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.     

A gas thrust bearing with pressurization

The Galerkin method was used to calculate a gas thrust bearing with pressurization. It is shown that, with an appropriate choice of approximating functions, the first approximation already gives satisfactory results. The data obtained are compared with the results of numerical integration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 25–31, January–February, 1973.     

Gas-lubricated sector thrust bearing with maximum load capacity

The problem of the optimization of the microgeometry of a gas-lubricated sector thrust bearing with respect to the maximum lift criterion is considered. The gas lubrication Reynolds equation is used for determining the pressure in the lubricating film. The case of low compressibility numbers is studied. The optimum profile is qualitatively determined on the basis of an analysis of the system of necessary conditions for an extremum. The results are presented for various values of the problem parameters. A comparison is made with the data of a number of gas thrust bearing optimization studies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 35–42, November–December, 1990.     

Optimum clearance of a gas hydrostatic thrust bearing with maximum load capacity

The optimum design of a gas hydrostatic thrust bearing clearance is obtained using the methods of calculus of variations. The variational problem of determining the clearance shape giving the maximum load capacity is solved for a given external pressurization and various journal speeds. The structure of the optimum solution is found on the basis of the gas lubrication approximation with and without constraints on the height of the bearing pad (pocket). The calculation results embrace all possible values of the parameters. A comparison with optimum liquid bearings is carried out. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 68–78, July–August, 2000. The work was carried out with support from the Russian Foundation for Basic Research (projects Nos. 99-01-01211 and 96-15-96158).     

Inertia effects in externally pressurized bearing with viscoelastic lubricant

A theoretical investigation of inertia effects on the load capacity of an externally pressurized bearing with a visco-elastic lubricant is presented. The methods of iteration and averaged inertia have been employed to find an approximate solution of the resulting non linear differential equation. Graphical representation of results together with those for the case without inertia has been shown. It is found that the elasticity of the liquid increases the load bearing capacity and the pressure at a point in the lubricant film. There is a negative contribution to the load capacity by inertia forces.     

Lubrication analysis of a cell sliding into a circular pore

The resistance to the blood cells at the entrance to capillaries and membrane pores contributes considerably to the peripheral resistance in the blood circulation. This paper proposes, for the first time, a simplified mechanical model in an attempt to treat the axisymmetric motion of a cell sliding into a circular pore. In this model, the shape of the cell is taken as given according to the microvideograph and the cell membrane is assumed to slide over its surface. The lubrication theory is applied to the thin layers of plasma between the membrane and the pore wall, yielding the pressure and shear stress distributions over the membrane as well as the resultant drag exerted on the cell. Our computations have simulated the process of the cell entering the pore, which is in qualitative agreement with the microvideographic observations.The Project Supported by National Natural Science Foundation of China.The authors wish to express their thanks to Professor S. Weinbaum of The City University of New York for his enlightening suggestions on the mechanical model of this problem during his visit to Peking University. The authors are also grateful to Professor Li Gui-shan of Institute of blood. The Chinese Academy of Medical Sciences, for providing the relevant microvideographes.     

Nonlinear natural responses of a rotating circular string

This study investigates the nonlinear dynamics of a rotating circular string subjected to a spring force fixed in space. The governing equation for out-of-plane vibration is developed using Hamilton's principle. The nonlinearities of the string deformation and the spring stiffness are considered in the governing equation. Applying Galerkin's method, the governing equation is transformed from a nonlinear partial differential equation into a set of coupled nonlinear ordinary differential equations through orthogonal trigonometric shape functions. Butenin's method is adopted to develop a closed-form analytical solution for single-mode oscillations of the system. Comparison shows that the closed-form solution is in a good agreement with the numerical results over a wide range of the nonlinearities. Multi-mode responses of the string are investigated by means of numerical integration. Based on the results, the nonlinear dynamics of the string are discussed.     

Thermofluiddynamic experiments with a heated and rotating circular cylinder in crossflow

Two-dimensional flow fields and temperature boundary layer profiles around a heated and rotating circular cylinder in crossflow were experimentally investigated for a subcritical freestream-Reynolds-number 5.6 · 10⁴ corresponding to a flow velocity of 7 m/s. Test parameter was the ratio of free stream velocity to peripheral speed, which encompasses the range between zero and 2.5. An electronically-controlled hot wire measurement technique, practicable for the requirements of 1–2 mm boundary layer thickness, was used. The numerous reliable test results confirm previous reported experiments. Characteristic features in heat transfer are discussed.List of symbols C _b correction factor for blockage - n rotation rate in rpm - r radial coordinate - R cylinder radius - Re Reynolds-number = U 2R/v - Re _R circumferential Reynolds-number = U _R 2R/v - T local temperature - U velocity - = U · C _b/U _R velocity ratio of air flow and cylinder surface, corrected for blockage - v kinematic viscosity - = T — T /T _w — T non-dimensional temperature Indices undisturbed flow conditions - w wall - R circumferential - c critical Dedicated to Alfred Walz on the occasion of his 80th birthday     

Thermofluiddynamic experiments with a heated and rotating circular cylinder in crossflow

Heat transfer experiments with a heated and rotating circular cylinder in an air crossflow were performed for subcritical freestream-Reynolds-number 8.3 · 10³ < Re _f < 7.1 · 10⁴ and for Nusselt-numbers up to 300. The experiments took into consideration important practical parameters such as freestream-turbulence intensity, aspect ratio, blockage ratio and temperature loading. The measured values were standardized to the ‘idealized case’ to allow adequate comparison with pertinent literature. The results encompass the range of velocity ratios α (circular/freestream-velocity) between zero and 2.5. For values α > 0.5, the typical Nusselt-number behaviour noted by some authors has been confirmed. For small velocity ratios α < 0.5, however, there are significant deviations from the heat transfer coefficients previously believed to be constant.     

Thermofluiddynamic experiments with a heated and rotating circular cylinder in crossflow

Two-dimensional boundary layer profiles of heated and rotating circular cylinders in crossflow were investigated for subcritical freestream-Reynolds-numbersRe = 3.4 · 104 andRe = 4.8 · 10⁴. The peripheral speed of the cylinder surface corresponds to velocity ratios 0 ≦ ξ ≦ 2(α, circular/freestream-velocity). Special attention was focused on the location of separation points, which was determined as a function of the velocity ratio a whereby a strong influence of wall temperature was revealed.     

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     

Hydrodynamic lubrication of an externally-pressurised bearing using a Reiner-Rivlin fluid as lubricant

Summary The use of a Reiner-Rivlin fluid as the lubricant in an externally-pressurised bearing has been considered. The effect of the coefficient of cross-viscosity has been shown to be advantageous for the increase in the load carrying capacity of the bearing.     

On the effects of a compressible viscous lubricant on the load‐bearing capacity of a journal bearing

A compressible viscous isothermal model is presented for studying journal‐bearing lubrication. The viscosity in the model thickens with increasing density. The governing equations are written in terms of velocity, the natural logarithm of the density and the kinematic extra‐stress tensor. A semi‐Làgrangian treatment of the material derivatives is combined with a spectral element discretization in space. The roles of the speed of sound and the eccentricity ratio on the load‐bearing capacity of the journal bearing are investigated. Compressibility is shown to enhance the load‐bearing capacity and this effect is amplified as the eccentricity ratio approaches unity. It is shown that for speeds of sound in the region of those of multigrade oils, the dominant component of the force on the journal acts along the line joining the centres of the bearing and journal and in the direction away from the narrow gap. Copyright © 2007 John Wiley & Sons, Ltd.     

Stokes flow of a rotating sphere around the axis of a circular orifice or a circular disk

This paper presents an infinite series solution to the creeping flow equations for the axisymmetric motion of a sphere of arbitrary size rotating in a quiescent fluid around the axis of a circular orifice or a circular disk whose diameters are either larger or smaller than that of the sphere. Numerical tests of the convergence are passed and the comparison with the exact solution and other computational results shows an agreement to five significant figures for the torque coefficients in both cases. The torque coefficients are obtained for the sphere located up to a position tangent to the wall plane containing either the orifice or the disk. It is concluded that the torque coefficients of the sphere and the disk are monotonically increasing with the decrease of the distance from the disk or the orifice plane in both cases.     

Vortex shedding flow behind a slowly rotating circular cylinder

This paper investigates flow past a rotating circular cylinder at 3600?Re?5000 and α?2.5. The flow parameter α is the circumferential speed at the cylinder surface normalized by the free-stream velocity of the uniform cross-flow. With particle image velocimetry (PIV), vortex shedding from the cylinder is clearly observed at α<1.9. The vortex pattern is very similar to the vortex street behind a stationary circular cylinder; but with increasing cylinder rotation speed, the wake is observed to become increasing narrower and deflected sideways. Properties of large-scale vortices developed from the shear layers and shed into the wake are investigated with the vorticity field derived from the PIV data. The vortex formation length is found to decrease with increasing α. This leads to a slow increase in vortex shedding frequency with α. At α=0.65, vortex shedding is found to synchronize with cylinder rotation, with one vortex being shed every rotation cycle of the cylinder. Vortex dynamics are studied at this value of α with the phase-locked eduction technique. It is found that although the shear layers at two different sides of the cylinder possess unequal vorticity levels, alternating vortices subsequently shed from the cylinder to join the two trains of vortices in the vortex street pattern exhibit very little difference in vortex strength.     

Effect of lubricant inertia in bearings with a non-Newtonian lubricant

Summary The inertia effects in externally pressurized and squeeze film bearings with lubricants obeying a power law are considered. It is found that the inertia forces decrease the load capacity of the externally pressurized bearing with a given flow rate and the inertia effect increases with the flow behaviour index. At a given feeding pressure, on the other hand, the inertia increases or decreases the load capacity when the flow behaviour index is smaller than or greater than 3, respectively. For squeeze films between circular plates and rectangular plates, the rate of squeeze is slowed down by the inertia and the inertia effect is larger in dilatant lubricants than in pseudoplastic lubricants.Nomenclature 2a diameter of the bearing, width of rectangular plates - 2b diameter of the recess - 2h film thickness - 2h ₀ initial thickness of squeeze films - l length of the rectangular plates - m consistency index - n flow behaviour index - p pressure - p _e external pressure - p _i feeding pressure - q flow rate - r radial distance - t time - u velocity of the lubricant - v squeeze velocity - w load capacity - W dimensionless load capacity - axial distance - viscosity of the lubricant - density of the lubricant     

Lubrication theory for micropolar fluids and its application to a journal bearing with finite length

In this paper, the field equation of micropolar fluid with general lubrication theory assumptions is simplified into two systems of coupled ordinary differential equation. The analytical solutions of velocity and microrotation velocity are obtained. Micropolar fluid lubrication Reynolds equation is deduced. By means of numerical method, the characteristics of a finitely long journal bearing under various dynamic parameters, geometrical parameters and micropolar parameters are shown in curve form. These characteristics are pressure distribution, load capacity, coefficient of flow flux and coefficient of friction. Practical value of micropolar effects is shown, so micropolar fluid theory further closes to engineering application.     

On the topological bifurcation of flows around a rotating circular cylinder

Flow fields around a rotating circular cylinder in a uniform stream are computed using a low dimensional Galerkin method. Reslts show that the formation of a Fopple vortex pair behind a stationary circular cylinder is caused by the structural instability in the vicinity of the saddle located at the rear of the cylinder. For rotating cylinder a bifurcation diagram with the consideration of two parameters, Reynolds numberRe and rotation parameter α, is built by a kinematic analysis of the steady flow fields. The project supported by the National Natural Science Foundation of China