Equations Of Magnetic Hydrodynamics Research Articles

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Пульсации течения жидкого металла, генерируемые переменным магнитным полем

This paper studies numerically a vortex flow of liquid metal driven by an electromagnetic force, which is generated by the interaction of the alternating magnetic field of a short solenoid with the induced electrical current. A short solenoid is placed coaxially to a cylindrical cell at half of its height. The mathematical model used to describe the process is based on the equations of magnetic hydrodynamics in the induction-free approximation. Calculations, which are carried out by the control volume method using the ANSYS Fluent package, show that the average flow has the form of two toroidal vortices. The calculated velocity fields are indicative of the oscillatory behavior of vortices, accompanied by a change in their sizes. In the examined range of the force parameter, the predominant flow pattern is the single-mode oscillatory flow. The dependences of characteristic frequency and Reynolds number on the force parameter, are obtained using spectral analysis. It has been found that the oscillation period is close to the period of rotation of a large-scale vortex. It has been established that the oscillations are of quasi-periodic character, and a distinct oscillation frequency is observed only in the flow region near the solenoid. The effect of velocity fluctuations is strong and can be detected in laboratory conditions when using gallium eutectic. Such experiment is planned for the near future. The flow rate of gallium eutectic will be measured by an ultrasonic Doppler anemometer. The results of numerical modeling and their verifications can be useful in determining the ways of reducing the intensity of vortex flows during the electromagnetic separation of impurities, which is based on the induction mechanism responsible for the generation of electromagnetic force that displaces particles. The data on the oscillation frequency of the unsteady flow can be used in the development of a non-contact technique for estimating the average electrical conductivity of a two-phase medium, such as a liquid metal with undesirable impurities.

Exact solutions for the description of nonuniform unidirectional flows of magnetic fluids in the Lin–Sidorov–Aristov class

The paper considers the exact integration of magnetic hydrodynamic equations for describing nonuniform unidirectional flows of viscous incompressible fluids. The construction of an exact solution is based on the well-known representation of hydrodynamic fields as the Lin–Sidorov–Aristov class. The 3d magnetic field is described by linear forms with respect to two spatial coordinates (longitudinal, or horizontal). The coefficients of the linear forms depend on the third coordinate and time. In view of the incompressibility condition, the 1D velocity field depends on two coordinates and time. The pressure is shown to be determined by a quadratic form with constant coefficients. These coefficients are determined by pressure distribution on the known (free) boundary. The exact solution is illustrated by the integration of non-1D hydrodynamic fields in the case of the steady motion of a conducting viscous incompressible fluid. This solution is polynomial, and it will be useful for the formulation of new problems of hydrodynamic stability.

Arbitrary amplitude electron acoustic solitary excitations with Cairns–Tsallis distributed hot electrons and positrons in magnetized space plasmas

A rigorous theoretical study is performed to understand the characteristics of nonlinear electron acoustic (EA) solitary waves propagating in collisionless magnetized plasma system comprising cold inertial electrons, nonthermal–nonextensive hot electrons and positrons and stationary ions. The propagation and evolution of two potential modes (fast and slow) are studied in the linear regime. By employing magnetic hydrodynamic equations and with the aid of travelling wave approximation, the nonlinear wave dynamics is demonstrated by the first integral of motion in the form of a pseudo-energy balance equation. In exploring the EA soliton characteristics, numerical results reveal that only negative polarity EA solitons exist in a restricted range of the physical parametric space of the plasma system. It is revealed that the effects of nonthermal and nonextensive parameters, hot to cold electron density ratio, Mach number and magnetic strength considerably modify the basic properties of solitary waves. The present results may be important in space plasmas as well as laboratory plasmas with non-Maxwellian particle population.

Electron-acoustic solitary potential in nonextensive streaming plasma

The linear/nonlinear propagation characteristics of electron-acoustic (EA) solitons are examined in an electron-ion (EI) plasma that contains negative superthermal (dynamical) electrons as well as positively charged ions. By employing the magnetic hydrodynamic (MHD) equations and with the aid of the reductive perturbation technique, a Korteweg-de-Vries (KdV) equation is deduced. The latter admits soliton solution suffering from the superthermal electrons and the streaming flow. The utility of the modified double Laplace decomposition method (MDLDM) leads to approximate wave solutions associated with higher-order perturbation. By imposing finite perturbation on the stationary solution, and with the aid of MDLDM, we have deduced series solution for the electron-acoustic excitations. The latter admits instability and subsequent deformation of the wave profile and can’t be noticed in the KdV theory. Numerical analysis reveals that thermal correction due to superthermal electrons reduces the dimensionless phase speed (bar{U}_{ph}) for EA wave. Moreover, a random motion spread out the dynamical electron fluid and therefore, gives rise to bar{U}_{ph}. A degree enhancement in temperature of superthermal (dynamical) electrons tappers of (increase) the wave steeping and the wave dispersion, enhancing (reducing) the pulse amplitude and the spatial extension of the EA solitons. Interestingly, the approximate wave solution suffers oscillation that grows in time. Our results are important for understanding the coherent EA excitation, associated with the streaming effect of electrons in the EI plasma being relevant to the earth’s magnetosphere, the ionosphere, the laboratory facilities, etc.

Analysis of Two-Fluid Plasma in the Electromagnetic Hydrodynamics Approximation and Discontinuous Structures in Their Solutions

Various versions of equations of two-fluid plasma called the equations of electromagnetic hydrodynamics are considered. They are a generalization of the classical magnetic hydrodynamics equations obtained by adding dispersive terms. Application of finite-difference methods for solving these equations is analyzed. The Riemann problem is solved numerically, and various types of discontinuity structures that expand with time are considered. These are fast and slow magnetosonic structures and Alfven structures. In the case of moderate amplitudes, fast and slow magnetosonic structures are typical in the theory of nondissipative structures. It is found that due to the disappearance of dispersion, short waves can overturn under certain initial conditions, which requires the analysis of discontinuous solutions or the inclusion of additional dissipative or dispersive terms into the equations. In the case when the fluid dynamic viscosity is added, a shock-type structure is discovered. The evolutionary nature of the discontinuity and the conditions on the discontinuity are studied.

On a modification of the Rusanov solver for the equations of special relativistic magnetic hydrodynamics

We describe the implementation of a modification of the Rusanov solver for the equations of special relativistic magnetic hydrodynamics. The particularity of the relativistic hydrodynamics equations, including a magnetic field, is a natural constraint on the admissible wave propagation velocity, which allows us to construct a fairly simple modification of the Rusanov solver using the maximum wave propagation velocity equal to the speed of light.

On a Modification of the Rusanov Solver for the Equations of Special Relativistic Magnetic Hydrodynamics

On a Modification of the Rusanov Solver for the Equations of Special Relativistic Magnetic Hydrodynamics

Physical Processes in the Lower Chromosphere of the Sun

A specificity of the boundary between the photosphere and the chromosphere of the Sun consists in considerable acceleration of plasma spicules above this boundary and appearance of wave motions (predominantly Alfven waves). Analytical solution to the equations of magnetic hydrodynamics is constructed in the vicinity of this boundary from the chromosphere side. Expressions describing the growth of velocity of plasma spicules and the Alfven waves in this vicinity are derived. It is demonstrated that the vertical motion of the Alfven waves and spicules is accompanied by horizontal motion of plasma in the presence of a vertical component of magnetic induction. The obtained expressions characterize the diversity of directions and complexity of motions in the lower chromosphere. They demonstrate that the horizontal component of magnetic induction considerably increases near the lower boundary of the chromosphere. The vertical velocity of spicules among which fast spicules II are dominant and the velocity of the Alfven waves substantially increase in a very thin layer of the lower chromosphere, by a factor of 1.8 and 1.5, respectively. Estimates of the horizontal motion of spicules and the vertical component of magnetic induction are obtained.

Stochastic Magnetic Hydrodynamic Hierarchy in a Strong External Magnetic Field

Based on a stochastic microscopic collisional model of the motion of charged particles in a strong external magnetic field, a hierarchy of equations of magnetic hydrodynamics is constructed. The transition to increasingly rough approximations occurs in accordance with a decrease in the dimensioning parameter, similar to the Knudsen number in gas dynamics. The result is stochastic and nonrandom macroscopic equations that differ from the magnetic analog of the Navier–Stokes system of equations, as well as from the systems of magnetic quasi-hydrodynamics. The main feature of this derivation is more accurate velocity averaging due to the analytical solution of stochastic differential equations with respect to the Wiener measure, in the form of which the intermediate meso model is presented in the phase space. This approach differs significantly from the traditional one, which uses not the random process itself but its distribution function. Emphasis is placed on the clarity of assumptions when moving from one level of detail to another, and not on numerical experiments that contain additional approximation errors.

Boundary Control Problems for the Stationary Magnetic Hydrodynamic Equations in the Domain with Non-Ideal Boundary

Boundary control problems for a stationary model of magnetohydrodynamics of a viscous incompressible fluid are considered in a domain with non-ideal boundary. The role of the control is played by a tangential component of a magnetic field specified on an entire boundary. In the case when the control function is square integrable, the solvability of the control problem is proved. Under assumption that control possesses a higher smoothness described by the space Hs(Γ)3 for any s > 0, an optimality system is derived. Based on the analysis of an optimality system, the local stability estimates of optimal solutions are established with respect to small perturbations of a cost functional.

Structure of nonlinear whistlers moving through plasma at an angle to the magnetic field

The paper presents solutions of two-fluid magnetic hydrodynamics equations describing small-scale fast magnetosonic stable waves — nonlinear whist-lers moving in a cold magnetized plasma at an angle α to the external magnetic field. At the fixed angle α, the Alfvén Mach number of the whistlers has a narrow range of allowed values. It has been found that when passing from extremely small Mach numbers to ex-tremely large ones, amplitudes and spatial structure of wave velocity components and whistler magnetic field change significantly. The range of angles of the motion direction of whistlers with respect to direction of the the external magnetic field vector is determined. Within this range, the obtained approximate analytical and numerical solutions are in satisfactory agreement.

Структура нелинейных вистлеров, движущихся в плазме под углом к магнитному полю

The paper presents solutions of two-fluid magnetic hydrodynamics equations describing small-scale fast magnetosonic stable waves — nonlinear whist-lers moving in a cold magnetized plasma at an angle α to the external magnetic field. At the fixed angle α, the Alfvén Mach number of the whistlers has a narrow range of allowed values. It has been found that when passing from extremely small Mach numbers to ex-tremely large ones, amplitudes and spatial structure of wave velocity components and whistler magnetic field change significantly. The range of angles of the motion direction of whistlers with respect to direction of the the external magnetic field vector is determined. Within this range, the obtained approximate analytical and numerical solutions are in satisfactory agreement.

Localized asymptotic solutions of the linearized system of magnetic hydrodynamics

We describe the asymptotic solutions of the Cauchy problem for the linearized system of equations of magnetic hydrodynamics with initial conditions localized near one point. It is shown that the structure of such solutions depends on whether the external magnetic field vanishes or not at this point. We discuss whether it is possible for the asymptotic solution to increase with time.

Problems on control for a steady-state magnetic-hydrodynamic model of a viscous heat-conducting fluid under mixed boundary conditions

The control problems for steady-state equations of magnetic hydrodynamics (MHD) for a viscous heat-conducting fluid considered under mixed boundary conditions for the magnetic field and temperature are investigated. Their solvability is proved, the optimality systems describing the necessary conditions of an extremum are derived, and the theorems of local uniqueness and stability of the optimum solutions for explicit quality functionals are formulated.

Spontaneous Rotation in the Exact Solution of Magnetic Hydrodynamic Equations for Flow between Two Stationary Impermeable Disks

Spontaneous Rotation in the Exact Solution of Magnetic Hydrodynamic Equations for Flow between Two Stationary Impermeable Disks

Numerical simulation of hydrodynamic flows in the jet electric

On the basis of concepts from magnetic hydrodynamics the mathematical model of hydrodynamic flows in the stream of electric arc plasma, obtained between the rod electrode and the target located perpendicular to the flat conductive, was developed. The same phenomenon occurs in the welding arc, arc plasma and other injection sources of charged particles. The model is based on the equations of magnetic hydrodynamics with special boundary conditions. The obtained system of equations was solved by the numerical method of finite elements with an automatic selection of the time step. Calculations were carried out with regard to the normal plasma inleakage on the solid conducting surface and the surface with the orifice. It was found that the solid surface facilitates three swirling zones. Interaction of these zones leads to the formation of two stable swirling zones, one of which is located at a distance of two radii from the axis and midway between the electrodes, another is located in the immediate vicinity of the continuous electrode. In this zone plasma backflow scattering fine particles is created. Swirling zones are not formed by using the plane electrode with an orifice. Thus, the fine particles can pass through it and consolidate.

Application of Analytical Model of the Electric Potential Distribution for Calculation of Charged Particle Dynamics in a Near-Wall Layer and Sputtering of the Plasma Facing Surfaces

Simple analytical formulas are derived for calculation of the electric field potential distribution in the magnetic pre-layer and the Debye layer near the plasma facing surfaces. It is shown that the calculated potential profiles are in good agreement with the dependences of the potential distribution on the magnetic field inclination obtained by solving the magnetic hydrodynamic (MHD) equations and modeling using the PIC code SPICE2. Dependences of the angular distribution of ions incident on the surface of plasma facing elements on the magnetic field inclination are obtained. Results of calculations demonstrate that the surface areas, on which the magnetic field is incident at sliding angles, are critical from the viewpoint of the increase of sputtering.

Influence of Thermal Parameters on the Structure and Intensity of Vortex Motion of a Melt in DC Arc Furnaces

The vortex motion of a metal melt in a dc bottom-electrode arc furnace has been modeled numerically with allowance for different thermal parameters. The motion of the melt was described by magnetic hydrodynamic equations for a nonisothermal liquid. An algorithm of solution of the problem within the framework of standard packages of applied programs has been developed. It has been shown that a determining contribution to the vortex motion of the metal melt is made by the electromagnetic Lorentz force. It has been established that a significant influence is exerted by convective flows resulting from the nonuniform distribution of heat that is released in the electric-arc region. A nonuniform distribution of the Joule heat from the electric current flowing over the melt exerts a minor influence on the structure and intensity of the vortex melt motion.

Numerical modeling of electro-vortical flows in a confined volume

The electro-vortical flow of a metal melt in a confined volume has been investigated for various electromagnetic and hydrodynamic regimes. The melt motion was described by magnetic hydrodynamic equations for an incompressible isothermal fluid. A strategy for solving the problem with the use of standard packages of applied programs has been developed. The laminar flow of the melt at small Reynolds numbers, for which there is a known solution, as well as the turbulent flow, for which there are experimental data, have been calculated. In calculating the turbulent flow, various turbulence models were used. It has been shown that the k–e and k–ω models describe satisfactorily the electro-vortical flow of the melt and agree well with experimental data. The good agreement between the results of calculations made with the use of various methods and packages of applied programs, as well as between theoretical and experimental data, point to the reliability of the developed methods and approaches and the validity of the obtained results.