Effect of radial and rotational lubricant inertia on the pressure distribution in externally pressurised thrust bearings

Expressions are obtained for the pressure distribution in an externally pressurised thrust bearing for the condition when one bearing surface is rotated. The influence of centripetal acceleration and the combined effect of rotational and radial inertia terms are included in the analysis. Rotation of the bearing causes the lubricant to have a velocity component in an axial direction towards the rotating surface as it spirals radially outwards between the bearing surfaces. This results in an increase in the pumping losses and a decrease in the load capacity of the bearing. A further loss in the performance of the bearing is found when the radial inertia term, in addition to the rotational inertia term is included in the analysis.Nomenclature r, z, cylindrical co-ordinates - V _r, V , V _z velocity components in the r, and z directions respectively - U, X, W representative velocities - coefficient of viscosity - p static pressure at radius r - p mean static pressure at radius r - Q volume flow per unit time - 2h lubricant film thickness - density of the lubricant - r ₂ outside radius of bearing = D/2 - angular velocity of bearing - R dimensionless radius = r/h - P dimensionless pressure = h ³ p/Q - Re channel Reynolds number = Q/h     

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     

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     

Thermal and inertia effects in externally pressurized conical thrust oil bearings

This paper presents theoretical and experimental investigations of thermal and inertia effects on the performance of externally pressurized conical thrust bearings. The analysis, as well as the experimental results, revealed that the increase in oil temperature due to pad rotation has a detrimental effect on the load carrying capacity, while it increases the flow rate. Increasing the speed of rotation, will increase or decrease the bearing load carrying capacity depending on the recess dimensions.Nomenclature c lubricant specific heat - F frictional torque - h film thickness - L load carrying capacity - P pressure - P pressure ratio (P/P ₁) - P ₁ inlet pressure - Q volume flow rate - r radius measured on cone surface - r radius ratio (R/R ₃) - R ₁ supply hole radius - R ₂ recess radius - R ₃ outside radius of bearing - S inertia parameter (0.15 ² R ₃ ² /P ₁) - T temperature - u, v, w velocity components (see Fig. 2) - z coordinate normal to cone surface - lubricant density - lubricant viscosity - 2 cone apex angle - rotational speed - recess depth     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

Inertia effects in hydromagnetic lubrication

Summary A theoretical investigation of inertia effects in a squeeze film bearing with an electrically conducting lubricant in the presence of a uniform transverse magnetic field is presented. The two cases of infinitely long rectangular plates and circular plates as bearing surfaces are considered. It is shown that the load supporting capacity of the bearing increases and the squeeze decreases if the lubricant inertia effects are taken into account. However, the inertia effect becomes smaller when the strength of the magnetic field increases.     

On the numerical computation of Generalized Burmester Points

This note describes a procedure for plane higher-curvature path analysis and synthesis. All coefficients have been written in terms of elementary instantaneous invariants. This facilitates the numerical computation of Generalized Burmester Points for a moving link of a planar mechanism in a non-symmetric position. FORTRAN subroutines have been written and a numerical example is provided.

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Sommario Si descrive una procedura di analisi e sintesi per meccanismi piani generatori di traiettoria con approssimazione del quarto ordine. Nella formulazione adottata, l'impiego degli invarianti istantanei elementari consente di valutare analiticamente i termini delle equazioni per la ricerca dei punti generalizzati di Burmester. Sono state implementate subroutines in linguaggio FORTRAN ed è stato sviluppato un esempio numerico.

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Nomenclature P _o velocity pole - inflection circle diameter - angle of rotation of the moving body - r _f ,r _m radii of curvature of the fixed and moving polodes, respectively - dl infinitesimal arc length measured on the polode - a,b coordinates ofP _o , in the canonical reference system¹ - a _i ,b _i i-th derivatives ofa andb, respectively, computed at the initial position (i.e. =0). These are the elementary instantaneous invariants - h,* polar coordinates of the moving point in the canonical reference system (0 ) - radius of curvature of the point-path trajectory - _E radius of curvature of the evolute of the point-path trajectory - _E ^/(2) radius of curvature of the evolute of the evolute of the point-path trajectory A canonical reference system is a rectangular right-handed cartesian system having they-axis directed toward inflexion pole, origin in the velocity pole.     

Rheodynamic lubrication of a journal bearing

Summary The problem of a journal bearing lubricated by a Bingham material has been solved. It has been found that the load capacity, and the moment of friction of the bearing are larger than in a journal bearing, lubricated with a Newtonian material.Nomenclature r radius of the journal - c radial clearance - r + c radius of the bearing - e eccentricity - w angular velocity of the journal - h thickness of the lubricant film at any point - thickness of the core - ø angular distance of a point, from the point, where film thickness is maximum - eccentricity ratio (e/c) - x distance along the bearing surface - y distance normal to the bearing surface - T shear stress in the lubricant - T ₀ yield value of a Bingham solid - viscosity of a Newtonian fluid - plastic viscosity of a Bingham solid - p fluid pressure in the lubricant film - Q volume flow of the lubricant - W ₀ load capacity of the bearing for ordinary lubricants - W load capacity of the bearing - M moment of friction - F coefficient of friction - ₁ maximum thickness of the inlet core - ₂ maximum thickness of the outlet core - ₁ circumferential extent of the inlet core in the journal bearing - ₂ circumferential extent of the outlet core in the journal bearing - h ₀ minimum hieght of core formation in the slider bearing - h _p maximum height of core formation in the slider bearing - u velocity of the fluid in the direction of x in the slider bearing - V velocity in the y direction - h ₁ height of the inlet core at the circumferential extent ₁ - h ₂ height of the outlet core at the circumferential extent ₂ - h ₃ height of the outlet core in the region ₂ - q Q/(cwr) - q ₀ value of q for Newtonian lubricants - p ₀ pressure at =0 - H h/c - H ₁ h ₁/c - H ₂ h ₂/c - B T ₀ C/wr = Bingham number     

An externally pressurised bearing with variable film thickness

Summary In this paper, a theoretical study of an externally pressurised bearing with variable film thickness has been presented. The load capacity and the frictional drag on the rotor have been determined. It has been shown that the frictional drag decreases as the angle, which the rotor surface makes with the stator, increases, but the load capacity and pressure do not depend upon it.Nomenclature angle between the rotor surface and the stator - angular velocity of the rotor - viscosity of the lubricant - h variable film thickness - L depth of the recess - p pressure - p _i inlet pressure - p _e exit pressure - Q rate of flow - r radial independent parameter - R ₀ radius of the recess - R outer radius of the stator - T frictional drag on the rotor - u radial velocity - v tangential velocity - w load capacity - z axial independent parameter     

A three-dimensional variational problem in the gasdynamic theory of a lubricant

In [1–3] optimal forms of the gap were found for one-dimensional aerodynamic sliding bearings. The coefficient of the bearing capacity is optimized under the condition that the one-dimensional Reynolds equation of a gas lubricant is used to determine the pressure in the bearing. In the present article the three-dimensional problem of finding the optimal profile of an aerodynamic sliding bearing in the case of small compressibility numbers is considered. The problem is solved by the methods of variational calculation. A qualitative investigation is made of the form of the optimal profile, the results of which are confirmed by a numerical solution of a system of Euler-Lagrange equations. The results of the calculations are given for different elongations of the bearing. On the basis of the profiles obtained, optimal profiles with a rectangular pocket, which are more practical to fabricate, are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 34–39, September–October, 1975.     

The load-bearing capacity of a continuous-flow squeeze film of liquid

A new form of squeeze film system is described in which the movement of one plate towards the other is simulated by the continuous volume generation of liquid over the plate area. The liquid exudes from 1580 holes distributed uniformly over the lower plate surface. An advantage of the system is that there are no moving parts, but it is important to evaluate the device using Newtonian liquids in order to compare the load bearing capacity with that predicted by equations developed for orthodox squeeze film systems. Liquid maldistribution is shown to be a problem which may be solved in various ways, one of which is to ensure that the pressure drop through the plate is high relative to that in the squeeze film.Results obtained using Newtonian liquids make satisfactory comparison with theoretical predictions, though liquid inertia probably makes a lower contribution to load bearing than is the case for an orthodox squeeze film. Liquid maldistribution is allowed for on a theoretical basis or corrected by the use of a distributor plate placed below the perforated surface.Preliminary tests using viscoelastic solutions (based on polyacrylamide of high molecular weight) suggest that the load bearing properties of the squeeze film are significantly enhanced. A load 600 per cent greater than the theoretical load is obtained in one case, the suggestion being made that this is due to stress of viscoelastic origin.Nomenclature D Exit diameter of holes in spinnerette - F ₁ to F ₆ Vertical force on top plate due to flow in squeeze film, defined by (1), (8), (11), (12), (13) and (14) respectively - h Plate separation - h _L Distance of distributor plate from lower surface of spinnerette (function of r) - I ₀ Modified Bessel function of first kind, order 0 - I ₁ Modified Bessel function of first kind, order 1 - K ₀ Modified Bessel function of second kind, order 0 - L Length of hole, based on diameter D, giving same pressure drop as actual spinnerette holes - dm/dt Mass flowrate of liquid - N Total number of holes in spinnerette (1580) - p Isotropic pressure in squeeze film - P _RES Isotropic pressure in reservoir behind lower plate of spinnerette - p–P _RES - (dp/dr)_s Pressure gradient in squeeze film - (dp/dr)_L Pressure gradient in lower film below spinnerette when distributor plate is used - Q Total liquid volume flowrate - q _s Volume flowrate through squeeze film at radius r - q _L Volume flowrate through lower film at radius r - r radial coordinate - R radius of upper disc - - v Velocity of upper disc relative to lower one (simulated by Q/R ² in continuous flow system) - V _R Average radial liquid velocity at radius R - V _S Liquid exit velocity from single hole - V _r V V _z Point velocity components in r, and z directions respectively - z Axial coordinate - Parameter in (8) (3ND ⁴/32LR ² h ³) - Viscosity of liquid - Density of liquid - _rz Shear stress     

Temperature variations across the lubricant film in hydrodynamic lubrication

Summary Temperature variations across the lubricant film in hydrodynamic lubrication have been taken into account. The consequent variations of viscosity cannot be neglected for high Prandtl number lubricants.Nomenclature A constant defined by eq. (13a) - b h ₀/h _L - c heat capacity - h film thickness - h ₀ h at X=0 - h _L h at X=L - H ₀ local heat transfer coefficient defined by (23a) - K thermal conductivity - L length of bearing - m* defined by (9) - Nu Nusselt number defined by (23b) - P pressure - Pe Peclet number (Re) (Pr) - q ₀ slider surface heat flux - q ₀ ^* dimensionless heat flux defined by (20) - Q ₀ slider surface total heat transfer - Q ₀ ^* dimensionless total heat transfer defined by (21) - Re Reynolds number Vh ₀/ - Pr Prandtl number c/ - T temperature - T ₀ slider surface temperature - T _B film bulk temperature - u longitudinal velocity - v transverse velocity - V slider velocity - x longitudinal coordinate - y transverse coordinate - x/h ₀ - u/V - h/h ₀ - y/h - density - viscosity - v/V - - - _S dimensionless stationary surface temperature - dimensionless average stationary surface temperature - _B dimensionless film bulk temperature     

Micropolar squeeze films between rough rectangular plates

In this paper, the lubrication theory for squeezing with micropolar fluids in smooth surfaces has been advanced to analyze the effects arising from roughness considerations using the stochastic approach. This theory is subsequently applied to the problem of squeezing between rough rectangular plates. It is observed that the roughness effects are more pronounced for micropolar fluids as compared to the Newtonian fluids.Nomenclature a x-dimension of rectangular plate - A area of rectangular plate - b z-dimension of rectangular plate - B non-dimensional roughness parameter, c/h _n (for load capacity), c/h _n1 (for squeeze time) - c maximum asperity deviation from nominal film height - E expectancy operator - f(N, l, h) defined by equation (4) - F(N, L, H) defined by equation (31) - F ₁(N, L, B) defined by equation (29) - F ₂(N, L, B) defined by equation (30) - F ₃(N, L, H _n , B) defined by equation (34) - F ₄(N, L, H _n , B) defined by equation (35) - g probability density distribution function - h film height, h=h _n +h _s - h _n nominal film height - h _s deviation of film height from nominal level - h _n1 initial (nominal) film height - H, H _n , H _s non-dimensional forms of h, h _n , h _s respectively - l characteristic material length, (/4)^1/2 - L length ratio, h _n /l (for load capacity), h _n1/l (for squeeze time) - n integer - N coupling number, (/(2+))^1/2 - p pressure - q _x , q _z flow components in x- and z-directions, respectively - t time - T non-dimensional time - w load capacity - W non-dimensional load capacity - x, z cartesian coordinates - angular coordinate - Newtonian viscosity - , micropolar viscosity coefficients - aspect ratio, b/a - standard deviation - /h _n - random variable - defined by equation (19) - defined in equation (28) - defined in equation (33)     

Analysis of the squeezing flow of dusty fluids

The current investigation deals with the study of the effect of introducing a small fraction of dust, by volume, to the fluid in a squeeze film on the viscous resistance to a steady moving disc. Expressions are obtained for the fluid-phase and the dust-phase velocity distributions and the dust particle number density. Analysis based on an iterative procedure indicates that the resistance to motion experienced by the moving disc increases due to the presence of dust.Nomenclature A arbitrary function of integration - B bulk concentration - F resistance to motion experienced by the disc (dusty fluid case) - F _c resistance to motion experienced by the disc (clean fluid case) - F* difference in resistance between the clean fluid and dusty fluid films - f mass concentration - h thickness of the squeeze film - K Stokes coefficient of resistance - m mass of a single dust particle - fluid viscosity coefficient - N dust particles number density - N ₀ dust particles number density at r=R - n iteration level - p fluid pressure in the squeeze film - P pressure in the surrounding - R radius of the disc - fluid density - (r, , y) cylindrical coordinates - t time - U fluid-phase velocity vector - V dust-phase velocity vector - ₁ fluid-phase radial velocity component - U ₂ dust-phase radial velocity component     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

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.     

Effect of slip on porous-walled squeeze films in the presence of a transverse magnetic field

The present investigation is devoted to study the effect of viscous resistance, arising due to sparse distribution of particles in porous media, on the load capacity and thickness time response of porous-walled squeeze films in the presence of a uniform magnetic field. The results of the analysis obtained by using Beavers and Joseph [1] slip-boundary condition show that the viscous resistance increases the load capacity and thickness time response of squeeze films when compared with the results of Chandrasekhara [2] obtained in the absence of viscous resistance. Hence, for efficient performance of a porous walled squeeze film a suitable porous media in which the material is loosely packed may be used.Nomenclature p pressure in the squeeze film - h thickness of the squeeze film at time t - h ₀ thickness of the film at t=0 - u streamwise velocity component in the squeeze film - v transverse velocity component in the squeeze film - P pressure in the porous material - H thickness of the porous material - U streamwise velocity component in the porous material - V transverse velocity component in the porous material - B B ₀+b - B ₀ impressed uniform magnetic field - b induced magnetic field - E electric field vector (E _x , E _y , E _z ) - m ₀ constant defined in (6), (B ₀ ² / _{m f m})^1/2 - v _h value of v at y=h - h/h ₀, the non-dimensional variable - _n eigen values - _f viscosity - _m magnetic permeability - density - _m magnetic diffusivity, 1/ _{m e} - dimensionless parameter, - _e electrical conductivity - q velocity vector (u, v) - L load capacity - I _n integral defined in (37) - M Hartmann number defined in (7), (m ₀ ² h ²)^1/2 - l length of the strips in x-direction - K permeability of the porous material - J current density vector (J _x , J _y , J _z ) - t time - G _n series coefficient appearing in equation (27)     

Squeeze flow of soft solids between rough surfaces

Newtonian liquids and non-Newtonian soft solids were squeezed between parallel glass plates by a constant force F applied at time t=0. The plate separation h(t) and the squeeze-rate were measured for different amplitudes of plate roughness in the range 0.3–31 m. Newtonian liquids obeyed the relation Vh ³ of Stephan (1874) for large plate separations. Departures from this relation that occurred when h approached the roughness amplitude were attributed to radial liquid permeation through the rough region. Most non-Newtonian materials showed boundary-slip that varied with roughness amplitude. Some showed slip that varied strongly during the squeezing process. Perfect slip (zero boundary shear stress) was not approached by any material, even when squeezed by optically-polished plates. If the plates had sufficient roughness amplitude (e.g. about 30 m), boundary slip was practically absent, and the dependence of V on h was close to that predicted by no-slip theory of a Herschel-Bulkley fluid in squeeze flow (Covey and Stanmore 1981, Adams et al. 1994).     

Elastohydrodynamic lubrication in a plane slider bearing

Summary The present work deals with the case of a two-dimensional slider bearing with a rigid pad and an elastic bearing. Fluid viscosity is assumed to be only a pressure function. We determined the bearing deformation, the pressure distribution and the load capacity at different values of the inclination angle of the slider, with a numerical integration of the system consisting of the elasticity and Reynolds equations. The results show that, with an iso-viscous fluid, bearing elasticity causes a load capacity decrease. Instead bearing elasticity together with the variation of fluid viscosity due to pressure causes a load capacity greater than that of the iso-viscous case (=0).

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Sommario Il presente lavoro studia il problema della coppia prismatica lubrificata con pattino rigido di allungamento infinito e cuscinetto deformabile; si suppone che la viscosità del fluido sia funzione della sola pressione. Il sistema di equazioni, costituito dall'equazione di Reynolds e dall'equazione dell'elasticità, è stato risolto numericamente, determinando la deformazione del cuscinetto, andamento della pressione e la capacità di carico per diversi valori dell'inclinazione del pattino. I risultati dimostrano che, con fluido isoviscoso, la deformabilità del cuscinetto determina una riduzione della capacità di carico. Se si considera, invece, effetto combinato dell'elasticità del cuscinetto e della variazione della viscosità del fluido, la capacità di carico risulta maggiore di quella che si ottiene con fluido isoviscoso (=0).

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Nomenclature /L - /L - x/L - x/L - - ¯C CZ/h ₁ - E elasticity modulus - h film thickness - H elastic deformation of the bearing - h ₁ minimum film thickness - h ₂ inlet thickness - inclination of the pad - h Z/h ₁ - HZ/h ₁ - L pad length - viscosity - ₀ viscosity with no over-pressure - p over pressure - p P _ec-P _rc where:ec=elastic caserc=rigid case - P h ₁ ² /6₀VL - h ₂/h ₁=1+L/h ₁ - FV bearing velocity - W load capacity per unit width - Wh ₂ ¹ /6₀ VL ² - Z E h ₃ ¹ /12 ₀ VL ² A first version of this paper was presented at the 7th National AIMETA congress, held at Trieste, October 2–5, 1984. This work was supported by C.N.R.     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel