Stability of plane-parallel MHD flows in transverse magnetic field

We study within the framework of linear theory the stability of plane-parallel flows of a viscous, electrically conducting fluid in a transverse magnetic field. The magnetic Reynolds numbers are assumed small. The critical Reynolds number as a function of the Hartmann number is obtained over the entire range of variation of the latter. The small perturbation spectrum is studied in detail on the example of Hartmann flow. Neutral curves are constructed for symmetric and antisymmetric disturbances. The destablizing effect of a magnetic field is studied in the case of modified Couette flow. The results obtained agree with the calculations of Lock and Kakutani (where they meet) and are at variance with the results of Pavlov.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 127–131, May–June, 1970.The authors wish to thank M. A. Gol'dshtik for his interest in this study.     

Vibrational convection in a plane layer with a longitudinal temperature gradient

An exact solution is obtained for the equations of vibrogravitational convection in an arbitrarily oriented plane fluid layer with a longitudinal component of the temperature gradient. It is shown that in the absence of a static field inclined vibration leads to the development of plane-parallel convective flow. On the other hand, gravitational plane-parallel convective flow can always be suppressed by an appropriate choice of the direction and amplitude of vibration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–15, July–August, 1990.     

Electrosolenoidal flows in a cylindrical cavity

The main difficulties in investigating three-dimensional magnetohydrodynamic (MHD) flows with vorticity arise, first, because it is necessary to solve an independent boundary-value problem in order to find the field of the electromagnetic forces and, second, because the regimes of these flows are strongly nonlinear for the majority of high-power technological MHD processes and a number of natural phenomena. Particular importance attaches to MHD flows generated by the interaction of an electric current applied to the fluid with the magnetic self-field. This class of MHD flows has become known as electrosolenoidal flows [1]. The presence of a definite symmetry in the distribution of the electromagnetic forces and the geometry of the region of the liquid conductor makes it possible to find a solution in self-similar form. The present paper is devoted to exact solutions of the nonlinear equations for axisymmetric electrosolenoidal flows of a conducting incompressible fluid in infinite cylindrical cavities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 48–53, May–June, 1991.     

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.     

Slow steady flows of a conducting fluid at large Hartmann numbers

We consider slow steady flows of a conducting fluid at large values of the Hartmann number and small values of the magnetic Reynolds number in an inhomogeneous magnetic field. The general solution is obtained in explicit form for the basic portion (core) of the flow, where the inertia and viscous forces may be neglected. The boundary conditions which this solution must satisfy at the outer edges of the boundary layers which develop at the walls are considered. Possible types of discontinuity surfaces and other singularities in the flow core are examined. An exact solution is obtained for the problem of conducting fluid flow in a tube of arbitrary section in an inhomogeneous magnetic field.The content of this paper is a generalization of some results on flows in a homogeneous magnetic field, obtained in [1–8], to the case of arbitrary flows in an inhomogeneous magnetic field. The author's interest in the problems considered in this study was attracted by a report presented by Professor Shercliff at the Institute of Mechanics, Moscow State University, in May 1967, on the studies of English scientists on conducting fluid flows in a strong uniform magnetic field.     

Stability of unsteady viscous flows

A study is made of the linear stability of plane-parallel unsteady flows of a viscous incompressible fluid: in the mixing layer of two flows, in a jet with constant flow rate, and near a wall suddenly set in motion [1]. The slow variation of these flows in time compared with the rate of change of the perturbations makes it possible to use the method of two-scale expansions [2]. The stability of nonparallel flows with allowance for their slow variation with respect to the longitudinal coordinate was investigated, for example, in [3–6]. The unsteady flows considered in the present paper have a number of characteristic properties of non-parallel flows [1], but in contrast to them are described by exact solutions of the Navier-Stokes equations. In addition, for unsteady planeparallel flows a criterion of neutral stability can be uniquely established by means of the energy balance equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, 138–142, July–August, 1981.I thank G. I. Petrov for suggesting the problem, and also S. Ya. Gertsenshtein and A. V. Latyshev for assisting in the work.     

On the set of steady regimes in chemical flow reactors

An extensive literature has been devoted (see, for example, [1–3]) to the question of the existence and uniqueness of steady regimes in chemical flow reactors. In the majority of cases, exact solutions to the corresponding problems cannot be obtained because of the nonlinearity of the functions that describe the kinetics of the chemical reactions. Investigations are therefore usually made by either approximate or numerical methods. In the present paper, exact solutions are obtained to a model nonlinear boundary-value problem of the steady distribution of the concentration in a one-dimensional isothermal chemical flow reactor with longitudinal mixing, and the question of the existence and the number of steady regimes is completely investigated in a three-dimensional space of the determining parameters. The function that describes the rate of the chemical reaction is taken to be a function that simulates approximately the dependence of the rate of the autocatalytic reaction on the concentration of the initial reactant.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 177–181, May–June, 1982.     

Long waves in two superposed layers of magnetic fluid

The equations and boundary conditions describing plane-parallel potential motions of two superposed layers of stably stratified magnetic fluid are formulated. The fluid is assumed to fill entirely a horizontal plane channel in the presence of a uniform longitudinal magnetic field induced by external sources. With reference to the case of long waves propagating over the interface between the upper and lower layers, it is shown that the action of the field may be interpreted as the result of an increase in the nondimensional surface tension by an amount proportional to the square of the undisturbed field. In the linear formulation the effect of the field on the evolution of a long-wave perturbation of the initially plane interface is investigated. Korteweg-de Vries equations with quadratic and cubic nonlinearities are derived and the action of the field on the internal solitary waves is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–133, May–June, 1993.     

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.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

Convection in an oscillating force field and microgravity

Convective flows in a plane layer of viscous fluid in the presence of an oscillating external force are investigated numerically [1 – 8]. The layer is assumed to be placed in a gravitational field. The cases in which the external field oscillations are generated by rotation about the horizontal axis or by vibration in the longitudinal direction are considered. The Navier-Stokes equations and the Boussinesq approximation are used for describing the fluid motion. The flows developing in the layer in the presence of a transverse temperature gradient are determined, the stability boundaries of these flows are found, and the supercritical motion regimes are studied. These investigations are carried out using the averaging method (in order to find the stability limits for high rotation velocities and vibration frequencies) and the Galerkin method.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–106, September–October, 1994.     

Electrical breakdown in air in a transverse magnetic field

The electrical breakdown of gases in a transverse magnetic field is discussed in references [1–16]. Attention has mainly been concentrated on the case of coaxial electrode geometry [1–10]. The existing experimental data on breakdown between plane-parallel electrodes [11–14] relate to a narrow range of variation of the parameters characterizing breakdown (P, d, H, U). The author has made an experimental study of the process of electrical breakdown in air in a transverse magnetic field between plane-parallel electrodes of finite size in the pressure interval from 650 to 5·10^–3 mm Hg at gap lengths of from 1 to 140 mm and magnetic inductions from 0 to 10 600 G.     

Flow in rotating radial channels at small Rossby and Ekman numbers

The problem of flow of a viscous fluid in a rotating channel is considered in the region of very small Rossby and Ekman numbers and moderately large Reynolds numbers. Asymptotic expressions with respect to the Ekman number are found for the velocity components and the longitudinal pressure gradient by solving a system of linear differential equations using Fourier series. The stability limits of such flow are predicted. Attention is drawn to a similarity between the velocity profiles of these flows and flows of a magnetic fluid and a fluid executing longitudinal oscillations in a fixed channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 11–15, January–February, 1984.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

Unsteady flows with a constant total pressure,described by the equations of ideal magnetohydrodynamics

Exact solutions of the equations of ideal magnetohydrodynamics describing the class of unsteady flows of an electrically conducting fluid with a constant total pressure are constructed. The solutions are written in the Lagrange coordinate system; arbitrariness in its choice was used to parameterize magnetic field lines. The wide functional arbitrariness the solutions provide a significant variation in the picture of the described fluid motions. An example of unsteady flow of an ideal electrically conducting fluid in a cylindrical channel with fixed magnetic tubes is given.     

On the obtaining of axisymmetric flows from plane-parallel flows

The obtaining of axisymmetric flows of incompressible and compressible fluids from plane-parallel flows by means of integral transformations relating harmonic and p-harmonic functions [1] is considered. A transformation is found that carries plane-parallel flows from elementary singularities into axisymmetric flows. It is shown that this transformation makes it possible to obtain the general form of the solution of axisymmetric problems of flow past bodies from the solution of plane-parallel problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1982.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.     

Magnetohydrodynamic transient flows of a non-Newtonian fluid

Some exact solutions of the time-dependent partial differential equations are discussed for flows of an Oldroyd-B fluid. The fluid is electrically conducting and incompressible. The flows are generated by the impulsive motion of a boundary or by application of a constant pressure gradient. The method of Laplace transform is applied to obtain exact solutions. It is observed from the analysis that the governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of the viscous fluid. Several results of interest are obtained as special cases of the presented analysis.     

Reduction of viscous flows having orthogonal magnetic and velocity field distributions

Summary Certain steady magnetohydrodynamic flows of a viscous incompressible fluid, in which the magnetic field is everywhere orthogonal to the velocity field, are related to viscous compressible flows having zero magnetic field. Examples are given to illustrate this relationship. Other linked magnetic and non-magnetic flows can be found using similar processes.