A new more consistent Reynolds model for piezoviscous hydrodynamic lubrication problems in line contact devices

Hydrodynamic lubrication problems in piezoviscous regime are usually modeled by the classical Reynolds equation combined with a suitable law for the pressure dependence of viscosity. For the case of pressure–viscosity dependence in the Stokes equation, a new Reynolds equation in the thin film limit has been proposed by Rajagopal and Szeri. However, these authors consider some additional simplifications. In the present work, avoiding these simplifications and starting from a Stokes equation with pressure dependence of viscosity through Barus law, a new Reynolds model for line contact lubrication problems is deduced, in which the cavitation phenomenon is also taken into account. Thus, the new complete model consists of a nonlinear free boundary problem associated to the proposed new Reynolds equation.Moreover, the classical model, the one proposed by Rajagopal and Szeri and the here proposed one are simulated through the development of some numerical algorithms involving finite elements method, projected relaxation techniques, duality type numerical strategies and fixed point iteration techniques. Finally, several numerical tests are performed to carry out a comparative analysis among the different models.     

A model for fluid flow in non-fully filled tribological interfaces – Part 1: Basics and Numerics

In literature, most contributions on starved lubrication focus on the occurring pressures in macroscopic devices. Hereby, usually the Reynolds equation is modified in different ways. In contrast to this proceeding, this paper's intention is the general investigation of this tribological regime to get a fundamental comprehension on the transition from boundary lubrication to mixed lubrication. The respective model describes the flow of the fluid through two rough surfaces moving relative to each other. The lack of fluid is regarded by the fact that elements may not be fully filled with the fluid. Only elements where the fluid fully fills the gap, generate a pressure. This effect is considered by a type of unilateral constraint in combination with a penalty function. The fluid flow is computed according to the Navier-Stokes equation. In combination with the continuity equation, a set of implicit nonlinear equations has to be solved. Its potential and basic application fields are finally discussed. A further paper will show applications of the algorithm towards different scenarios. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elrod–Adams cavitation model for a new nonlinear Reynolds equation in piezoviscous hydrodynamic lubrication

In previous studies, different cavitation models have been incorporated into the classical Reynolds equation in piezoviscous regimes. The advantages of the Elrod–Adams cavitation model compared with the Reynolds model have been demonstrated in this classical framework. Recently, a new nonlinear Reynolds equation was rigorously justified [15] for lubricated line contact problems by introducing the piezoviscous Barus law into the departure Navier–Stokes equations before passing to the thin film limit. In addition, the corresponding nonlinear first order ordinary differential equation (ODE) has been proposed.In the present study, we incorporate the Elrod–Adams model for cavitation and we pose the free boundary problem associated with the nonlinear first order ODE, which involves a multivalued Heaviside operator for the relationship between the lubricant pressure and saturation. After analyzing the qualitative properties of the solution, we propose suitable numerical techniques for solving the problem as well as obtaining the lubricant pressure, saturation, and viscosity. Finally, we give some numerical results to illustrate the performance of the proposed numerical methods as well as comparisons with alternative models.     

Solvability of the Reynolds Equation of Gas Lubrication

The first boundary-value problem for the Reynolds equation of the theory of gas lubrication with smooth data is studied. The existence and uniqueness of a solution are established. Bibliography: 7 titles.     

Effect of rotation in lubrication problems: Existence of more fundamental solutions

A generalized Reynolds equation is derived in the present paper by taking into account the effect of rotation in lubrication problems. The existence of certain fundamental solutions is shown in this extended framework which is not allowed in the classical Reynolds theory. Results concerning the pressure and the load capacity of the resulting bearing system are obtained and interpreted in the respective cases when the film thickness is a linear or an exponential functions of the coordinate along the bearing length. One of the important results is that while the load capacity decreases with increasing values of α for an exponentially inclined slider in the classical Reynolds theory, it increases with increasing values of α in the present context.     

SOR方法的拓展与应用

对流体润滑的压力控制方程,在有限差分法的基础上,通过对SOR超松弛因子和迭代精度的选择,采用SOR逐次超松弛迭代法对控制方程进行了数值求解.在保证方程求解精度的基础上,还具有收敛快、稳定性好,计算工作量小等特点.     

Indirect boundary integral formulation for the solution of shear‐thinning flow inside single rotor mixers

An efficient indirect boundary integral formulation for the evaluation of inelastic non‐Newtonian shear‐thinning flows at low Reynolds number is presented in this article. The formulation is based on the solution of a homogeneous Stokes flow field and the use of a particular solution for the nonlinear non‐Newtonian terms that yields the complete solution to the problem. Matrix multiplications are reduced in comparison to other means of handling nonlinear terms in boundary integral formulations such as the dual reciprocity method. The iterative solution of the nonlinear system of equations has been performed with a modified Newton‐Raphson method obtaining accurate results for values of the power law index as low as 0.4 without domain partitioning. Geometries such as Couette flow and a typical industrial polymer mixer have been analyzed with the proposed method obtaining good results with a reduction in computational cost compared with other equivalent formulations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1610–1627, 2011     

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.     

Initial-Boundary Value Problem for the Reynolds Lubrication Equation

The nonstationary Reynolds equation of the theory of gas lubrication is considered. The existence and uniqueness of a solution to the initial-boundary value problem for this equation are established in the case of sufficiently smooth data. Estimates for the solution are obtained for large bearing numbers. Bibliography: 8 titles.     

Some asymptotic results in hydrodynamic lubrication theory

Exact and asymptotic solutions to Reynolds equation for one-dimensional hydrodynamic lubrication are obtained for the case of rigid bearing surfaces and pressure-dependent viscosity.     

润滑理论中的广义雷诺方程和不等变分问题

本文运用张量分析工具和S-坐标系,推导了润滑理论中的广义雷诺方程及相应的不等变分问题,它计及了润滑流动中的弯曲效应,轴和轴瓦曲面内蕴性质对流动的影响.     

流变学流体的蠕动传输:食道中食物块的运动模型

研究食道中蠕动传输的流体力学.对任意的波形和任意的管道长度,建立起流变学流体蠕动传输的数学模型.用粘性流体的Ostwald-de Waele幂定律,描述非Newton流体的流动特性.解析公式化模型,详细且精确地给出食物块在食道中蠕动传输相关的一些重要性质.分析中应用了润滑理论,本研究特别适合于Reynolds数不大的情况.将食道看作环形的管道,通过食道壁周期性的收缩来传输食物块.就单个波和周期性收缩一组波的传播,研究与传输过程有关变量的变化,如压力、流速、食物颗粒轨迹以及流量等.局部压力的变化,对流变指数n有着高度的敏感性.研究结果清晰地表明,食物块在食道中蠕动传输时,Newton流体或流变学流体构成的连续流体,以组合波传播比大间隔单波传播,传输效率要高得多.     

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.     

A Suggested Mechanism for Nonlinear Wall Roughness Effects on High Reynolds Number Flow Stability

The interactions between an uneven wall and free stream unsteadiness and their resultant nonlinear influence on flow stability are considered by means of a related model problem concerning the nonlinear stability of streaming flow past a moving wavy wall. The particular streaming flows studied are plane Poiseuille flow and attached boundary-layer flow, and the theory is presented for the high Reynolds number regime in each case. That regime can permit inter alia much more analytical and physical understanding to be obtained than the finite Reynolds number regime; this may be at the expense of some loss of real application, but not necessarily so, as the present study shows. The fundamental differences found between the forced nonlinear stability properties of the two cases are influenced to a large extent by the surprising contrasts existing even in the unforced situations. For the high Reynolds number effects of nonlinearity alone are destabilizing for plane Poiseuille flow, in contrast with both the initial suggestion of earlier numerical work (our prediction is shown to be consistent with these results nevertheless) and the corresponding high Reynolds number effects in boundary-layer stability. A small amplitude of unevenness at the wall can still have a significant impact on the bifurcation of disturbances to finite-amplitude periodic solutions, however, producing a destabilizing influence on plane Poiseuille flow but a stabilizing influence on boundary-layer flow.     

两个非同心旋转圆柱间粘性流动的广义雷诺方程及其基本流

运用张量分析方法及修正双极坐标系,建立了轴承润滑流动所应满足的广义Reynolds方程.应用薄流层中的Navier-Stokes方程的渐近分析方法和张量分析工具,得到了两个非同心旋转圆柱之间粘性流动的基本流所应满足的方程.这个基本流可以表示为两个同心旋转圆柱之间的Taylor流加上一个扰动项,并且给出了数值计算例子.     

On a variational inequality on elasto-hydrodynamic lubrication

We consider the problem of a deformable surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The mathematical model consists of the Reynolds variational inequality with nonlocal coefficients given by an integral operator which depends on the fluid pressure. The nonlocal operator represents the deformation of the lubricated surfaces. The problem considers the vertical displacement of the elastic surface from its reference configuration. The goal of the paper is to obtain the range of these admissible displacements. We present general results for nonlocal coefficients with applications to particular problems in elasto-hydrodynamic lubrication.     

各向异性湍浮力模型的研究

通过对隐式代数应力模型应力结构的分析,认为湍浮力流动的应力由应变和浮力作用引起,从而利用应力输运的线性近似和平衡近似提出了非线性各向异性湍浮力模型·　该模型避免了数值奇异,其可靠性通过计算和实验的对比得到了验证·     

Wavelet-multigrid analysis of squeeze film characteristics of poroelastic bearings

The combined study of effects of poroelasticity and couple stresses on the performance of lubrication aspects of porelastic bearings in general and that of synovial joints in particular are analyzed. The modified form of Reynolds equation which incorporates the elastic nature of cartilage and Stokes couple-stress fluid as lubricant is derived and solved using a recently developed wavelet multigrid method. This method has greatest advantage of minimizing the errors using wavelet transforms in obtaining accurate solution as grid size tends to zero. It is found that, 6–7 cycles are required to obtain a reasonably accurate solution in the multigrid scheme, whereas, only one cycle is required to obtain the solution in the wavelet-multigrid method. Also, matrix of discrete wavelet-transform acts as a natural preconditioner producing rapid convergence. It is observed that, the poroelastic bearings with couple-stress fluid as lubricant provide enhancement in pressure and ensure the increased load carrying capacity compared with viscous fluids. This may be one of the reasons in the efficient lubrication and proper functioning of synovial joints.     

Φ—伪压缩型映象的具误差的lshikawa和Mann迭代程序的收敛性问题

引入Φ－伪压缩型映象的概论,并研究了此类映象的具误差的Ｉｓｈｉｋａｗａ和Ｍａｎｎ迭代程序的收敛性问题。本文结果改进和发展了许多人的最新结果。