The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection)

An analysis is carried out to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. The localized heating or cooling introduces a finite discontinuity in the mathematical formulation of the problem and increases its complexity. In order to overcome this difficulty, a non-uniform distribution of wall temperature is taken at finite sections of the plate. The nonlinear coupled parabolic partial differential equations governing the flow have been solved by using an implicit finite-difference scheme. The effect of the localized heating or cooling is found to be very significant on the heat transfer, but its effect on the skin friction is comparatively small. The buoyancy, magnetic and suction parameters increase the skin friction and heat transfer. The positive buoyancy force (beyond a certain value) causes an overshoot in the velocity profiles.A mass transfer constant - B magnetic field - C_fx skin friction coefficient in the x-direction - C_p specific heat at constant pressure, kJ.kg^–1.K - C_v specific heat at constant volume, kJ.kg^–1.K^–1 - E electric field - g acceleration due to gravity, 9.81 m.s^–2 - Gr Grashof number - h heat transfer coefficient, W.m².K^–1 - Ha Hartmann number - k thermal conductivity, W.m^–1.K - L characteristic length, m - M magnetic parameter - Nu_x local Nusselt number - p pressure, Pa, N.m^–2 - Pr Prandtl number - q heat flux, W.m^–2 - Re Reynolds number - Re_m magnetic Reynolds number - T temperature, K - T_o constant plate temperature, K - u,v velocity components, m.s^–1 - V characteristic velocity, m.s^–1 - x,y Cartesian coordinates - thermal diffusivity, m².s^–1 - coefficient of thermal expansion, K^–1 - , transformed similarity variables - dynamic viscosity, kg.m^–1.s^–1 - ₀ magnetic permeability - kinematic viscosity, m².s^–1 - density, kg.m^–3 - buoyancy parameter - electrical conductivity - stream function, m².s^–1 - dimensionless constant - dimensionless temperature, K - w, conditions at the wall and at infinity     

Finite-difference analysis of MHD Stokes problem for a vertical infinite plate in dissipative fluid

Finite-difference solution of MHD flow past an impulsively started vertical infinite plate in an electrically conducting fluid has been presented on taking into account the viscous dissipative heat. Results for velocity and temperature are shown graphically whereas the numerical values of the skin-friction and the rate of heat transfer are entered in the table. The results are discussed in terms of the parameters M (the Hartmann number), G (the Grashof number, G>0, cooling of the plate by free convection, G<0, heating of the plate by free convection currents), E (the Eckert number) and P (the Prandtl number).Nomenclature B ₀ applied magnetic field - c _p specific heat at constant pressure - g acceleration due to gravity - k thermal conductivity - t time - T temperature of the fluid near the plate - T temperature of the fluid far away from the plate - U ₀ velocity of the plate - u velocity of the fluid - coefficient of volume expansion - kinematic viscosity - scalar electrical conductivity - coefficient of viscosity - density of the fluid     

Heat transfer in magnetohydrodynamic flow near a stagnation point

Summary Steady, axisymmetric, magnetohydrodynamic flow with a stagnation point on an infinite plane wall is considered with a magnetic field applied normal to the wall. Solutions are obtained in the form of series for the velocity, magnetic field and temperature when the magnetic field parameter () and the ratio of viscosity to magnetic diffusivity () are small. The case=O(1) is considered briefly when solutions which Meyer³⁾ obtained by physical order-of-magnitude arguments are derived mathematically as expansions in. Some remarks are made on the consistency of extending the results to flow within the boundary layer near the nose of a bluff body.     

Determination of gradients of flow variables behind three-dimensional stationary and pseudo stationary curved shock waves in conducting gases

Summary In this paper we have obtained the gradients of magnetic field, velocity, pressure and density behind a shock wave in three dimensional steady motion of a conducting gas. For the shock configuration, we take a continuous differentiable function of coordinates and it is assumed that the components of the magnetic field H ⁱ , velocity components u ⁱ , pressure p and density behind the shock-surface are differentiable functions. Moreover we take H ⁱ , u ⁱ , p and in front of the shock-wave as constant quantities. In § 4 we have obtained the gradients of flow and field quantities behind the pseudostationary shockwave. § 5 is devoted to the calculation of gradients of flow and field quantities in cases where the normal component of the magnetic field is zero on both sides of the shock wave. In § 6 the relation between the curvature k of the shock-surface and the curvature K of the stream line just behind the shock surface in two dimensional steady motion has been derived. § 7 deals with the determination of the ratio K/k for an attached shock in the case of a wedge.     

Thermal radiation effects on MHD-Rayleigh flow with constant surface heat flux

Thermal radiation heat transfer effects on the Rayleigh flow of gray viscous fluids under the effect of a transverse magnetic field are investigated. The free convection heat transfer problem from constant surface heat flux moving plate is selected for study. It is found that the increasing of the magnetic field number M= H₀² / U₀²decreased velocities inside boundary layer, the increasing of the conduction–radiation parameter Rd=k__R/4aT³ decreased both temperatures and heat transfer rates. It is also found that the increasing of the dimensionless surface heat flux parameter q₀^*=q₀ /(kU₀T₎ increased the temperatures inside the boundary layer and increased the heat transfer rates. Comparison with previous works shows excellent agreement. Different transient velocity profiles, temperature profiles and local Nusselt numbers against different dimensionless groups are drawn.     

Rayleigh problem in slip flow with transverse magnetic field

An analytical continuum solution of the Rayleigh problem in slip flow with applied magnetic field is obtained using a modified initial condition and slip boundary conditions. The results are uniformly valid for all times and show that the velocity slip and the local skin friction coefficient remain almost unaffected by the imposition of the magnetic field for small times. They increase however with the magnetic field for large times. The present results reduce to the corresponding results of the hydrodynamic case when there is no magnetic field.Nomenclature A constant - b characteristic length - B magnetic field vector - B ₀ magntidue of the applied magnetic field normale to the plate - B _x magnitude of the induced magnetic field parallel to the plate - C slip coefficient, (2–f)/f - C _f skin friction coefficient, - C _D average drag coefficient - erfc(_x) complementary error function, - E electric field vector - f Maxwell's reflection coefficient - H _a Hartmann number, (B ₀ ² b ²/)^1/2 - nondimensional magnetic parameter - J current vector - Kn=L/b Knudsen number - L mean free path - M Mach number - p constant parameter - P _m magnetic Prandtl number, Re _m/Re= ₀ - q velocity vector - Re Reynolds number, Ub/ - Re _m magnetic Reynolds number, ₀ Ub - t time - nondimensional time, tU/b - u velocity of the fluid parallel to the plate - nondimensional velocity, u/U - U velocity of the plate - Laplace transform of - x, y coordinates along and normal to the plate respectively - y nondimensional distance, y/b - Z nondimensional parameter, 1/Re ^1/2 Kn - ratio of specific heats - boundary layer thickness - velocity slip - viscosity - ₀ magnetic permeability - kinematic viscosity - nondimensional time parameter, ( /Re)^1/2/Kn - density - electrical conductivity     

Magnetic field effects on mechanical behaviour of ferrosuspensions

Summary In a measuring cell of a rotational magnetorheometer a rheological investigation of a number of ferrosuspensions is carried out over wide ranges of their concentration, temperature, intensity and orientation of the magnetic field. The obtained data are correlated by a new universal magnetorheological characteristic invariant for the ferrosuspension type and temperature, intensity and orientation of the external magnetic field.An experimental investigation is made of peculiarities of the mechanical behaviour of ferrosuspension shear flows in a coaxial cylinder system, with their internal structure dynamically controlled by a rotating magnetic field. Some estimates are made of the momentum transfer mechanism in ferrosuspensions affected by magnetic and shear fields.

---

Zusammenfassung In der Meßzelle eines Magneto-Rotationsrheometers wurden rheologische Untersuchungen an einer Reihe von Ferrosuspensionen in einem weiten Bereich von Konzentration und Temperatur, sowie Intensität und Orientierung des Magnetfeldes durchgeführt. Die gewonnenen Daten werden in Form einer universellen magnetorheologischen Charakteristik zusammengefaßt, die sowohl bezüglich des Typs und der Temperatur der Ferrosuspension als auch der Intensität und der Orientierung des äußeren Magnetfeldes invariant ist.Weiter wurde eine experimentelle Untersuchung der mechanischen Besonderheiten bei Ferrosuspensionen in der Scherströmung eines Couette-Viskosimeters infolge der dynamischen Veränderung ihrer inneren Struktur durch ein rotierendes Magnetfeld durchgeführt. Einige Abschätzungen der Impulsübertragung in Ferrosuspensionen als Funktion des Magnet- und des Scherfeldes sind angegeben.

---

Notations volumetric concentration of the disperse ferrosuspension phase, % - I magnetization, T - H magnetic field intensity, A/m - thickness of the ferrosuspension layer under investigation, mm - shear stress, N/m² - shear rate, s^–1 - angular velocity of the field, s^–1 - angular velocity of the rotor, s^–1 - i current in the inductor windings, A - apparent ferrosuspension viscosity, N s/m² - T temperature, K With 9 figures and 1 table     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

Thermal Capacity Effect on Transient Free Convection Adjacent to a Vertical Surface in a Porous Medium

In this paper we analyse how the presence of the thermal capacity of a vertical flat plate of finite thickness, which is embedded in a porous medium affects the transient free convection boundary-layer flow. At the time t = 0, the plate is suddenly loaded internally with a constant heat flux rate q, so that a transient boundary-layer flow is initiated adjacent to the plate. Initially, the transient effects due to the imposition of the uniform heat flux rate at the plate are confined to a thin fluid region near to the surface and are described by a small time solution. These effects continue to penetrate outwards and eventually evolve into a new steady state flow. Analytical solutions have been derived for these transient (small time) and steady state (large time) flow regimes, which are then matched by a numerical solution of the full boundary-layer equations. It has been found that the non-dimensional fluid temperature (or fluid velocity) profiles are reduced when the thermal capacity effects, described by a parameter Q ^*, are reduced. For small values of Q ^*, the approach of these profiles to their steady state values is monotonic. However, for large values of Q ^*, the temperature profiles are observed to locally exceed (pass through a maximum value) the final steady state values at certain distances from the plate. In general, the maxima in the temperature profiles increase in size as Q ^* increases and the time taken to approach the steady state solutions increases significantly.     

Flow transitions in a rotating magnetic field

Critical Rayleigh numbers have been measured in a liquid metal cylinder of finite height in the presence of a rotating magnetic field. Several different stability regimes were observed, which were determined by the values of the Rayleigh and Hartmann numbers. For weak rotating magnetic fields and small Rayleigh numbers, the experimental observations can be explained by the existence of a single non-axisymmetric meridional roll rotating around the cylinder, driven by the azimuthal component of the magnetic field. The measured dependence of rotational velocity on magnetic field strength is consistent with the existence of laminar flow in this regime.List of symbols B ₀ magnitude of magnetic induction - B_r, B radial and azimuthal magnetic induction components - C wall admittance - d cell diameter - d _w wall thickness - g gravity at earth's surface - Ha Hartmann number - h cell height - k _f thermal conductivity of fluid - k _w thermal conductivity of wall - L1, L2, L3, L4 thermistor temperatures - Ra Rayleigh number - Ra ^c critical Rayleigh number for the transition from no flow to laminar flow - Ra ^t critical Rayleigh number for the transition from time-independent to time-dependent flow - r radial coordinate - T _a temperature at top of cell - T _b temperature at bottom of cell - T temperature difference between cell bottom and cell top - T_c critical temperature difference between cell bottom and top time - t time - U1, U2, U3, U4 thermistor temperatures - z vertical coordinate - volumetric thermal expansion coefficient - skin depth - k thermal diffusivity - magnetic permeability - kinematic viscosity - density - electrical conductivity - azimuthal coordinate - angular frequency of magnetic induction This work was supported by the Microgravity Science and Applications Division of the National Aeronautics and Space Administration.     

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     

Study of thermal flows from two-dimensional,upward-facing isothermal surfaces using a laser speckle photography technique

A laser specklegram or speckle photography technique allows a direct measurement of surface temperature gradients and provides a full field interrogation with an extremely high resolution from a single data taking. The specklegram technique has been successfully applied to investigate the natural convection heat transfer from an upward-facing isothermal plate. For a plate with a large aspect ratio of 15, both local and global Nusselt numbers have been determined from the direct measurement of local temperature gradients. The Rayleigh number, based on the length scale equivalent to the ratio of the surface area to the perimeter, has been varied from 9.0 × 10³ to 4.0 × 10⁴. The present result for the global heat transfer has shown that a 1/5-power law, i.e., Nu = C₁ Ra ^1/5, correlates the data more properly whilst previously published results showed a large scatter in the exponent, ranging from 1/8-power to 1/4-power. The proportional constant, C₁ has been determined to be 0.56 which shows a fairly good agreement with previously published theoretical results. The laser specklegram technique has shown a strong potential as a powerful and convenient method for an experimental assessment of natural convection heat transfer problems. The specklegram technique at the same time has eliminated the deficiencies of both the mass transfer analogy technique and the classical heat transfer measurement technique.List of symbols a characteristic length scale defined as a = A/P where A is the surface area and P is the perimeter of the plate edge [mm] - AR aspect ratio [L/H] - c defocusing distance [mm] - d image distance of Young's fringes from speckle negative - h thermal convection coefficient [W/m² · K] - average thermal convection coefficient [W/m² · °C] - H width of the test section measured perpendicular to the optic axis [mm] - k thermal conductivity [W/m · K] - L length of the test section measured parallel to the optical axis [mm] - n index of refraction - Nu local Nusselt number [ha/k] - global Nusselt number - Pr Prandtl number [v/] - q heat flux per unit area [W/m² · s] - Ra Rayleigh number - s fringe spacing [mm] - Sc Schmidt number [v/D] - T temperature [K] Greek symbols thermal diffusivity [m²/s] - volumetric coefficient of expansion (1/T) - v kinematic viscosity of air [m²/s] - wavelength of helium-neon laser [632.8 nm] - amount of speckle dislocation     

On the effect of magnetic field on viscous lifting and drainage of conducting fluid

This paper concerns with obtaining the solution of the problem of viscous lifting and drainage of a thin liquid film clinging to a vertical plane surface moving with a velocityf(t) in the presence of a transverse magnetic field. Specializing to the case when the surface moves with a constant acceleration, it has been found that the film thickness, for large magnetic fields, increases with the increase in magnetic field.Nomenclature a acceleration of the plate - A non-dimensional acceleration, =a/g - B magnetic induction vector - B ₀ applied magnetic field - f(t) any function oft - Laplace transform off(t) - g gravitational acceleration - h film thickness - H non-dimensional film thickness, =h(g/ ²)^1/3 - J current density vector - k (/)^1/2 B ₀ - M k( ²/g)^1/3 - n summation index - q mass flow rate - Q non-dimensional mass flow rate, =q/ - t time - T non-dimensional time, =t(g ²/)^1/3 - Laplace transform ofv(x, t) - V fluid velocity vector, =[0,v(x, t), 0] - (x, y, z) space coordinates - Y non-dimensionaly-coordinate, =y(g/ ²)^1/3 Greek symbols _n (n+1/2) - conductivity - density - kinematic viscosity     

Natural convection in a square enclosure with an internal isolated vertical plate

Numerical computations were performed for the average Nusselt number at an internal vertical plate situated in a square enclosure, with the inner plate and the bounding wall of the enclosure maintained at uniform but different temperatures. Natural convection occurred in the air which occupied the enclosure space. The position of the inner vertical plate within the enclosure was varied parametrically. The plate height-cavity height ratio was 0.513. For narrow distance between the inner plate and the bounding wall the inner plate Nusselt number was enhanced. Aside from this, the plate average Nusselt number was remarkably insensitive to the plate position. The effect of the Rayleigh number on the velocity and temperature fields and local Nusselt numbers are also discussed. The agreement between the predicted flow pattern forRa=1.1×10⁶ and the flow visualization result was reasonably good.

---

Natürliche Konvektion in einem quadratischen Horizontalschacht, der eine freistehende, senkrechte Platte enthält

Zusammenfassung Eine numerische Untersuchung liefert mittlere Nußelt-Zahlen an einer, in einem quadratischen Horizontalschacht freistehenden, senkrechten Platte, wobei deren Temperatur und die der umgebenden Wände jeweils konstant gehalten werden. Im Luftraum dazwischen stellte sich freie Konvektion ein. Die Position der Platte war veränderlich, ihre Höhe blieb mit 51.3% der Schachthöhe konstant. Rückte die Platte nahe an eine Schachtwand, so erhöhte sich die Nußelt-Zahl auf der dieser zugewandten Seite, während die Gesamt-Nußelt-Zahl bezüglich der Platte fast konstant bleibt. Es wird auch der Einfluß der Rayleigh-Zahl auf das Geschwindigkeitsund Temperaturfeld diskutiert. BeiRa=1.1·10⁶ stimmten die Ergebnisse aus der Berechnung gut mit den experimentellen Befunden einer Strömungsvisualisation überein.

---

Nomenclature a distance between vertical plate and side-wall of enclosure thermal diffusivity (in definition ofu _r) - b distance between vertical plate and bottom of enclosure - g gravitational acceleration - G characteristic flow rate - H height of vertical plate - k thermal conductivity - k _f fluid thermal conductivity - K relative thermal conductivity,k/k _f - L width of square enclosure - M _res mass residual - Nu local Nusselt number - Nu _m average Nusselt number - Nu _L local Nusselt number of left side of vertical plate - Nu _R local Nusselt number of right side of vertical plate - Nu _B local Nusselt number of bottom side of vertical plate - Nu _T local Nusselt number of top side of vertical plate - p effective pressure - P dimensionless pressure,P=p/[(Ra Pr)(a/H)²] - Pr Prandtl number - Ra Rayleigh number,Ra=gTH ³ Pr/ ² - T temperature - T _i temperature of internal plate - T _o temperature of enclosure surface - u, v velocity components inx-, y-direction - U, V dimensionless velocities,U=u/u _r, V=v/u_r - u _r reference velocity,u _r=(Ra Pr)^1/2/(a/H) - X, {iyY} dimensionless coordinates,X=x/H, Y=y/H Greek symbols heat transfer coefficient - volume expansion coefficient - thickness of plate - kinematic viscosity - density - dimensionless temperature, (T _i–T)/(T _i–T_o)     

Rotation of axisymmetric bodies in hydromagnetics

Summary This paper deals with the distrubance due to steady rotation of axisymmetric bodies in a viscous incompressible fluid of infinite conductivity in which the uniform ambient flow field is collinear with the uniform magnetic field. The known results of Sowerby¹) for the problem Couple on a rotating spheroid in a slow stream in an incompressible viscous fluid are generalized to apply to all ellipsoids of revolution and a special case of the circular disc has also been investigated in detail. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. The coefficient of couple has been found to the first order approximation in terms ofR(1–S) orR(S–1) whereR is the Reynolds number andS=H ₀ ² /4W ² the pressure number which plays an important part. The couple exerted on the body is computed for various values ofS andR.     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

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.