbed electron trajectories, and diffusion due to the finite conductance of the plasma. The time of establishment of the flow is equal, to the time of flight of the plasma through the channel (the region occupied by the magnetic field). The present report is a generalization of the analysis of steady disturbances conducted in [1],Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 19–29, January–February, 1975.