Diffusion to a particle at large péclet numbers and small Reynolds and Hartmann numbers

A study is made of the influence of a homogeneous magnetic field on the mass transfer for a spherical solid particle and a liquid drop in a flow of a viscous electrically conducting fluid. The previously obtained [1] velocity field of the fluid is used to calculate the concentration distribution in the diffusion boundary layer, the density of the diffusion flux, and the Nusselt number, which characterizes the mass transfer between the particle and the surrounding medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 189–192, January–February, 1980.     

The effect of induced magnetic fields on convective stability of a conductive liquid carrying a current

The stability of equilibrium in a plane layer of an electrically conductive liquid located in electric and magnetic fields is considered. The effect of an unperturbed induced magnetic field and its perturbations on stability of this type is considered. The stability is analyzed nonlinearly.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 102–110, January–February, 1974.     

An electromagnet with Hall excitation

In 1911 Corbino showed that if a disk with a current flowing to the axis is placed in a magnetic field parallel to the axis of the disk, then due to the Hall emf the initially straight line of electric current is turned into a spiral. This leads to an increase in the length of the current line and thus to an increase in the disk resistance. The change in the disk resistance in a magnetic field was used in [1] to switch the current in the circuit of an inductive energy store. If the electric current carriers move from the edge of the disk to the axis, the azimuthal Hall current is accompanied by an increase in the magnetic field inside the disk compared with that outside it [2]. The same processes occur in a hydromagnet [3–5], in which the radial flow of a conducting liquid in an axial magnetic field is used to amplify a magnetic field. In the papers mentioned earlier the transients which occur when the steady magnetic field is established were not considered. To produce a magnetic field, and particularly for switching, the switch-on time of the device is of considerable importance. Hence, in this paper we consider the nonstationary problem of the amplification of a magnetic field. The amplification of the field is obtained and the time taken for the stationary state to build up is found. Both quantities depend exponentially on the magnetic Reynolds number. For a hydromagnet it is shown that the steady-state magnetic field differs considerably from that obtained in [4, 5]. The disagreement between the results is due to the fact that the boundary conditions in [4, 5] were arbitrarily chosen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 44–48, May–June, 1979.The author thanks E. I. Bichenkov and R. L. Rabinovich for useful discussions and advice.     

Some exact solutions of the ferrohydrodynamics equations

The theoretical study of nonisothermal flows of magnetizable liquids presents serious matheical difficulties, which are intensified as compared to to the study of normal liquids by the necessity of simultaneous solution of both the hydrodynamics and Maxwell's equations, with corresponding boundary conditions for the magnetic field. Thus, in most cases problems of this type are solved by neglecting the effect of the liquid's nonisothermal state on the field distribution within the liquid, and also, as a rule, with use of an approximate solution for Maxwell's equations and fulfillment of the boundary conditions for the field [1–3]. The present study will present easily realizable practical formulations of the problem which permit exact one-dimensional solutions of the complete system of the equations of thermomechanic s of electrically nonconductive incompressible Newtonian magnetizable liquids with constant transfer coefficients. A common feature of the formulations is the presence of a longitudinal temperature gradient along the boundaries along which liquid motion is accomplished. Plane-parallel convective flows of this type in a conventional liquid and their stability were considered in [4–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–133, May–June, 1979.     

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.     

Forces acting on a weakly conductive dielectric in an electric field

The nature of the forces acting on a weakly conductive liquid dielectric in an electric field will be considered. In the general case there act upon the liquid dielectric a Coulomb force related to space charge and a polarization force [1]. In many studies the motion of a conductive liquid dielectric has been explained by the presence of the polarization force, with the Coulomb force being ignored. In the present study it will be demonstrated that the force related to space charge may be larger than or of the same order as the forces connected with polarization of the medium and, generally speaking, must be considered in describing the equations of motion in concrete cases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 155–157, January–February, 1977.     

Inductive acceleration of an electrically conductive particle in a viscous liquid

The characteristics of the motion of a particle in an electrically conducting liquid with constant crossed electric and magnetic fields present have been investigated in connection with the problem of MHD-separation in many papers (for example, see the bibliography in [1]). The separation of electrically conducting particles contained in a dielectric liquid, which can be accomplished with the help of a variable magnetic field [2], is also of practical interest. The ponderomotive force acting on a spherical conducting particle near a straight conductor through which the discharge current of a capacitor bank is flowing is found in this paper, and the motion of a particle in a viscous liquid under the action of this force is investigated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 31–34, November–December, 1984.     

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.     

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.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

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.     

Slow motions of a liquid drop in a viscous liquid

The problem of the fully established slow translational motion of a round drop (bubble) in a viscous liquid was solved by Adamar and Rybchinskii [1, 2]. The results of experimental measurements are rarely in agreement with the Adamar-Rybchinskii formula. This is connected with braking of the flow due to surface-active impurities, which are usually rather numerous in liquids. Nevertheless, we shall consider the problem of the not fully established motion of a drop in the simplest case, assuming that there are no surface-active substances. The article discusses problems of the vibrations and motions of a spherical drop in a viscous liquid, with arbitrary accelerations. An analysis is made of a formula for the force of resistance of a drop of liquid with a high viscosity, an elastoviscous drop, and a particle with slipping-through.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 32–37, November–December, 1975.     

Instabilities of nonuniform flows of acoustically active media

The analytic methods and results of investigating the acoustic instability of nonuniform steady channel flows are reviewed. The study is based on the system of equations describing the motion of an electrically conducting gas at low magnetic Reynolds numbers [25]. This makes it possible to consider the acoustic effects in plasma and nonconducting gas flows within the framework of a unified approach.Based on paper presented to the fluid mechanics sections of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 31–46, September–October, 1992.     

Motion of a magnetizable liquid in a rotating magnetic field

Moskowitz and Rosensweig [1] describe the drag of a magnetic liquid — a colloidal suspension of ferromagnetic single-domain particles in a liquid carrier — by a rotating magnetic field. Various hydrodynamic models have been proposed [2, 3] to describe the macroscopic behavior of magnetic suspensions. In the model constructed in [2] it was assumed that the intensity of magnetization is always directed along the field so that the body torque is zero. Therefore, this model cannot account for the phenomenon under consideration. We make a number of simplifying assumptions to discuss the steady laminar flow of an incompressible viscous magnetizable liquid with internal rotation of particles moving in an infinitely long cylindrical container in a rotating magnetic field. The physical mechanism setting the liquid in motion is discussed. The importance of unsymmetric stresses and the phenomenon of relaxation of magnetization are emphasized. The solution obtained below is also a solution of the problem of the rotation of a polarizable liquid in a rotating electric field according to the model in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–43, July–August, 1970.     

The determination of the crater for an explosion in ground with angular and curvilinear free boundaries

One of the simple mathematical models in the theory of the deformation of continuous media in an explosion is the solid—liquid model [1, 2]. This does not describe the dynamics of the ground and so enables us to determine only approximate characteristics of the crater. This model has now been used to study a wide range of problems in determining a crater in a continuous medium with various tensile properties and various positions of the explosive [3–5]. We consider below within the framework of the solid—liquid model boundary-value problems in determining a crater in the explosions of point explosives and uniformly distributed explosives on the surface and deep within an isotropic ground with angular and curvilinear free boundaries. The desirability of studying problems such as these was pointed out by Il'inski.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1985.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Reflection of magnetoacoustic waves

The problem of the reflection of magnetoacoustic waves at the boundary dividing an elastic medium from a fluid medium with infinite conductivity in the presence of an arbitrary constant magnetic field was treated in [1]. In writing down the boundary conditions the continuity of the tangential component of the magnetic field was used. This condition is valid when the conductivity of the medium is finite but not when the conductivity is infinite. In this connection a problem similar to that in [1] is solved, without employing this particular boundary condition. The amplitude conversion coefficients found for the limiting cases of weak and strong magnetic fields coincide with the respective coefficients given in [2,3] for media with a finite conductivity, if we allow the conductivity in the latter expressions to become infinite.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, January–February, pp. 56–61, 1970.     

Surface shape of a magnetic liquid drop near the edge of a magnetic wedge

The two-dimensional problem of the shape of the free surface of a magnetic fluid in a gravity field, a uniform external magnetic field and the nonuniform field of a magnetized metal wedge is considered. The results of numerically calculating the shape of the free surface of a magnetic liquid drop retained on an inclined plane by the field of a magnetizing wedge are presented. The changes in the shape of the free surface of an infinite volume of magnetic liquid near the edge of a wedge with increase in the external field are investigated. It is shown that for a certain critical field some of the magnetic liquid separates and adheres to the edge of the wedge. Experimental data on the determination of the maximum cross-sectional area of a drop retained by the magnetic field of a wedge and the critical rise of the magnetic liquid relative to the level outside the field are presented. The experimental and theoretical results are in agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 113–119, July–August, 1992.The authors wish to thank V. V. Gogosov for useful discussions and his interest in the work.     

Transverse waves and discontinuities in an ideal magnetized fluid

Suppose that a constant and uniform field B₀ exists within an ideal fluid medium, that is, a medium in which dissipative processes are absent. If this medium is a conducting one and its magnetization and polarization can be neglected, then finite perturbations of the transverse (perpendicular to B₀) components of the velocity and the magnetic field propagate a-long b₀ with a constant velocity without change of form [1], These plane transverse waves, called Alfven waves, are linear and cannot lead to discontinuities if there are no discontinuities in the initial conditions. In this paper we shall consider plane transverse waves in an ideal fluid medium which is not only electrically conducting, but which can also be magnetized by the magnetic field. In such a medium transverse waves are no longer linear and they can develop into jumps in the magnitudes of the field and the velocity. Like Alfven waves, these waves leave the density of the medium unchanged, so that they can also exist in an incompressible fluid. This circumstance is reflected in the nature of the discontinuity, which is analyzed in Section 5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 116–124, November–December, 1973.The author thanks A. G. Kulikovskii and A. A. Barmin for their valuable remarks contributed during discussions of the work.     

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.