Fully Coupled Thermo-Hydrodynamic Simulation Model for Journal Bearings

The isothermal form of Reynolds fluid film equation is used to predict the pressure generation in hydrodynamic journal bearings if temperature effects are neglected. Often, however, temperature effects may be important and cannot be neglected, because oil viscosity significantly varies with temperature. Also, thermal expansion of journal shaft and bearing housing must be taken into account since the bearing clearance changes with increasing temperature. Hence, the Reynolds pressure field equation, the energy equation for the fluid film and the heat transfer equations for journal and bearing housing have to be solved simultaneously. The coupled thermo-hydrodynamic fluid flow problem is mathematically defined by a system of nonlinear integro-differential equations. The governing equations are discretized and solved by a finite element approach. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

微极流体润滑在径向轴承中应用

本文从微极流体场方程出发,在润滑层的通常假设下,把它化简为两个独立的常微分方程组,并求得速度、微转动角速度的解析表达式.推导了微极流体润滑的雷诺方程,把它应用于有限长径向轴承的求解.通过数值计算得到了微极效应对各种动力参数、几何参数下轴承的压力分布、承载力、流量系数和摩擦系数的影响,并析了它的实际意义,使微极流体理论应用到工程问题又接近了一步.     

转动与摆动复合运动下脂润滑关节轴承的数值分析

建立了球面轴承的三维润滑模型,该模型将内圈的转动运动、轴颈倾斜引起的内圈倾斜和内圈的摆动运动等因素纳入考虑,推导出球坐标下适用于非Newton(牛顿)流体润滑的Reynolds(雷诺)方程.应用该模型,并考虑使用润滑脂的Ostwald流变模型,对向心关节轴承的润滑问题进行了数值计算,研究了在不同的幂律指数、内圈倾斜角度和摆动角速度下,脂润滑膜的压力分布、最大压力、承载力和流量.结果表明:在合适的操作条件下,脂润滑能产生明显的流体动压效应;在其它参数不变时,幂律指数对脂润滑膜的最大压力和承载能力影响显著,相对于Newton流体,剪切稠化流体可提高润滑膜的最大压力和承载能力,并增加周向流量,而剪切稀化流体的影响效果则相反;内圈倾斜角度对脂润滑膜最大压力和承载能力的影响较小,内圈摆动角速度的影响则较为明显.     

Effects of partial slip on a fourth-grade fluid with variable viscosity: An analytic solution

In this article we endeavour to analyse the flow and heat transfer characteristics in a fourth-grade fluid with variable viscosity. Two illustrative models of variable viscosity namely the Reynolds and Vogels are considered. The flow in a stationary cylinder is induced by a constant pressure gradient. Partial slip is taken into account at the wall of the cylinder. Analytic solutions of velocity and temperature are first developed and then discussed in each case.     

Junction between a 3-D Stokes flow and a 2-D thin film with pressure boundary conditions

A 3-D Stokes equation modelling the flow in the inlet region of a lubricated device is studied. Boundary conditions on both the pressure and the velocity are considered. The geometry induces that the flow region is divided into two parts: the first one is a thin gap with height ?h, where a change of scale is performed, and the second one is a cavity whose dimensions are large with respect to ?. Two equivalent variational formulations are proposed and lead to accurate a priori estimates. The resulting asymptotic equation for the pressure in the thin gap is the currently used Reynolds equation, whose boundary conditions are obtained via the junction with the 3-D cavity. Both hydrostatic and hydrodynamic cases are considered.     

Application of stabilized term in free boundary problems for optimizing bi-directional-rotation herringbone-grooved journal bearings

In order to deal with the free boundary problems occurred in cavitation region, this work incorporates an artificial term in Reynolds equation to stabilize the solution. The current result shows that adding the extra term gives accurate pressure distribution and benefits the selection of the groove shape and size. In addition, this study utilizes herringbone grooves on a reversible rotating journal bearing (RHGJB) and numerically analyzes the characteristics of miniature journal bearings with an inner diameter of 0.6 mm. Miniature journal bearings are limited in that the load capacity is significantly reduced with decreased bearing size, and therefore, the performances (load capacity and side leakage) of miniature RHGJBs are optimized. The optimum geometrical appearance of miniature RHGJBs is calculated by evaluating several groove parameters. Using the Taguchi parameter design methodology greatly reduces the number of full-factorial experimental design tests required to determine the optimal design parameters of the RHGJB from 5⁹ (1,953,125) to 450 runs.     

A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves

A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves under the cavitation boundary condition is proposed. The pressure distribution of the Reynolds equation of a finitely long journal bearing with axial grooves is expressed as a particular solution and a homogeneous solution. The particular solution and the homogeneous solution are separated, respectively, into an additive form and a multiplicative form by the method of separation of variables. The circumferential separable function of the homogeneous solution can be expanded on the basis of the infinite series of trigonometric functions. The pressure distribution of the particular solution is obtained by the Sommerfeld transformation. The termination positions of the fluid film are determined by the continuity condition. The analytical expressions for the nonlinear fluid film forces of a finitely long journal bearing with two axial grooves are obtained. The fluid film forces calculated by the proposed method agree well with the results obtained by the finite-difference method. The effects of the bearing parameters on the nonlinear fluid film forces are analyzed.     

Thermally Enhanced Motions of Variable-Inflow Surface Gravity-Driven Flows

In this paper, we report on theoretical and numerical studies of models for suddenly initiated variable-inflow surface gravity currents having temperature-dependent density functions when these currents are subjected to incoming radiation. This radiation leads to a heat source term that, owing to the spatial and temporal variation in surface layer thickness, is itself a function of space and time. This heat source term, in turn, produces a temperature field in the surface layer having nonzero horizontal spatial gradients. These gradients induce shear in the surface layer so that a depth-independent velocity field can no longer be assumed and the standard shallow-water theory must be extended to describe these flow scenarios. These variable-inflow currents are assumed to enter the flow regime from behind a partially opened lock gate with the lock containing a large volume of fluid whose surface is subjected to a variable pressure. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through a small opening under a lock gate at one end of a large rectangular tank containing the deep slightly more dense ambient fluid. Finding this time-dependent inflow velocity, which will then serve as a boundary condition for the solution of our two-layer system, involves solving a forced Riccati equation with time-dependent forcing arising from the surface pressure applied to the fluid in the lock.
The results presented here are, to the best of our knowledge, the first to involve variable-inflow surface gravity currents with or without thermal enhancement and they relate to a variety of phenomena from leaking shoreline oil containers to spring runoff where the variable inflow must be taken into account to predict correctly the ensuing evolution of the flow.     

MHD free convective flow through porous medium in the presence of hall current, radiation and thermal diffusion

An unsteady free convective flow through porous media of viscous, incompressible, electrically conducting fluid through a vertical porous channel with thermal radiation is studied. A magnetic field of uniform strength is applied perpendicular to the vertical channel. The magnetic Reynolds number is assumed very small so that the induced magnetic field effect is negligible. The injection and suction velocity at both plates is constant and is given by v ₀. The pressure gradient in the channel varies periodically with time along the axis of the channel. The temperature difference of the plates is high enough to induce the radiative heat. Taking Hall current and Soret effect into account, equations of motion, energy, and concentration are solved. The effects of the various parameters, entering into the problem, on velocity, temperature and concentration field are shown graphically.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Two classes of sub-optimal shapes for one dimensional slider bearings with couple stress lubricants

Shape optimization of slider bearings operating with couple stress lubricants is performed here for the first time by using a novel direct optimal control approach, which defines the gradient of the film height as a control. The bearing load is maximized. One dimensional Reynolds and energy equations are used. Several constraints are taken into consideration. They avoid the occurrence of cavitation and ensure the validity of the Reynolds equation. The model is validated against a known analytical solution (the Rayleigh step bearing). Two simple design rules are inferred, which yield two different classes of sub-optimal shapes: the multi-stepped bearings and the multi-sloped bearings, respectively. Multi-stepped bearings consist of several steps and the couple stress parameter may affect the constant value of the film height between steps. Multi-sloped bearings consist of several inclined regions and the couple stress fluid parameter may affect the constant value of the film height between regions. The slider bearings operation under variable load is stable. A sensitivity analysis identified the design parameters which have the highest impact on bearing performance. The optimal slider bearing shapes obtained for Newtonian lubricants do not change when most common couple stress fluids are used. Isothermal models may be used successfully at lower values of the couple stress parameter.     

The transition between the Stokes equations and the Reynolds equation: A mathematical proof

The Reynolds equation is used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. Using the asymptotic expansion in the Stokes equations, we show the existence of singular perturbation phenomena whenever the two surfaces are in relative motion. We prove that the Reynolds equation is an approximation of the Stokes equations and that the kind of convergence is strongly related with the boundary conditions on the velocity field.     

Numerical and experimental study of the flow in an eccentric Couette-Taylor system with small gap

Various papers were dedicated to the technical and industrial aspects of the Taylor–Couette flow, by analyzing the flow structure in dependence of characteristic parameters. Also some studies concerning the influence of an eccentrical position of the rotating inner cylinder relatively to the fixed outer cylinder exist in the case of a large gap width. The principal technical application of such studies is the flow in journal bearings where the gap width is very small, of the order 0.1\% relative to the inner cylinder radius. Analytical and numerical investigations of the flow in journal bearings are generally based on the two–dimensional Reynolds approximation. This investigation in aiming at a validation of the range where the Reynolds equation is applicable in relation to the characteristical parameters: gap width, eccentricity and Reynolds number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of variable properties on the unsteady Couette flow with heat transfer considering the Hall effect

The influence of temperature dependent viscosity and thermal conductivity on the transient Couette flow with heat transfer is studied. An external uniform magnetic field is applied perpendicular to the parallel plates and the Hall effect is taken into consideration. The fluid is acted upon by a constant pressure gradient. The two plates are kept at two constant but different temperatures and the viscous and Joule dissipations are considered in the energy equation. A numerical solution for the governing non-linear equations of motion and the energy equation is obtained. The effect of the Hall term and the temperature dependent viscosity and thermal conductivity on both the velocity and temperature distributions is examined.     

异质材料有限长微通道电渗流热效应

采用数值方法,分析有限长PDMS/玻璃微通道电渗流热效应．数值求解双电层的Poisson-Boltzmann方程,液体流动的Navier-Stokes方程和流-固耦合的热输运方程,分析二维微通道电渗流的温度特性．考虑温度变化对流体特性（介电系数、粘度、热和电传导率）的反馈效应．数值结果表明,在通道进口附近有一段热发展长度,这里的流动速度、温度、压强和电场快速变化,然后趋向到一个稳定状态．在高电场和厚芯片的情况下,热发展长度可以占据相当一部分的微通道．电渗流稳定态温度随外加电场和芯片厚度的增加而升高．由于壁面材料的热特性差异,在稳定态时的PDMS壁面温度比玻璃壁面温度高．研究还发现在微通道的纵向和横向截面有温度变化．壁面温升降低双电层电荷密度．微通道纵向温度变化诱发流体压强梯度和改变微通道电场特性．微通道进流温度不改变热稳定态的温度和热发展长度．     

Non-isothermal lubrication problem with Tresca fluid–solid interface law. Part II Asymptotic behavior of weak solutions

We study in this paper the asymptotic analysis of an incompressible Newtonian and non-isothermal problem, when one dimension of the fluid domain tends to zero. We prove the strong convergence of the unknowns which are the temperature, the velocity and the pressure of the fluid, we obtain the limit problem with the specific Reynolds equation, and we also prove the uniqueness of the limit temperature velocity and pressure distributions.     

Nonlinear reynolds equation for lubrication of a rapidly rotating shaft

Using the homogenization theory, we derive the nonlinear Reynolds equation governing the process of lubrication of a slipper bearing with rapidly rotating shaft. We prove that this nonliner lubrication law is an approximation of the full Navier-Stokes equations in a thin cylinder with periodic roughness. The analyticity of the nonlinear function giving the relation between the velocity and the pressure drop is proved. The first term in its Taylor's expansion is the classical linear Reynolds law. Boundary layer correctors are computed.     

Couple stress fluid flow with variable properties: A second law analysis

The present work examines the combined influence of variable thermal conductivity and viscosity on the irreversibility rate in couple stress fluid flow in between asymmetrically heated parallel plates. The dimensionless fluid equations are solved by using homotopy analysis method (HAM) and validated with Runge‐Kutta shooting method (RKSM). The convergent series solution is then used for the irreversibility analysis in the flow domain. The effects of thermal conductivity and viscosity variation parameters, couple stress parameter, Reynolds number, Grashof number, Hartmann number on the velocity profile, temperature distribution, entropy production, and heat irreversibility ratio are presented through graphs, and salient features of the solutions are discussed. The computations show that the entropy production rate decreases with increased magnetic field and thermal conductivity parameters, whereas it rises with increasing values of couple stress parameter, Brinkman number, viscosity variation parameter, and Grashof number. The study is relevant to lubrication theory.     

Reduction Approaches for Thermogasdynamic Lubrication Problems

Gas thrust bearings are often used in low-load applications, e.g. in air cycle machines, in micro gas turbines or in rotor systems for fuel cell applications, to support a shaft in axial direction. The pressure and temperature distribution in a gas thrust bearing pad are described by the generalized Reynolds equation according to Dowson and the 3D energy equation. In this paper, two different approaches are presented in order to reduce the dimension of the governing nonlinear integro-differential equation system and in order to stabilize the solution process. In the first reduction approach, the temperature in the fluid is averaged across the fluid film according to Lee and Kim. In the second approach, Legendre polynomials are used to approximate temperature, density and fluidity across the fluid film according to Elrod, Brewe and Moraru. The reduction techniques are compared with respect to numerical efficiency, accuracy and convergence behavior. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

螺旋槽球形轴承的承载能力

本文运用J·H·Vohr和C·H·T·Pan的理论和方法,建立了广义坐标系下的螺旋槽轴承的压力雷诺方程.然后在球轴承的边界条件下采用参数摄动法导出了动压螺旋槽球轴承润滑油膜的雷诺方程的近似解析解.由此,对各轴承槽型参数关于承载能力的影响作了计算和讨论,给出的最佳槽型参数与实验结果是一致的,与当前已发表的国内外资料相比较也是一致的.由于目前业已发表的文章均为计算机数值解,因此,本文对螺旋槽球轴承的特性研究提供了一个新的方法和途径.本文承中国科学院力学研究所林同骥,付仙罗同志及上海651研究所丁世德,蔡建中同志审阅,并提出了宝贵意见,作者谨在此表示衷心的感谢.