Determination of current flow in a plasma by a simulation method

A method is proposed which, for specific assumptions, allows us to determine the density distribution of a constant current flowing between electrodes in a plasma for plane parallel or radially symmetric electric and magnetic fields, allowing for anisotropic conductivity.Notation e_r, e, e_z unit vectors in a cylindrical coordinate system - E, e_r, e_z electric field strength vector and its components - V electric field potential - H, H_r, H, H_z magnetic field strength and its components - j current density vector - e electron charge - m electron mass - c velocity of light - momentum transfer time - ₀ normal plasma conductivity - _e electron cyclotron frequency - h unit vector in the direction of the magnetic field     

Two-dimensional flow of a perfect conducting gas in crossed electric and magnetic fields

Reference [1, 2] give a solution of the problem of the two-dimen-, sional flow of an inviscid thermally-nonconducting gas with constant conductivity in a channel of constant cross section for particular forms of the given applied magnetic field. The present paper obtains a solution of the problem of the two-dimensional flow of a gas with variable conductivity in crossed electric and arbitrary magnetic fields by means of the small parameter method. The magnetic Reynolds number R_m and the magnetohydrodynamic interaction parameter S are chosen as parameters. The international system of units is employed.Notation V flow velocity - j electric current density - p pressure in the flow - E electric field strength - gas density - electrical conductivity of the gas - T gas temperature - ratio of specific heats at constant pressure and volume - L channel half-height - ] permeability (magnetic) - B magnetic induction vector - B₀ applied magnetic field     

Approximate solution of the boundary problem for the equations of a stationary electrostatic charged-particle beam

The solution is given of the equations of a three-dimensional stationary electrostatic beam of charged particles of like sign filling the region between two nearby curvilinear surfaces. We assume that the flow is nonrotational and nonrelativistic and that the velocity vector is a single-valued function. The solution is constructed in the form of an asymptotic series in powers of the small parameter , which is the ratio of the characteristic transverse (a) and longitudinal (l) dimensions of the problem. The first dimension is taken to be the distance between the electrodes, andl defines the scale at which the geometric and physical parameters (emitter curvature, electric field E on the emitter, and the emission current density J) change noticeably. The emission regimes limited by the space charge (-regime), temperature (T-regime), and the case of nonzero initial velocity (U-regime) are studied. The asymptotic behavior is given by the formulas for the corresponding one-dimensional flow between parallel surface.The solution of the boundary problem for emission in the-regime reduces to determination of the emission current density J for fixed electrode geometry and given accelerating voltage. The corresponding formulas are presented, retaining terms of order ³.Two approximations with respect to are performed for the T- and U-regimes. Here the unknown quantity for given properties of the emitting surface (J) will be the electric field E.The results provided by the constructed expansions are compared with the exact solution for flow from a planar emitter along circular trajectories [1]. As an example we examine the two-dimensional problem of flow between two nearby circular cylindrical electrodes with disruption of the coaxiality.The conventional tensor notations are used.     

Asymptotic Solutions of Problems of One-Dimensional Time-Dependent Combustible Gas Flows in the Presence of Thermal Action

In a development of studies [1, 2], asymptotic solutions of the Navier-Stokes equations are found for one-dimensional combustible gas flows in the presence of various forms of thermal action on a moving surface (x=x _w(t)). In the problems considered, the temperature or the heat flux q _w(t) is specified on the surface or the surface is the interface between a combustible gas and a moving heated piston or another gas (for example, in a shock tube). Use is made of the fact that, as t , in many cases the values of v _w=(dx/dt)_w and q _w are bounded. This leads to a steady-state flow in the flame zone in the coordinate system moving with its front and homogeneous uniform flow ahead of and behind it. Solutions of all these problems are given for the burnt-gas boundary layer region adjacent to the surface. The numerical calculations performed confirm the results obtained. A velocity law leading to time invariability of the flow pattern obtained with allowance for the interaction between the boundary layer and the burnt-gas homogeneous flow is found, including in the problem of the breakdown of an arbitrary discontinuity. The results are generalized to include the case of motion at an angle of incidence with an additional velocity component aligned with the surface.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

A general solution of 3-D quasi-steady-state problem of a moving heat source on a semi-infinite solid

The well-known Jaeger–Rosenthal asymptotic particular solution for the quasi-steady-state problem of moving heat source is proven to be inconsistent with the source constant intensity, especially at dimensionless trailing edge coordinates vx/a < −2. The problem is reduced to an equivalent Poisson’s equation by exponential transformation of moving coordinate scale. Using the method of images, the fundamental solution is found; the temperature rise function exponentially approximates to 0 along negative semi-axis. The temperature field in a semi-infinite solid for the general case of surface power intensity distribution is expressed, using the found Green’s function. The cases of point, line, and circular heat sources are considered. The found fundamental solution and particular solution for moving circular heat source explain the phenomena of martensite transformation in low-carbon steel substrate at relatively low source velocity 1.7 cm/s.     

A rotating field mhd hydrostatic thrust bearing

It is found that the load capacity of a magnetohydrodynamic thrust bearing with a rotating disk can be increased by rotating the axial magnetic field at a suitable speed in a direction opposite to that of the disk rotation. This method of improving the bearing performance is considered to be efficient if the Hartmann number is not too large. Thus for a given load, the size and weight of the magnet to be used in a thrust bearing with rotating field can be reduced considerably.Nomenclature a radius of plenum recess - b outside disk radius - B ₀ magnetic induction of applied axial magnetic field - hE ₀ ^1/2/a ^1/2, nondimensionalized electric field - E ₀ radial electric field at r=a - E _r radial electric field - h half of lubricant film thickness - M (B ₀ ² h ²/)^1/2, Hartmann number - P pressure - P _e pressure at r=b - P ₀ pressure at r=a - Q volume flow rate of lubricant - Q ₀ volume flow rate of a nonrotating bearing in the absence of applied magnetic field - r radial coordinate - u, v fluid velocity components in radial and circumferential directions, respectively - W load capacity of bearing - W ₀ load capacity of a nonrotating bearing in the absence of a magnetic field having a flow rate which the same bearing would have at Hartmann number M - z axial coordinate - azimuthal coordinate - coefficient of viscosity of lubricant - _e magnetic permeability - fluid density - electrical conductivity - angular velocity of rotating disk - _C critical disk velocity at which W=0 - _M angular velocity of axial magnetic field - optimum angular velocity of magnetic field On leave of absence from Department of Aero-Space Engineering, University of Notre Dame, Notre Dame (Ind.), U.S.A.     

Solution of the linearized couette-flow problem in a rarefied gas by the integral-diffusion method

In ordinary diffusion theory the transfer of properties is determined by the local gradients of the corresponding fields. As the mean free path increases, the flux density becomes an integral quantity and is determined by a neighborhood of the point under consideration of the order of a few mean free paths. In a previous article [1], the author proposed a model for a one-dimensional transfer process in linear rarefield-gas problems based on the analogy with radiative transfer. The same approach, though without directional averaging, is used in the present paper to analyze the linearized Couette flow problem. The solution obtained here has the properties of the solution obtained by more exact methods based on the solution of the Boltzmann equation [3-4].Nomenclature p_xy shear stress - c mean thermal velocity of molecules - 2/3 A mean free path - d half-width of channel - ±w₀ plate velocity - c nonequilibriumvalue of momentum flux density - y transverse coordinate - ratio of specific heats - W dimensionless velocity - P_xy shear stress scaled with respect to the shear stress in free-molecule flow - Y dimensionless coordinate - W₁(y) velocity distribution according to Millikan's solution - coefficient of viscosity - R Reynolds number - K Knudsen number     

Strong Solutions to the Navier-Stokes Equations Around a Rotating Obstacle

We study the existence of strong solutions to the three-dimensional Navier-Stokes initial-boundary value problem in the domain, , exterior to a rigid body that rotates with constant angular velocity, . We show that when the initial data, u₀, are prescribed in an appropriate functional class, a strong solution exists at least in some finite time interval. Moreover, the solution exists for all times, provided u₀, in suitable norm, and the magnitude of do not exceed a certain constant depending only on the kinematic viscosity and on the regularity of . In this latter case, we also show that the velocity field converges to the velocity field of the corresponding steady-state solution.     

Flow of a multicomponent gas between parallel permeable surfaces

The present paper gives an exact solution of the equations describing the flow of a multicomponent gas between two parallel permeable planes, one of which moves relative to the other with constant velocity (i. e., we study a flow of the Couette type).Notation y coordinate - u, v velocity components - density - c_i mass concentration of i-th component - I_i diffusional flux of i-th component - H enthalpy - T temperature - m molecular weight - viscosity coefficient - heat conduction coefficient - c_p mixture specific heat - D_ij the binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - N total number of components - n number of components in injected gas - l distance between planes Indices i, j component numbers - w applies to quantities for y=0 - ^* applies to quantities for y=l     

Couette flow of a multicomponent ionized gas

We consider equilibrium flow of a multicomponent ionized gas between two catalytic plates of infinite length, one of which moves parallel to the other with constant velocity. The results of [1] are generalized for ionized gaseous mixtures which are in local thermodynamic equilibrium. Formulas are presented for calculating the thermal flux and the effective thermal conductivity for ambipolar diffusion.Then a special ionization case is discussed.Notation A_i chemical symbol of the i-th component - W_i projection of the molar diffusive flux vector of the i-th component on the y-axis - x_i molar concentration - H_i enthalpy - m_i molecular weight - Q_s heat of the s-th reaction - K_ps(T) equilibrium constant of the s-th reaction - W_i mass formation rate of the i-th component per unit volume - Z_i charge number - e unit charge (electron charge) - E electric field intensity - distance between the plates - N number of components - v _sl stoichiometric coefficients - density - T temperature - p pressure - u projection of average velocity on y-axis - viscosity - thermal conductivity - D_ij binary diffusion coefficient - R universal gas constant - k Boltzmann constant In conclusion, the author wishes to thank G. A. Tirskii for proposing the study and for suggestions made in the course of the investigation.     

Magneto-hydrodynamically lubricated externally pressurised bearing with variable film thickness

Summary This paper studies theoretically the use of a conducting lubricant in an externally pressurised bearing with variable film-thickness in the presence of an axial magnetic field. The flow and other characteristics are determined and it is shown that the pressure and load capacity can be increased by increasing the strength of the applied magnetic field at a given flow rate. But at a given feeding pressure the load capacity and pressure do not depend upon the magnetic field. The load capacity of this bearing is greater than that of a bearing having a constant film-thickness. It is also pointed out that the frictional drag on the rotor can be minimised by supplying electrical energy to the system.Nomenclature angle which the rotor surface makes with the stator (see fig. 1) - angular velocity of the rotor - _t terminal voltage between the electrodes - _t.o.c. open circuit voltage - viscosity of the lubricant - conductivity of the lubricant - B ₀ strength of the applied magnetic field - E _r radial component of the electric field - h variable film-thickness - h ₀ minimum film-thickness - I total current - I _s.c. short circuit current - L depth of the recess - M Hartmann number - p pressure - p _e exit pressure - p _i inlet pressure - Q rate of volume flow - r radial coordinate - R ₀ radius of the recess - R outer radius of the stator - R _i internal resistance - T frictional drag - u radial velocity - v tangential velocity - W load capacity - normalised load capacity - z axial coordinate     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Effect of angular inertia of lubricant in MHD hydrostatic thrust bearings

Summary Circumferential motion of a conducting lubricant in a hydrostatic thrust bearing is caused either by the angular motion of a rotating disk or by the interaction of a radial electric field and an axial magnetic field. Under the assumption that the fluid inertia due to radial motion is negligibly small in comparison with that due to angular motion, it is found analytically that the rotor causes an increase in flow rate and a decrease in load capacity, while both are increased by the application of an electric field in the presence of an axial magnetic field. The critical angular speed of the rotor at which the bearing can no longer support any load is obtained, and the possibility of flow separation in the lubricant is discussed.Nomenclature a recess radius - b outside disk radius - B ₀ magnetic induction of uniform axial magnetic field - E ₀ radial electric field at r=a - E _r radial electric field - h half of lubricant film thickness - M Hartmann number = (B ₀ ² h ²/)^1/2 - P pressure - P ₀ pressure at r=a - P _e pressure at r=b - Q volume flow rate of lubricant - Q ₀ flow rate of a nonrotating bearing without magnetic field - r radial coordinate - r _s position of flow separation on stationary disk - u, v fluid velocity components in radial and circumferential directions, respectively - W load carrying capacity of bearing - W ₀ load capacity of a nonrotating bearing without magnetic field - z axial coordinate - coefficient of viscosity - _e magnetic permeability - fluid density - electrical conductivity - electric potential - angular speed of rotating disk - _c critical rotor speed at which W=0     

Method of calculating supersonic three-dimensional flow past smooth bodies

A method is suggested in [1] for calculating supersonic flow past smooth bodies that uses an analytic approximation of the gasdynamic functions on layers and the method of characteristics for calculating the flow parameters at the nodes of a fixed grid. In the present paper this method is discussed for three-dimensional flows of a perfect gas in general form for cylindrical and spherical coordinate systems; relations are presented for calculating the flow parameters at the layer nodes, results are given for the calculation of the flow for specific bodies, and results are shown for a numerical analysis of the suggested method. Three-dimensional steady flows with plane symmetry are considered. In the relations presented in the article all geometric quantities are referred to the characteristic dimension L, the velocity components u, v, w and the sonic velocitya are referred to the characteristic velocity W, the density is referred to the density of the free stream, and the pressure p is referred to w².     

Leakage effects in laminar duct flow: a numerical study

The influence of transverse leakage into a pressure-driven laminar flow in an infinitely long square duct is investigated. By a simple decomposition of the resulting three-dimensional pressure field, the leakage-induced secondary flow problem decouples from the primary flow problem. The numerical study reveals that two qualitatively different secondary flow patterns may occur, depending on the leakage flow rate. For a given streamwise pressure gradient it is observed that the axial mass flow rate may reduce by about 30 percent under certain leakage conditions, accompanied by a corresponding 50 percent increase in the Darcy-Weisbach friction factor.Nomenclature D duct height and width - D _h hydraulic diameter - D _i,j cell divergence - F dimensionless pressure force - h dimensionless slit height,H/D - H slit height - i,j indices - K streamwise kinematic pressure gradient - n summation index, time level - p dimensionless pressure - p pressure increment - P pressure - cross-sectional pressure variation - q dimensionless volumetric axial flow rate - Q leakage flow rate in m²/s - Re leakage Reynolds number,U ₀ D/ - Re _q primary flow Reynolds number - t time increment - x, y dimensionless cross-sectional coordinates - x, y cell widths - X, Y cross-sectional coordinates - z dimensionless axial coordinate - Z axial coordinate - u, v, w dimensionless velocity components - U, V, W velocity components - U ₀ leakage velocity,Q/H - V cross-sectional average velocity - W ₀ dummy velocity scale Greek letters density - kinematic viscosity - overrelaxation factor     

An experimental study of the behavior of transient isolated convection in a rigidly rotating fluid

The temperature field of starting thermal plumes were measured in a rotating annulus with various rotation rates and buoyancies. The experiments revealed many details of the internal structure of these convective phenomena and also significant horizontal displacements from their source. Measurements show an increase in the maximum temperature observed in the thermal caps with increasing rotation and a more rapid cooling of the buoyancy source.List of symbols D angle relating inward centripetal acceleration to buoyant acceleration, defined by tan D = R/g - g gravitational acceleration - P total pressure of ambient fluid - R radial coordinate measured from rotation axis - R ₀ distance from rotation axis to buoyancy source - u velocity of fluid parcel along the radial direction - velocity of fluid parcel along the azimuthal direction - w velocity of fluid parcel along the axial direction - z axial coordinate, measured upward from the plane containing the buoyancy source - density of a buoyant parcel of fluid - ₀ density of the ambient fluid - azimuthal angle measured from the radial line passing through the buoyancy source - rotation rate of the R––z coordinate system in radians/second     

Supersonic responses induced by point load moving steadily on an anisotropic half-plane

Supersonic responses of an anisotropic half-plane solid induced by a point load moving steadily on the half-plane boundary are investigated. Analytic expressions for the responses of the displacements and stresses for field points either inside or on the surface of the half-plane solid are given for general anisotropic materials. For the special cases of monoclinic materials with symmetry plane at $x_3=0$ and orthotropic materials, the supersonic as well as subsonic responses of the displacements and stresses are further expressed explicitly in terms of elastic stiffnesses. Responses for the case of isotropic materials known in the literature are recoverable from present results.     

INTERFACE CRACK IN BIPIEZOTHERMOELASTIC MEDIA

By using the extended version of Eshelby-Stroh's formulation and the method of analyt-ical continuation,the problems of interface cracks are reduced to a Hilbert problem of vector form.Ageneral explicit closed form solution for the piezothermoelastic interface crack problem is then ob-tained,the whole field solutions of temperature,heat flux,displacements,electric field,stress andelectric induction are given,the explicit expressions for the crack opening displacements and electricpotential are also provided.