Magnetohydrodynamics is a branch of fluid dynamics which studies the movement of an electrically-conducting fluid in a magnetic field.
Faraday first pointed out an interaction of sea flows with the Earth is magnetic field (1832). In the beginning of the 20th century the first proposals for applying electromagnetic induction phenomenon in technical devices with electrically-conducting liquids and gases appeared. Systematic studies of magnetohydrodynamic (MHD) flows began in the 30s when the first exact solutions of MHD equations were obtained and experiments on liquid metal flows in MHD channels were performed [J. Hartmann and F. Lazarus (1937)]. The discovery of Alfven waves finalized the establishment of magnetohydrodynamics as an individual science [H. Alfven (1942)].
Currently magnetohydrodynamics is applied in astrophysics and geophysics, fission and fusion, metallurgy and direct energy conversion, etc.
MHD applications fall within the conditions of the MHD approximation, according to which nonrelativistic, low frequency movement of an electrically-conducting fluid is considered when displacement and convection currents, electric body force and energy density of electric field can be neglected.
The MHD equations consist of the mass, momentum and energy conservation laws:
Maxwell's equations and Ohm's law:
and equations of fluid properties:
For many working fluids permittivity and permeability are equal to those of a vacuum (ε_{0} = 8.854·10^{−12} Farad/m, μ_{0} = 4π·10^{−7} Henry/m). The following quantities enter into MHD equations: ρ, p and u are fluid density, pressure and internal energy, respectively, is fluid velocity, is the viscous stress tensor, is the heat flux vector, is the electric field, is magnetic induction, is current density, ρ_{e} is electric charge density, σ is fluid electrical conductivity, β is the Hall coefficient.
The conservation equations are related to Maxwell's equations and Ohm's law through electromagnetic body force ( × ) and power density of work done on the fluid by the electromagnetic field. is related with power density of electromagnetic body force ( × ) by the relationship
where j^{2}/σ is the rate of Joulean dissipation.
Magnetohydrodynamics is characterized by dimensional parameters which include, in addition to the conventional hydrodynamic parameters (Re, Pr, etc.), new ones containing electromagnetic variables: A = υ/υ_{a} is the Alfven number where υ_{a} = B/ is the Alfven velocity; Re_{m} = μσυL is the magnetic Reynolds number; K = E/υB is the parameter of electric field (or load parameter); β = ωτ is the Hall coefficient being the ratio of electron cyclotron frequency and mean frequency of electron collisions with neutrals; S = σB^{2}L/ρυ is the parameter of MHD interaction; Ha = BLρ is the Hartmann number and some other parameters.
Boundary conditions used in MHD equations are formulated by traditional hydrodynamics and electrodynamics methods. External conditions represent the paths of electrical current and magnetic field lines outside the flow volume, and in particular, the configuration of external electrical circuit and the type of magnet system.
Exclusion of variables and from electrodynamic equations leads to an equation containing only one variable , called the equation of induction. In the case of σ = const and β = 0 the equation of induction can be written as:
where ν_{m} = 1/μσ is the magnetic viscosity.
The first term in the right hand side of the equation determines the convective transport of magnetic field by fluid particles, the second term describes the diffusion of the magnetic field in the fluid. The relative role of convection and diffusion is determined by magnetic Reynolds number Re_{m}. At Re_{m} = 0 diffusion of magnetic field only takes place at finite velocity. In this case, magnetic viscosity ν_{m} determines either the characteristic time of magnetic field variation t ~ l^{2}/υ_{m} at distance l or the characteristic depth of magnetic field penetration l ~ during time t, defined in electrical engineering as the skin layer.
At Re_{m} = 0 fluid movement does not influence the applied magnetic field, which is induced by currents circulating outside flow volume.
At Re_{m} = ∞ the effect of magnetic field freezing is observed when magnetic flux through any closed liquid contour is conserved and all liquid particles—initially at the magnetic field line— continue to be on the line.
These properties of MHD flows are analogous to those of conventional hydrodynamics derived from theorems on vortex and circulation of velocity. In an ideally-conducting fluid (σ = ∞), when there is no dissipation of energy, small perturbations travel as nonattenuating MHD waves. For the plane wave of the type exp i(kx - wt) traveling in a uniform magnetic field, the linearized MHD equations are separated into two independent subsystems of equations which define Alfven and magnetoacoustic waves.
Alfven waves are characterized by transversal oscillations of perturbations of velocity and magnetic field traveling with velocity
In Alfven waves there is no density perturbation, so these waves can propagate both in compressible and incompressible fluids.
Magnetoacoustic waves include perturbations of density ρ' and also variables , and . The solution of the dispersion equation reveals two types of magnetoacoustic waves, denoted as fast waves and a slow wave, respectively, which propagate at corresponding velocities
where a is the sonic speed.
In Figure 1, the phase velocity diagram shows the dependence of the wave velocity value upon the angle θ between the undisturbed magnetic field and the wave vector .
For flows of ideally-conducting fluids, discontinuous solutions are possible. The relationships between flow parameters are obtained from the equations of mass, momentum and energy conservation and also from the boundary conditions for the electromagnetic field. Analysis of these relationships reveals four types of discontinuities.
Contact discontinuities. In a contact discontinuity, the normal component of velocity does not exist; velocity, pressure and magnetic fields are continuous. Density, temperature and also entropy have different values across the interface. Conditions for flow parameters at a contact discontinuity are written in the form:
Here, the brackets denote the difference of values at the interface.
Tangential discontinuities. Conditions for tangential discontinuities are:
The velocity and magnetic field are tangent to the discontinuity surface and have arbitrary jumps in magnitude and direction. Density discontinuity is also arbitrary.
Rotational discontinuities. In rotational discontinuities, fluid thermodynamic properties and the normal component of velocity are continuous while the magnetic field vector rotates around the normal direction, being constant in magnitude:
At the rotational discontinuity the jump of the tangential component of velocity is coupled to the jump of vector by the equation { } = { }/ . The normal component of velocity is equal to the velocity of Alfven wave: υ_{n} = B_{n}/ . The weak rotational discontinuity goes over into Alfven wave.
Shock waves. At the discontinuity surface, hydrodynamic parameters and magnetic field have jumps:
It can be shown that vectors , , and lie in the plane normal to the discontinuity surface. The Hugoniot equation of MHD shock wave has the following form:
The condition that entropy must decrease (S_{2} > S_{1}) determines the possible existence of only compression shocks (ρ_{2} > ρ_{1}). As Bt1 increases the degree of gas compression decreases at a given intensity of shock wave.
Two types of shock wave can occur: fast and slow shocks. In both types, the normal velocities are higher than υ_{+1} and υ_{−1} respectively in front of the Shockwave and less than υ_{+2} and υ_{−2} behind the shock. The tangential component of magnetic field BT rises across a fast shock and drops across a slow shock. Weak MHD shock waves transform into the corresponding magnetoacoustic waves.
The following are possible transitions between the four discussed MHD discontinuities:
Contact discontinuities may transform into tangential discontinuities and vice versa.
Rotational discontinuities and shock waves may transform into tangential discontinuities and vice versa.
Rotational discontinuities may transform into shock waves and vice versa.
The influence of an electromagnetic body force upon stationary MHD flows is distinctly demonstrated by the solution of the Hartmann problem. This problem considers a flow of conducting viscous incompressible fluid between two parallel insulating planes in a uniformly applied magnetic field = . In fully-developed flow the quantities = u ; = B_{x} + + ; = j depend only on coordinate z in the direction normal to the planes placed at z = ±b and pressure p = p(x) + p'(z) depends also on coordinate x in the direction of the velocity vector. The solution of the z-component of the momentum equation has the following form:
and reflects the balance between the z-component of the electromagnetic body force ( × )_{z} and the transverse pressure gradient (pinch effect).
The velocity distribution u(z) is determined by forces of viscous friction and ( × )_{x}
where the Hartmann number Ha = B_{0}b , u is average in z-direction velocity, which is interrelated with longitudinal pressure gradient dp/dx. In Figure 2, the velocity profiles (solid lines) at various values of the Hartmann number Ha are presented. When Ha increases the velocity profile becomes fuller, leading to a rise in shear force. This occurs due to nonuniform distribution of electromagnetic body force ( × )_{x} = σ(E_{0} - uB_{0})B_{0}, where E_{0} is the constant electric field determined by external circuit. It can be seen that the body force accelerates the near wall layers of the fluid relative to the core layers.
The solution of the equation of induction for boundary conditions B_{x} (±b) = 0, corresponding to the open circuit (E_{0} = u B_{0}), yields the distribution of B_{x}(z):
where Re_{m} = μσ b. The induced magnetic field B_{x} is proportional to Re_{m} and produces a bend of magnetic field lines, in the direction of fluid flow (connection effect). The broken curves in Figure 2 present vector lines for the magnetic field for Re_{m} = 5.
The analysis of the combined influence of electromagnetic body force ( × ) and electrical power density upon the flow of compressible fluid is performed by use of the equations of reverse action. These equations are derived from one-dimensional stationary MHD equations when magnetic and electrical field = B and = E are considered to be given quantities and effects of viscosity and thermal conductivity are negligible:
These equations determine the velocity and Mach number variations in subsonic and supersonic flows in the presence of MHD interaction. As in conventional gasdynamics physical action with a given sign exerts opposite influences on subsonic and supersonic MHD flows. The effect of MHD upon the flow velocity is variable in direction and changes sign twice: at u = u_{1}, when the body force and power actions, influencing oppositely the velocity variation, are balanced, and at u = u_{3}, when the current density j = −σB(u − u_{3}) equals zero.
In Figure 3, the family of lines u(Ma) passing through an arbitrary point Ma_{0}, u_{0} at the condition E/B = const is plotted. The arrows on the lines indicate the direction of u and Ma variations downstream. The broken lines separate regions of different variations of u and Ma. The line u = u_{3} divides the plane u — Ma in two parts: for u < u_{3} the flow occurs in acceleration mode with consumption of energy from the external circuit (jB > 0,jE > 0), for u > u_{3} the flow acts as a generator (jB < 0,jE < 0). The u/Ma diagram determines the properties and limit regimes of flow in MHD devices of constant cross section. Thus, in an MHD generator, (u > u_{3}) subsonic flow is accelerated and in the limit mode is choked at the outlet (Ma_{2} = 1), while supersonic flow is decelerated and is choked or asymptotically approaches minimum velocity u_{3} (as x → ∞). These limitations can be excluded by switching on an additional physical action, e.g., a geometric action by using an MHD channel of variable cross section. In MHD channels of constant cross section, it is possible for the flow to pass smoothly through the sonic velocity. Such transition can occur when the sign of action is changed at Ma = 1. In Figure 3, there are two transitions which are possible for flow regimes described by u(Ma) lines passing through the points Ma = 1, u = u_{1} and Ma = 1, u = u_{3}.
The influence of MHD interaction on volume structure of channel flow is determined by nonuniformities of current and potential distributions. These nonuniformities, in turn, are related to the viscous and thermal boundary layers at the walls and also with configuration of electrode and insulator wall elements and the form of the external electrical circuit. At the insulator walls, the Hartmann effect dominates. In compressible fluid at high velocities, the relative acceleration of wall layers causes a nonmonotonic velocity profile with a maximum point inside the boundary layer. At the electrode walls, deceleration of flow is possible in the case when the body force is directed against the flow (e.g. in the generator regime). This effect increases the boundary layer thickness and causes boundary layer separation. The energy input into the boundary layer increases the temperature gradient, which consequentially increases the wall heat flux.
In cases where electromagnetic body force has a transverse component ( × )_{y}, secondary flows can arise. For Re_{m} << 1, secondary flow occurs when the axial current density j_{x} is nonuniform over the channel cross section and the transverse body force is not potential. From the equation for the velocity vortex ω_{x} an estimation for the occurrence of secondary flow is as follows:
where 1 is the perimeter of the vortex cell. In an MHD generator, the condition j_{x} > 0 takes place. In this case, the negative pressure gradient ∂p/∂y in the channel cross section, caused by ( × )_{y}, induces a two-cell vortex structure in which the fluid moves in the boundary layer at the insulator wall in the positive direction of y-axis (toward the anode) and then to the middle of the anode. Fluid particles moving along the walls are gradually cooled and push electric current to the central part of the anode. A positive feedback exists between the anode current concentration, leading to the elevated local Joule heating, and the intensity of vortex flow. At sufficiently large values of the parameters of MHD interaction, an arcing occurs and overheating of the anode central part takes place.
Plasma flows in MHD channels undergo various types of instabilities. An analysis of the linearized system of MHD equations for Re_{m} << 1 shows the following main types of instabilities: acoustic, superheating and vortex MHD instabilities.
The acoustic instability is manifested in an enhancement of sonic waves under influence of current density fluctuations caused by perturbances of thermodynamic parameters and corresponding electrical conductivity. A positive contribution to the growth of short sonic waves is produced by two factors: electromagnetic body forces and Joule heating variations caused by conductivity fluctuations. The destabilizing influence of these mechanisms is largest when the wave travels in the direction of the nondisturbed body force ( × ), for the condition (∂σ/∂T)_{s} > 0 and where the damping action of magnetic friction is overcome.
The superheating instability is characterized by the growth of entropy disturbances caused by the positive feedback between fluctuations of entropy and Joule heating. This type of instability occurs for the condition (∂σ/∂T)_{p} > 0 and has the maximum increment at = 0. The development of finite amplitude disturbances for electrical conductivity growing with temperature may result in moving localized regions with elevated temperature named T-layers.
Vortex instability is manifested by the growth of velocity vortex disturbances induced by the electrical conductivity and electromagnetic body force gradients in nonuniformly conducting fluid. For the condition (∂σ/∂T)_{p} > 0, two mechanisms of the instability initiation are possible: a type of Rayleigh-Taylor instability relating to departures from equilibrium of heavy liquid superimposed over a light liquid and the convective instability of a liquid layer heated on the underside.
In turbulent flows, the magnetic field dampens the vortex vector components normal to the field. The attenuation decrement of the velocity pulsations is estimated as . The decrease in the velocity pulsations causes a decrease in turbulent shear stresses. The available experimental data on structure of turbulent MHD flows in channels show that with an increase of magnetic field, the pulsations of velocity, temperature and electric field decay, averaged velocity profiles are laminarized, the hydraulic resistance decreases (except when Hartmann effect prevails) and the transition from a laminar regime of flow to a turbulent one is made more difficult.
REFERENCES
Kulikovsky, A. G. and Lyubimov, G. A. (1965) Magnitohydrodynamics, Addison-Wesley, Reading.
Landau, L. D. and Lifshitz, E. M. (1960) Electrodynamics of Continuous Media, Addison-Wesley, Reading.
Shercliff, J. A. (1965) A Textbook of Magnetohydrodynamics, Pergamon Press, Oxford.
References
- Kulikovsky, A. G. and Lyubimov, G. A. (1965) Magnitohydrodynamics, Addison-Wesley, Reading.
- Landau, L. D. and Lifshitz, E. M. (1960) Electrodynamics of Continuous Media, Addison-Wesley, Reading.
- Shercliff, J. A. (1965) A Textbook of Magnetohydrodynamics, Pergamon Press, Oxford.