Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

End effects in magnetohydrodynamic channels at finite magnetic reynolds numbers

The effects of the magnetic Reynolds number have been examined via the distribution of the magnetic fields induced by the motion of a medium in a rectangular channel with conducting walls in the presence of an inhomogeneous magnetic field; the effects of wall conductivity and geometry of the external field are also examined as regards the distribution of the induced currents, the Joule loss, and the electric and magnetic fields over the cross section. The problem has previously been considered for a channel with insulating walls [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 19–27, May–June, 1971.We are indebted to A. B. Vatazhin for his interest.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

Linear approximation for the flow of a conducting gas with a high magnetic reynolds number through a channel

In this paper, we discuss the flow of a nonviscous and non-heat-conducting gas through a channel of variable cross section under the influence of a transverse magnetic field. For high magnetic Reynolds numbers, the flow is shown to consist of a core and current layers at the electrodes and at the fixed channel walls. The distributions of currents and other parameters in the core and in the current layers are found analytically, in a linear approximation. The Joule dissipation in the current layers may be more intense than that in the core. The longitudinal currents and Joule dissipation increase with increasing Hall parameter in the electrode layers. Zhigulev [1] has shown that magnetic boundary layers may form in the flow of a conducting gas when there is a high magnetic Reynolds number (R_m«1). He illustrated this situation by the shielding of a plasma flow from the magnetic fields produced near a plate which is electrically isolated from the plasma and through which a current is flowing. In an incompressible fluid, the layer thickness is proportional to R_m ^?1/2. Morozov and Shubin [2] have offered a linear-approximation treatment of the structure of the electromagnetic near-electrode layers which arise during the flow of a nonviscous plasma with a high R_m and a small “exchange” parameter ξ≈H/R_m, for flow transverse to a magnetic field and near a corrugated wall. They pointed out the possible formation of “dissipationless” near-electrode layers with thicknesses on the order of the Debye or electron Larmor radii, and a “dissipative” layer whose thickness increases along the length of the electrodes and is proportional to (R_mc_B ²/c_T ²)^?1/2, where c_B and c_T are the magnetic and thermal sound velocities. Morozov and Shubin studied the properties of dissipationless and dissipative electromagnetic layers at segmented accelerator electrodes through which a current is passing, for an arbitrary “exchange” parameter, in [2] and [3], respectively. The exchange parameter ξ was found in [4]. Such layers should also exist at solid electrodes and at the nonconducting walls of an accelerator channel. Study of the two-dimensional flow in a channel is significantly simplified when such layers are present.     

Stationary supersonic conducting gas flow in a channel with nonconducting walls under weak magnetohydrodynamic interaction

A solution of the problem of flow in a channel with nonconducting walls for a small magnetohydrodynamic interaction parameter N is obtained by numerical methods. In the 0–10 range of variation of the Hall and magnetic Reynolds number parameters the distributions of the electrical parameters and the average (over the cross section) and local gasdynamic flow parameters are computed for two different geometries of the applied magnetic field. It is shown that an increase in the Hall and magnetic Reynolds number parameters is accompanied by a diminution in the Joule dissipation and the perturbation of the average (over the cross section) gasdynamic flow characteristics. It is disclosed that the distribution of the gasdynamic parameters over the channel cross section is extremely nonmonotonic in the end current zones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 20–29, July–August, 1970.In conclusion, the author is grateful to A. B. Vatazhin for useful comments and constant attention to the research and to I. U. Tolmach for valuable comments.     

Combined influence of wall conductance,pressure work,viscous dissipation and joule heating on MHD channel flow heat transfer

To investigate the combined influence of viscous dissipation, pressure work, Joule heating, arbitrary voltage ratio, unequal wall conductances and wall heat fluxes on the fully developed laminar MHD channel flow heat transfer, the exact solution of the energy equations for fluid and channel walls are derived assuming the Hartmann velocity profile. It is concluded that there can be a substantial difference, depending upon Hartmann number, electric field and Brinkman number, between the Nusselt number considering the wall conductance and that neglecting it. Representative results are presented in diagrams.     

Wall functions for numerical modeling of laminar MHD flows

A general wall function treatment is presented for the numerical modeling of laminar magnetohydrodynamic (MHD) flows. The wall function expressions are derived analytically from the steady-state momentum and electric potential equations, making use only of local variables of the numerical solution. No assumptions are made regarding the orientation of the magnetic field relative to the wall, nor of the magnitude of the Hartmann number, or the wall conductivity. The wall functions are used for defining implicit boundary conditions for velocity and electric potential, and for computing mass flow and electrical currents in near wall-cells. The wall function treatment was validated in a finite volume formulation, and compared with an analytic solution for a fully developed channel flow in a transverse magnetic field. For the case with insulating walls, a uniform 20×20 grid, and Hartmann numbers Ha={10,30,100}, the accuracy of pressure drop and wall shear stress predictions was {1.1%,1.6%,0.5%}, respectively. Comparable results were obtained also with conducting Hartmann walls. The accuracy of predicted pressure drop and wall shear stress was essentially independent of the resolution of the Hartmann layers. When applied also to the parallel walls, the wall functions reduced the errors by a factor two to three. The wall functions can be implemented in any general flow solver, to allow accurate predictions at reasonable cost even for complex geometries and nonuniform magnetic fields.     

Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

Nonstationary flow of a conducting gas in crossed electric and magnetic fields

The plasma is inviscid, cool, and not thermally conducting; it flows in a channel of constant cross section. The solution is derived by the small parameter method, for which purpose the magnetic interaction N is used. There have been previous studies of the transient-state flow of an inviscid and thermally nonconducting plasma in crossed electric and magnetic fields [1–3]. A plasma of infinite conductivity has been considered [1], as well as flow involving entropy change in an MHD system with strong electromagnetic fields [2, 3].     

Combined influence of Hall effect,ion slip,viscous dissipation and Joule heating on MHD heat transfer in a channel

To investigate the combined influence of Hall effect, ion slip, viscous dissipation and Joule heating on the fully developed laminar MHD channel heat transfer, the exact solution of the energy equation is derived assuming a constant wall heat flux, finely segmented electrodes and a small magnetic Reynolds number. It is concluded that there can be a substantial difference, depending upon Hartmann number, electric field intensity and Brinkman number, between the Nusselt number considering the Hall effect and that neglecting it. Representative results are presented in diagrams and in tables.     

Integral characteristics of MHD channel with nonconducting baffles

The electromagnetic characteristics of an MHD channel with insulated walls, two quite long electrodes, and nonconducting baffles are obtained for arbitrary magnetic field distribution law along the channel. Some specific cases of magnetic field and baffle location specification are examined in detail.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 30–39, November–December, 1969.     

Steady MHD Couette flow with temperature-dependent physical properties

The influence of variation in physical variables on the steady magnetohydrodynamic (MHD) Couette flow with heat transfer is studied. An external uniform magnetic field is applied perpendicular to the parallel plates and the fluid is acted upon by a constant pressure gradient. The viscosity and the thermal as well as electric conductivities are assumed to be temperature dependent. The two plates are kept at two constant but different temperatures, and the viscous and Joule dissipations are considered in the energy equation. A numerical solution for the governing nonlinear coupled equations of motion and the energy equation is obtained. The effect of the temperature-dependent viscosity, thermal conductivity, and electrical conductivity on both the velocity and temperature distributions is examined. H.A. Attia - On leave from: Dept. of Eng. Mathematics and physics, El-Fayoum University, El-Fayoum, Egypt     

MHD stagnation point flow towards heated shrinking surface subjected to heat generation/absorption

The magnetohydrodynamic(MHD) stagnation point flow of micropolar fluids towards a heated shrinking surface is analyzed.The effects of viscous dissipation and internal heat generation/absorption are taken into account.Two explicit cases,i.e.,the prescribed surface temperature(PST) and the prescribed heat flux(PHF),are discussed.The boundary layer flow and energy equations are solved by employing the homotopy analysis method.The quantities of physical interest are examined through the presentation of plots/tabulated values.It is noticed that the existence of the solutions for high shrinking parameters is associated closely with the applied magnetic field.     

Buoyant MHD flows in a vertical channel: the levitation regime

Buoyant magnetohydrodynamic (MHD) flows with Joulean and viscous heating effects are considered in a vertical parallel plate channel. The applied magnetic field is uniform and perpendicular to the plates which are subject to adiabatic and isothermal boundary conditions, respectively. The main issue of the paper is the levitation regime, i.e., the fully developed flow regime for large values of the Hartmann number M, when the hydrodynamic pressure gradient evaluated at the temperature of the adiabatic wall is vanishing. The problem is solved analytically by Taylor series method and the solution is validated numerically. It is found that the fluid velocity points everywhere and for all values of M downward. For small M’s, the velocity field extends nearly symmetrically (with respect to the mid-plane) over the whole section of the channel between the adiabatic and the isothermal walls. For large values of M, by contrast, the fluid levitates over a broad transversal range of the channel, while the motion becomes concentrated in a narrow boundary layer in the neighborhood of the isothermal wall. Accordingly, the fluid temperature is nearly uniform in the levitation range and decreases rapidly within the boundary layer in front of the isothermal wall. It also turns out that not only the volumetric heat generation by the Joule effect, but also that by viscous friction increases rapidly with increasing values of M, the latter effect being even larger than the former one for all M.     

Numerical simulation of magnetohydrodynamic flow in a toroidal duct of square cross-section

We present numerical simulation results of the quasi-static magnetohydrodynamic (MHD) flow in a toroidal duct of square cross-section with insulating Hartmann walls and conducting side walls. Both laminar and turbulent flows are considered. In the case of steady flows, we present a comprehensive analysis of the secondary flow. It consists of two counter-rotating vortex cells, with additional side wall vortices emerging at sufficiently high Hartmann number. Our results agree well with existing asymptotic analysis. In the turbulent regime, we make a comparison between hydrodynamic and MHD flows. We find that the curvature induces an asymmetry between the inner and outer side of the duct, with higher turbulence intensities occurring at the outer side wall. The magnetic field is seen to stabilize the flow so that only the outer side layer remains unstable. These features are illustrated both by a study of statistically averaged quantities and by a visualization of (instantaneous) coherent vortices.     

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.     

Joule loss due to compressibility of gas in a channel of variable section

It is known that closed electric currents arise in a conducting medium moving in a non-uniform magnetic field. These currents lead to additional energy loss and adversely affect the characteristics of magnetohydrodynamic channels. (The numerous investigations of these effects are dealt with in the review [2, 3].) Eddy electric currents are also formed, however, when a medium flows in a uniform magnetic field perpendicular to the to the plane of motion if the channel has a variable cross section and the medium is compressible [1], This paper is devoted to an investigation of some features of these flows. It is assumed in the analysis that the gas flows in channels whose geometry varies slightly.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 3–9, July–August, 1968.     

Low speed MHD Couette flow of a slightly rarefied and electrically conducting gas

The effects of an applied magnetic field on the steady, laminar, low speed plane Couette flow of a slightly rarefied and electrically conducting gas are studied. Consideration is given to the slip-flow regime, wherein the gas rarefaction begins to play its important role. The generally accepted method of analysis for slip flows is utilized, i.e. the continuum magnetohydrodynamic equations of motion are used throughout the gas, together with the first and the second order slip velocity and temperature jump boundary conditions. Considerations are further given to (1) the case of zero electric field and (2) the case of a nonconducting channel in which the net current across the channel is zero.     

Development of magnetohydrodynamic boundary layers

Many authors have studied the problem of the development of a hydrodynamic boundary layer when a body is suddenly set in motion. The results obtained are presented most fully in the monographs of H. Schlichting [1] and L. G. Loitsyanskii [2]. In magnetohydrodynamics the development of the boundary layer over the surface of an infinite flat plate for uniform oncoming flow has been closely studied (for example [3, 4]). Below, the problem of the development of a plane magnetohydrodynamic boundary layer is considered in a different formulation. We shall suppose that the distributions of velocity U(x) and enthalpy h(x) are given along the body contour for t=0. At that moment the viscosity and thermal conductivity mechanisms are instantaneously switched on. Viscous and thermal boundary layers begin to grow in a direction normal to the body. The medium in the boundary layer interacts with the magnetic field. This formulation corresponds to the development of a magnetohydrodynamic boundary layer on a body which is set in motion with a jerk, in the case where the rate of establishment of magnetohydrodynamic flow of the inviscid, thermally nonconducting fluid around the body is much less than the rate of development of the boundary layer. Then U(x) and h(x) are found by solving the problem of stationary magnetohydrodynamic flow of an inviscid thermally nonconducting fluid around a body, or simply the hydrodynamic flow if the medium interacts with the field only in the boundary layer.     

MHD equations for paramagnetic media

The conservation laws are used to obtain phenomenologically the complete system of equations of motion of a conductive paramagnetic fluid in a magnetic field. In addition to the usual MHD equations (with additional terms accounting for the magnetization of the medium), this system includes the equation for the rate of change of the magnetic moment.The hydrodynamic equations for a fluid with internal rotation have been obtained in [1] and extended in [2] to the case of the paramagnetic properties resulting from this rotation: here the fluid was considered nonconducting. The analysis of [2] is extended to the case of a fluid with nonzero electrical conductivity. This will be the same extension of MHD as the theory of [1, 2] is for conventional hydrodynamics.