Electromagnetic disturbance created by an expanding conducting cylinder in a magnetic field

We examine the electromagnetic disturbance created by a linearly expanding, ideally conducting cylinder in an external uniform magnetic field with account for effects of order v/c (v is the boundary velocity).The problem of uniform magnetic field compression in a contracting cylindrical cavity with ideally conducting walls with radius decreasing linearly with time was solved in [1]. As far as we know the corresponding external problem has not been investigated.     

Propagation of a shock wave across a current sheet without reflection

A class of exact solutions is obtained for the magnetohydrodynamic equations with plane waves which describe the solid-body motion of an ideally conducting gas in a given uniform gravitational field. The gas motion is initiated by the impact of a piston which generates a shock wave propagating through the initial equilibrium state with a decreasing density. Propagation of the shock wave across a current sheet is considered within the framework of the results obtained.     

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.     

Frictional moving contact problem between a conducting rigid cylindrical punch and a functionally graded piezoelectric layered half plane

Meccanica - In this study, a frictional moving contact problem between an electrically conducting rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to a...     

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.     

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.     

Wave propagation on a charged surface of a cylindrical liquid column surrounding a long porous core

A mathematical model of wave propagation and instability on a charged surface of an infinite cylindrical column of a conducting fluid, surrounding a coaxial infinite porous core, is formulated and studied. The conditions are found under which the disturbances of the liquid column become unstable and result in its fragmentation into a chain of connected droplets. It is shown that the length of the droplets decreases with increase in the electric field.     

Model of Nonlinear Evolution of Long-Wave Perturbations on an Ideally Conducting Jet with Current in a Longitudinal Magnetic Field. Collision of Magnetized Jets

Within the framework of the magnetohydrodynamic approach, a system of equations is derived for nonlinear evolution of long-wave axisymmetric perturbations on a conducting fluid jet with surface electric current, located along the axis of a conducting solid cylinder in a longitudinal magnetic field. The fluid is assumed to be inviscid, incompressible, and ideally conducting, like the cylinder walls. It is shown that, if the longitudinal field is uniform and the axial flow is shear-free, this system can be either hyperbolic or elliptic-hyperbolic, depending on problem parameters. The boundaries of hyperbolicity and ellipticity regions in the space of solutions are determined. In the hyperbolicity region, equations of characteristics and conditions on them are obtained. The problem of the decay of velocity discontinuity on the jet is considered. Conditions are found for the existence of a continuous self-similar solution in the hyperbolicity region, corresponding to collision of jets.     

Two-dimensional spreading of a cloud of conducting gas in a magnetic field self-similar solution

A self-similar solution of the problem on the spreading in a magnetic field of a cloud of conducting gas, having the shape of a cylinder of noncircular cross section, is constructed. The cylindrical surface of the gas is restrained by a nonconducting sheath that spreads according to a prescribed law. The shape of the transverse cross section of the cylindrical cloud is determined from the solution. Cross sections obtained for a concrete case are represented in graphic form.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 29–36, March–April, 1973.In conclusion the author thanks V. I. Khonichev for assistance with the numerical calculations.     

Interaction Between a Slow Magnetohydrodynamic Shock Wave and a Tangential Discontinuity

The self-similar problem of the oblique interaction between a slow MHD shock wave and a tangential discontinuity is solved within the framework of the ideal magnetohydrodynamic model. The constraints on the initial parameters necessary for the existence of a regular solution are found. Various feasible wave flow patterns are found in the steady-state coordinate system moving with the line of intersection of the discontinuities. As distinct from the problems of interaction between fast shock waves and other discontinuities, when the incident shock wave is slow the state ahead of it cannot be given and must to be determined in the process of solving the problem. As an example, a flow in which the slow shock wave incident on the tangential discontinuity is generated by an ideally conducting wedge located in the flow is considered. The basic features of the developing flows are determined.     

Channel flow of an anisotropically conducting medium in a zone of entry into a magnetic field

A considerable number of papers are devoted to the problem of the deformation of a plane flow of a conducting liquid moving through a channel |x| < , 0 y h=const in a zone of entry into a magnetic field B=(0, 0, B. (x)), where (x) is the Heaviside unit function((x)=0 for x < 0 and (x)=i for x < 0). Apparently the first paper in this direction was that of Shercliff [1, 2] in which the asymptotic (for x .o- )profile of a perturbed velocity was. determined for a flow of an isotropic conducting liquid in a channel with nonconducting walls. The flow considered by Shemliff takes place in magnetohydrodynarnic flowmeters. Complete calculation of the perturbed flow of an isotropie conducting liquid in the channel of a magnetohydrodynamic generator is carried out in [3]. Asymptotic velocity profiles in the channel of a magnetohydrodynamic generator, with ideally segmented electrodes and the flow of an anisotropically conducting medium along them, were found in [4]. General formulas for the calculation of the asymptotic velocity profile, from the known distribution of the perturbing forces along the channel, are presented in [5]. In [6, 7] the Green function is proposed for the solution of the equation for the stream function of the perturbed flow. Finally, in [8], the solution for the perturbed flow of an anisotropically conducting liquid in a channel with continuous electrodes is described by means of the Green function, and the asymptotic profiles of the velocity are calculated.In this paper the flow of anauisotropically conducting liquid is determined in a channel with ideally segmented electrodes. The solution is set up with the aid of the Fourier method. The resulting series, in which the slowly converging part can be related to the asymptotic profile [4] calculated from the solution of an ordinary differential equation, make it possible to determine the velocity field rapidly. A detailed deformation pattern of the velocity profile is set up. Certain general properties of the flow in a zone of entry into a magnetic field are noted; with the aid of these it is possible to discover the error in the calculations [8].     

Spontaneous swirling in MHD flows with circular streamlines

This paper considers the problem of the evolution of azimuthal perturbations in axisymmetric magnetohydrodynamic. flows of an ideally conducting inviscid fluid with circular streamlines. The fluid is. in a toroidal gap between two surfaces with constant values of the stream function. The equations of. fluid motion are derived in the approximation of infinitely a narrow gap. The parameters at which. spontaneous swirling is possible are determined numerically, and the properties of secondary swirling. flows resulting from instability of the initial steady-state poloidal flow are established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 89–97, March–April, 2009.     

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.     

Mathematical analysis of the radiation from a magnetic line source on a cylindrically capped wedge covered by a plasma sheath

Summary Radial eigenfunctions appropriate for the dielectric (or plasma) layered medium and air are used to obtain an exact solution of the problem of radiation from a magnetic line source (or a slotted antenna) on a conducting infinite wedge with a cylindrical cap covered by a dielectric (or plasma) sheath. In order to satisfy the boundary conditions at the interfaces between the dielectric (or plasma) and the air, it is necessary to expand the radial eigenfunctions of one medium in terms of those of the other.This method can be used to solve any other two dimensional diffraction or radiation problem involving a cylindrically capped wedge surrounded by a dielectric (or plasma) sheath.     

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.     

Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids

In this paper, based on three-dimensional linear generalized thermoelasticity, an exact analysis of free vibration of a simply supported homogeneous isotropic, thermally conducting, cylindrical panel with voids initially at uniform temperature and undeformed state has been presented. Three displacement potential functions are introduced for solving the equations of motion, heat conduction and volume fraction field. The purely transverse wave gets decoupled from rest of motion and is not affected by thermal and volume fraction (voids) fields. After expanding the displacement potentials, volume fraction and temperature functions with orthogonal series, the equations of the considered vibration problem are reduced to five-second order coupled ordinary differential equations whose formal solution can be expressed by using Bessel functions with complex arguments. The corresponding results for thermoelastic panel without voids, elastic panel with and without voids have been deduced as special cases from the present analysis. In order to illustrate the analytical results, the numerical solutions of various relations and equations have been obtained to compute the lowest frequency as function of different cylindrical panel parameters. The computer simulated results have been presented graphically.     

Stationary cylindrical vortex in a viscous electrically conducting fluid

An exact solution of the magnetohydrodynamic equations is constructed which describes steady vortex flow in a stationary cylinder on the axis of which a conductor carrying a known current is located. The solution is obtained under the assumption that the fluid is viscous and has finite electrical conductivity and that the magnetic field has only the axial and azimuthal components in a cylindrical coordinate system. It is found that the action of the Lorentz force is compensated by changing the pressure. Fluid flow occurs from the periphery to the axis of the cylinder under a pressure gradient, with flow rotation and swirling. The fluid flow causes a concentration of the magnetic lines near the axis of the cylinder, providing an exponential decrease in the magnetic field strength with distance from the axis. This flow can be considered as a model of a local increase in the magnetic field strength due to the transfer of its force lines by the flow of the electrically conducting fluid.     

Plastic deformation of a wedge by a sliding punch

We present a self-similar solution of the problem of deformation of an ideally plastic wedge by a sliding punch with regard to contact friction; such a solution generalizes the well-known solutions of the problem of wedge penetration into a plastic half-space and of compression of an ideally plastic wedge by a plane punch. The problem is of interest for modeling the processes of plastic deformation of rough surfaces of metal pieces by a rigid tool.     

Self-similar solution to the problem of the expansion of a cylindrical column of conducting gas in a longitudinal magnetic field

The nonstationary radial motion of a long cylindrical column of conducting gas in a time-varying longitudinal magnetic field is considered. Exact solutions are found by the method of separating the variables for the system of equations of magnetohydrodynamics on the assumption that the statistical pressure of the plasma at the boundary of the column is proportional to the external magnetic pressure. Some numerical computations are performed and the energetic characteristics of the interaction process are calculated. The ratio of the useful work done by the gas over an infinite time interval to the initial energy of the column is given as a function of the magnetic Reynolds number. We note that a similar method was applied in [1], where not only was the average temperature taken over the cross section, but the inertia of the medium was also neglected. When the inertia is taken into account, we have the additional requirement that the statistical pressure be proportional to the magnetic pressure at the boundary of the column.A physically similar model may be interpreted, for example, as the expansion of a compressible conducting gas column in a nonconducting incompressible fluid situated in a permeable cylinder of some radius R infinite along the axis of symmetry. The requirement that the statistical pressure be proportional to the magnetic pressure reduces to the condition that the external pressure on the boundary of the permeable cylinder of radius R should vary according to a specific law, which may easily be determined.     

Orientation effect on the Boussinesq indentation of a transversely isotropic piezoelectric material

Three-dimensional finite element analysis was used to study the effect of the angle between the loading direction and the axisymmetric direction on the indentation behavior of a transversely isotropic piezoelectric half-space by a cylindrical indenter of flat end. Two cases were considered in the analysis, which included (a) the indentation by an insulating indenter, and (b) the indentation by a conducting indenter. Both the indentation load and the indentation-induced potential were found to be proportional to the indentation depth. Using the simulation results and the analytical relationship for the indentation by a rigid, insulating indenter, semi-analytical relationships were developed between the indentation load and the indentation depth and between the indentation-induced potential on the indenter and the indentation depth for the conducting indenter, respectively. The proportionality between the indentation-induced potential and the indentation depth is only a function of the angle between the loading direction and the poling direction, independent of the type of indenters, which may be used to measure the relative direction of the loading axis to the axisymmetric axis of transversely piezoelectric materials from the indentation test.