Motion of an incompressible conducting liquid in a strong self-consistent magnetic field

One of the common regimes of operation of many laboratory and industrial magnetohydrodynamic (MHD) devices using liquid metals as working medium is the regime for which the Alfvén number A, the ratio of the magnetic and kinetic energy densities, appreciably exceeds unity. For example, for a typical MHD device [1] with characteristic length 0.1 m of the working region, velocity 1 m/sec of the medium, and magnetic induction 1 T (the medium is molten sodium at temperature 330°C) the Alfvén number is A - 900. To simplify the investigation of the processes in such devices, one can use the approximation of a strong magnetic field proposed by Somov and Syrovatskii [2] to describe certain types of hydrodynamic flows of a dissipationless plasma in a magnetic field. In the present paper, the approach to the analysis of the self-consistent magnetohydrodynamic problem in this asymptotic approximation is extended to the case of an incompressible liquid with finite conductivity. A study is made of the closed reduced system of MHD equations obtained from the complete model in the zeroth order in the small parameter A^–1, in which the magnetic field is a force-free field. An investigation is made of the free diffusion of force-free magnetic field with constant coefficient a of proportionality between the current density and the magnetic induction in a spatially unbounded liquid, and the kinematic properties of a velocity field of the liquid in which the force-free nature of the magnetic field is maintained during the damping process are determined. It is shown that the complete class of such velocity fields is represented by the group of rigid-body motions of the liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1991.     

Shape of an incompressible,weakly conducting jet in a strong electric field

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 20–25, March–April, 1991.     

Laminar slip flow of an electrically conducting incompressible rarefied gas in a channel with a transverse magnetic field

Summary The effects of a constant external magnetic field on the laminar, fully developed flow of an electrically conducting incompressible rarefied gas in a nonconducting parallel-plate channel are studied. Consideration is given to the slip-flow regime, wherein a gas velocity discontinuity occurs at the channel walls. It is found that the magnitude of the slip velocity is unaffected by the magnetic-field strength for a given pressure drop, but that the mean gas velocity and wall friction coefficient are functions of both the velocity slip coefficient and the magnetic-field strength. The effect of a second-order slip-flow boundary condition is briefly discussed.     

Dynamic strain of a conducting half space with a cavity in a strong magnetic field

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 16–20, November–December, 1991.     

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.     

Stability of a perturbed conducting gas flow in a magnetic field for arbitrary magnetic Reynolds numbers

A numerical calculation is made which describes the conversion into a T-layer of a finite perturbation in electrical conductivity imposed on a one-dimensional supersonic flow of a compressible medium for a finite value of the magnetic Reynolds number. The development of the injected perturbation is significantly affected by the magnetic Reynolds number of the unperturbed flow, and to each value of this number there corresponds a particular boundary region in which the perturbation is taken up by the magnetic field into an induced T-layer. The stability is investigated in the linear approximation for a minimal perturbation, and the dispersion equation is solved with allowance for gradients in the unperturbed parameters. It is shown that an overheating instability can arise in the system and lead to the formation of a T-layer.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 3, pp. 3–9, May–June, 1973.The authors thank L. M. Degtyarev, L. A. Zaklyaz'minskii, and A. P. Favorskii for useful discussions and advice during the completion of this work.     

Calculation of the free surface of a conducting liquid in a strong electric field

A study is made of the static shape of a conducting liquid in a strong inhomogeneous electric field at the end of a capillary. Numerical solution of a self-consistent problem is used.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 165–167, November–December, 1983.     

Charged jet of an incompressible liquid in an electric field

The problem of the jet flow of an incompressible liquid with free boundaries in an electric field is solved in the approximation of a laminar boundary layer. An exact solution for a round jet is found in the class of self-similar solutions. In the case of a flat slit jet, a solution is constructed in the form of a series in powers of the coordinate transverse to the plane of symmetry. The dependence of the radius (half-width) on the longitudinal coordinate is given. Branch of the Karpov Physicochemistry Institute, State Science Center, Obninsk 249020. Karpov Physicochemistry Institute, State Science Center, Moscow 115523. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 12–16, July–August, 1998.     

Convective motion of a conducting liquid carrying a current in a transverse magnetic field

The convective instability of a layer of conducting liquid carrying a current and lying in a magnetic field perpendicular to the current is considered. The problem of the nonconductive approximation in a linear setting is solved. The relationships between the Rayleigh number and the Hartmann number (determining the neutral stability) are derived.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza. No. 2, pp. 20–26, March–April, 1971.     

Current distribution in a flat magnetohydrodynamic channel for the motion of an electrically conducting medium in a strong magnetic field

In a strong magnetic field the ions as well as the electrons acquire Larmor rotation. In this case the electric field in the channel and its integral characteristics depend both on the geometry of the channel and the external magnetic field strength, as well as on the physical state and chemical composition of the moving medium. Phenomena occur which are observed in a channel when only the spiral motion of the electrons is taken into account (current concentration on the ends of the electrodes, distortion of the streamlines in the center of the channel etc.).However, the presence of ion slipping exerts a much more marked effect on the integral characteristics of channels. Thus, for example, the power of a magnetohydrodynamic generator in a strong magnetic field becomes constant and remains so as the field increases still further. This property of an energy converter is explained by the fact that the internal resistance of the generator has a square law dependence on the magnetic field.Another important phenomenon is connected with the slipping of ions. In a strong magnetic field the Hall electromotive force disappears, the electrical conductivity of the medium becomes a scalar instead of being a tensor, and the current distribution pattern in the channel assumes the same form as in the case in which there is no Larmor rotation of the electrons. The electrical conductivity of the medium is then taken to mean its effective value which is a function of the magnetic field. In the present paper the task of finding the current distribution in a channel is reduced to solving a boundary value problem for a special class of periodic functions. For this we use the theory of boundary -value problems for the class of automorphic functions.The author is grateful to A. B. Vatazhin for a useful discussion of the paper.     

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.     

Convective stability of a weakly conducting liquid in an electric field

In an inhomogeneously heated weakly conductive liquid (electrical conductivity 10^–12–1 cm^–1) located in a constant electric field a volume charge is induced because of thermal inhomogeneity of electrical conductivity and dielectric permittivity. The ponderomotive forces which develop set the liquid into intense motion [1–6]. However, under certain conditions equilibrium proves possible, and in that case the question of its stability may be considered. A theoretical analysis of liquid equilibrium stability in a planar horizontal condenser was performed in [2, 4]. Critical problem parameters were found for the case where Archimedean forces are absent [2]. Charge perturbation relaxation was considered instantaneous. It was shown that instability is of an oscillatory character. In [4] only heating from above was considered. Basic results were obtained in the limiting case of disappearingly small thermal diffusivity in the liquid (infinitely high Prandtl numbers). In the present study a more general formulation will be used to examine convective stability of equilibrium of a vertical liquid layer heated from above or below and located in an electric field. For the case of a layer with free thermally insulated boundaries, an exact solution is obtained. Values of critical Rayleigh number and neutral oscillation frequency for heating from above and below are found Neutral curves are constructed. It is demonstrated that with heating from below instability of both the oscillatory and monotonic types is possible, while with heating from above the instability has an oscillatory character. Values are found for the dimensionless field parameter at which the form of instability changes for heating from below and at which instability becomes possible for heating from above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 16–23, September–October, 1976.In conclusion, the author thanks E. M. Zhukhovitskii for this interest in the study and valuable advice.     

Equilibrium states of an elastic conducting rod in a magnetic field

Our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. We assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains. Fairly standard methods give existence of a minimum (not necessarily unique) with e.g., L₂-regularity. The assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal necessary conditions for optimality. A new general technique is developed to justify these conditions; it then follows that minimizers satisfy the equilibrium conditions in the classical sense. (A feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class.) In the symmetric case, which admits trivial (straight and untwisted) solutions, we show that nontrivial solutions also exist if the field is strong enough.     

Hydraulic resistance of rough channels with the flow of an incompressible current-conducting liquid in the presence of a constant transverse magnetic field

The article discusses new results of an investigation of the hydraulic resistance of channels with artificial roughness. On the basis of the experimental data, a conclusion is drawn with respect to the insignificant role of the forces of inertia in comparison with other forces under laminar flow conditions. An analogy is brought out between Hartmann flow around elements of the roughness and the flow of a free MHD stream around bodies, for cases where the height of the elements of the roughness is much greater than the thickness of the gradient layer 〈k〉?H^?1. This opens up additional possibilities for studying MHD flow around bodies with large Reynolds and Hartmann numbers. Dependences are given for determining the resistance coefficient. It is shown that, in a turbulent flow zone, for a majority of rough channels, the experimental data on the resistance coefficient are correlated satisfactorily in the form of a linear dependence on the Stuart number.     

Effect of Joule dissipation on the convective instability of a conducting liquid with flow in a magnetic field

The results of [1] are extended to the case when the Joule dissipation leads to a nonlinear profile of the unperturbed temperature of the liquid. Convective instability of a conducting liquid, with flow in a magnetic field directed perpendicular to the flow, with a temperature-dependent distribution of the conductivity which is nonhomogeneous in the direction of action of the electromagnetic force, was discussed in [1], neglecting Joule dissipation. This type of approach permitted investigating an energy equation without electromagnetic terms, which to a certain degree facilitated the solution of the problem. In many cases, however, the Joule dissipation is considerable and may exert a considerable effect on the development of convective instability. Thus, without taking account of Joule evolution of heat, instability can arise only with positive values of the Rayleigh number, exceeding some critical value, while, at the same time, Joule dissipation may lead to a situation in which instability will develop also with negative values of the Rayleigh number, i.e., under conditions when the state without the evolution of Joule heat is absolutely stable.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 19–22, January–February, 1972.     

Stability of a falling conducting liquid film in an alternating electric field

The flow of a thin viscous conducting liquid film falling along one of the plates of a vertically positioned plane capacitor is studied. The capacitor is connected to an alternating current power supply. It is shown that the presence of the electric field leads to flow destabilization; moreover, the parametric resonance of capillary waves is observed.