Integral characteristics of MHD channel with nonconducting baffles

The electromagnetic characteristics of an MHD channel with insulated walls, two quite long electrodes, and nonconducting baffles are obtained for arbitrary magnetic field distribution law along the channel. Some specific cases of magnetic field and baffle location specification are examined in detail.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 30–39, November–December, 1969.     

Stationary flow of a conducting liquid in a rectangular channel with two nonconducting walls and two walls having an arbitrary conductivity parallel to an external magnetic field

The flow of a conducting liquid in a channel of rectangular cross section with two walls (parallel to the external magnetic field) having an arbitrary conductivity, the other two being insulators, is considered. The solution of the problem is presented in the form of infinite series. The relationships obtained are used for numerical calculations of the velocity distribution and the distribution of the induced magnetic field over the cross section for several modes of flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostt i Gaza, No. 5, pp. 46–52, September–October, 1970.     

Effect of the form of the electrodes on the current distribution in a magnetogasdynamic channel

A solution is given to the plane problem of the flow of a conducting gas across a homogeneous magnetic field in a magnetogasdynamic channel taking account of the Hall effect at small magnetic Reynolds numbers. The channel is formed by two long electrodes, and the cross section of the channel varies slightly and periodically along the gas flow. It is assumed that the electromagnetic forces are small. It is shown that the current distribution in the channel is nonuniform to a consider able degree and that inverse currents can form at the electrodes, with both subsonic and supersonic flows of the conducting gas. Transverse motion of the gas, due to a change in the cross section of the channel, leads to an increase of Joule energy losses. In [1] the current distribution was obtained in a flat channel formed by infinite dielectric walls, with the flow of a steady-state stream of plasma through the channel across a homogeneous magnetic field. With interaction between the flow and the magnetic field, closed current loops develop in the channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–33, November–December, 1970.     

The Stress–Strain State of an Elastic Ferromagnetic Medium with an Ellipsoidal Inclusion in a Homogeneous Magnetic Field

A stress–strain problem is solved for an infinite elastic magnetically soft medium with an ellipsoidal inclusion in an external magnetic field. The main characteristics of the stress–strain state and induced magnetic fields in the medium and the inclusion are determined and their distribution over the surface of the inclusion is analyzed     

Influence of hall effect and ion slip on velocity and temperature fields in an MHD channel

The combined influence of viscosity, Hall effect and ion slip on hydrodynamic fields and on heat transfer is investigated. The exact solutions for velocity, induced magnetic field and temperature are derived for the laminar MHD flow in a flat channel assuming a small magnetic Reynolds number, finely segmented electrodes, fully developed flow and uniform heat flux at channel walls. The internal generation of heat is not considered. The Kantorowitsch method of variational calculus is employed to approximate the complicated velocity distribution.     

The Stress State of a Transversely Isotropic Ferromagnetic with a Parabolic Crack in a Homogeneous Magnetic Field

The magnetoelastic problem for a transversely isotropic ferromagnetic body with a parabolic crack in the plane of isotropy is solved explicitly. The body is in an external magnetic field, which is perpendicular to the plane of isotropy. The field induces elastic strains and a magnetic field in the body. The characteristics of the stress–strain distribution and induced magnetic field are determined; and their singularities in the neighborhood of the crack are analyzed. Formulas for the stress intensity factors of the mechanical and magnetic fields near the crack tip are presented     

Kinetic model of the electron plasma component in plasma accelerators with closed drift

The dynamics of the electron plasma component in plasma accelerators with closed drift and a long acceleration zone are considered under certain simplifying assumptions (strong uniform magnetic field, elastic diffuse electron scattering by the channel walls, etc.). Exact solutions of the kinetic equation for electrons in different limiting cases are found. The results are generalized to include the case of a weakly nonuniform magnetic field.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 187–193, November–December, 1994.     

Nonstationary conducting free flow in a transverse magnetic field

In magnetohydrodynamic flow the viscous friction at the walls can be substantial. The role of viscous friction can be considerably reduced by using a free or a semirestricted flow of the conducting fluid. Nonstationary phenomena in one-dimensional motion of a free plane incompressible fluid flow in a transverse magnetic field are examined. The narrow sides of the flow come into contact with the sectional electrodes connected through external circuits with an active-inductive load. The magnetic Reynolds number and the magnetody-dynamic interaction parameter are assumed to be large. When the electric field due to electromagnetic induction in the channel is much smaller than the field due to the external circuits, the problem can be reduced to the characteristic Cauchy problem for a quasilinear hyperbolic system of first-order equations which can be solved by the method of characteristics using a computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 34–39, July–August, 1970.     

Stress–Strain State of a Ferromagnetic with a Paraboloidal Inclusion in a Homogeneous Magnetic Field

The stress–strain state of an infinite elastic soft ferromagnetic medium with an elliptic paraboloidal inclusion is analyzed. The material of the inclusion is a soft ferromagnetic too. The medium is in a magnetic field directed along the minor axis of the elliptic section of the paraboloid by a plane perpendicular to its axis. The main characteristics of the stress–strain state and induced magnetic fields in the medium and inclusion are determined. The features of the stress distribution over the inclusion boundary are studied     

Stationary supersonic conducting gas flow in a channel with nonconducting walls under weak magnetohydrodynamic interaction

A solution of the problem of flow in a channel with nonconducting walls for a small magnetohydrodynamic interaction parameter N is obtained by numerical methods. In the 0–10 range of variation of the Hall and magnetic Reynolds number parameters the distributions of the electrical parameters and the average (over the cross section) and local gasdynamic flow parameters are computed for two different geometries of the applied magnetic field. It is shown that an increase in the Hall and magnetic Reynolds number parameters is accompanied by a diminution in the Joule dissipation and the perturbation of the average (over the cross section) gasdynamic flow characteristics. It is disclosed that the distribution of the gasdynamic parameters over the channel cross section is extremely nonmonotonic in the end current zones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 20–29, July–August, 1970.In conclusion, the author is grateful to A. B. Vatazhin for useful comments and constant attention to the research and to I. U. Tolmach for valuable comments.     

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.     

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

Flow of a conductive gas in a circular tube in a strong axiosymmetric magnetic field

The flow of a conductive gas along a channel in an external axiosymmetric magnetic field with a finite value of the magnetogasodynamic parameter N is examined. Numerical flow calculations are performed for a circular tube in such a field. Gas dynamic parameter fields, total pressure losses, and electric current intensities with the presence of transsonic zones and highly compressed regions are determined. Through comparison of the results obtained with linear theory data, the range of applicability of the latter is determined. Of the works dedicated to study of flow in external magnetic fields with N1, we should take note of [1], in which the process of entry of the gas into a transverse magnetic field was examined; [2], which studied one-dimensional transient motion with shock waves; and [3], where mixed flow in a Laval nozzle with an axiosymmetric homogeneous magnetic field was studied. Flow in a circular tube was examined in [4]; but the analysis performed by the characteristic method permitted calculation of only the initial supersonic flow zone. Motion in circular tubes in the presence of an axiosymmetric, magnetic field was studied in the linear formulation in [4, 5].Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–155, September–October, 1972.     

Two-dimensional magnetic fields in magnetohydrodynamic channels with steel walls at finite magnetic reynolds numbers

Plane problems on the distribution of a two-dimensional magnetic field in magnetohydrodynamic channels with ferromagnetic walls at appreciable magnetic Reynolds numbers and prescribed flow hydrodynamics are studied. An integral representation for the total magnetic induction is constructed with the use of a complex influence function describing the field resulting from a unit current. This makes it possible to obtain arbitrarily close approximations to exact solutions of the problems on a digital computer. Influence functions for various channels can be determined by mirror reflections and conformai mappings. The method is illustrated by numerical calculations of the distribution of the magnetic field for the flow of a conducting fluid along a plane ferromagnetic wall and the flow of a fluid in the space between ferromagnetic walls. Calculations are carried out on the effect of an external circuit and an inhomogeneous transverse velocity profile on the distribution of the magnetic field.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–11, September–October, 1971.In conclusion the authors thank G. A. Lyubimov, A. B. Vatazhin, V. V. Gogosov, and A. E. Yakuberiko for useful discussion of the formulation of the problem and of results of the study.     

Features of the electric current and field distributions in flows with narrow layers of highly conductive gas

A solution obtained by Fourier's method provides the basis for analyzing the influence of a narrow gas layer, of higher conductivity than the rest of the flow, on the Joule dissipation and current distribution in the terminal zone of a plane magnetohydrodynamic channel with nonconducting walls. The MHD interaction parameter, Reynolds magnetic number, and Hall parameter are assumed small. It is shown that a narrow, highly conductive layer can on occasions be replaced by a surface of discontinuity, on which well-defined relations between the electric quantities are satisfied. The presence of such a layer leads to an increase in the Joule dissipation and a reduction in the lengths of the current lines. A hopeful arrangement for a magnetohydrodynamic energy converter is one in which an inhomogeneous flow is used, consisting of a continuous series of alternating very hot and less hot zones [1,2]. For this arrangement, it is worth examining the influence of the stratified conductivity distribution of the working body on the Joule dissipation and the electric currents in the channel. Numerous papers have discussed the case of inhomogeneous conductivity in the context of MHD system electrical characteristics. A general solution was obtained in [3] for the stationary problem on the electric field in a plane MHD channel with nonconducting walls when the magnetic field and conductivity are arbitrary functions of the longitudinal coordinate. In [4], where the braking of undeformed conducting clusters was investigated, the Joule dissipation, linked with the appearance of closed eddy currents in the cluster as it enters and leaves the magnetic field, was evaluated. The relationships between the electrical quantities, on moving through a narrow layer of low-conductivity liquid, were considered in [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 1, pp. 39–43, January–February, 1970.In conclusion, the author thanks A. B. Vatazhin for valuable advice and discussion.     

The Stress State of a Ferromagnetic with an Elliptic Crack in a Homogeneous Magnetic Field

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing an elliptic crack is considered. It is assumed that the body is under an external magnetic field perpendicular to the crack plane. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the elliptic crack are studied. Formulas are given for the stress intensity factors for the force and magnetic fields near the crack tip     

Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

Flows of a conducting incompressible liquid in an arbitrary region with a strong magnetic field

A general solution is obtained describing the flow of a conducting liquid in an arbitrary region, which is bounded by nonconducting walls, in the case when it is possible to neglect inertial and viscous forces in comparison with the magnetic field, as well as the induced magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 144–150, May–June, 1973.     

Calculation of the flow of an electrically conducting liquid in a duct of rectangular cross section with electrodes

A numerical solution is given for the problem of the flow of an electrically conducting liquid in a duct of rectangular cross section whose walls in the direction at right angles to the applied magnetic field are nonconducting, whereas those parallel to the field are perfect conductors. It is assumed that all the quantities except the pressure are independent of the coordinate along the axis of the duct, that the applied magnetic field is homogeneous, and that the induced current is diverted into an external circuit. The total current in the external circuit and the difference of the potentials of the conducting walls are found as functions of the external load, the Hartmann number, and the ratio of the lengths of the sides of the duct. It should be noted that problems of this kind have already been considered on many occasions and by many different approximate methods. The most complete bibliography on this question can be found in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–45, September–October, 1970.     

Production of strong toroidal fields by magnetic fluxcompression

Experiments conducted to produce megaGauss toroidal magnetic fields by flux compression are described. The possibility of intensifying the field by more than 20 times is shown experimentally. The experimental results are compared with theory. It has been shown previously experimentally that when compressing a longitudinal magnetic field by a liner which is also accelerated by magnetic pressure it is possible to obtain stable motion and good symmetry of the collapse [1–3]. These conditions are necessary for effective field intensification. However a configuration with closed lines of force is necessary from the viewpoint of the possibility of plasma containment and heating by a magnetic field. With this objective we examined the possibility of intensifying a toroidal field by magnetic flux compression.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3 pp. 47–50, May–June, 1970.In conclusion the authors wish to thank G. I. Budker for his support and continued interest in the project, P. Z. Chebotaev for assistance in the computer calculations, V. N. Karasyuk for constructing the apparatus, and also A. V. Kutovenko and P. G. Zubov for help in carrying out the experiments.