Reynolds Lubrication Equation Research Articles

Non-monotonic frictional behavior in the lubricated sliding of soft patterned surfaces.

We study the lubricated contact of sliding soft surfaces that are locally patterned but globally cylindrical, held together under an external normal force. We consider gently engineered sinusoidal patterns with small slopes. Three dimensionless parameters govern the system: a speed, and the amplitude and wavelength of the pattern. Using numerical solutions of the Reynolds lubrication equation, we investigate the effects of these dimensionless parameters on key variables such as contact pressure and the coefficient of friction of the lubricated system. For small pattern amplitudes, the coefficient of friction increases with the amplitude. However, our findings reveal that increasing pattern amplitude beyond a critical value can decrease the friction coefficient, a result that contradicts conventional intuition and classical studies on the lubrication of rigid surfaces. For very large amplitudes, we show that the coefficient of friction drops even below the corresponding smooth case. We support these observations with a combination of perturbation theory and physical arguments, identifying scaling laws for large and small speeds, and for large and small pattern amplitudes. This study provides a quantitative understanding of friction in the contact of soft, wet objects and lays theoretical foundations for incorporating the friction coefficient into haptic feedback systems in soft robotics and haptic engineering.

Modified Reynolds Equation for Thin Film Lubrication with Arbitrarily Enhanced Viscosity Surface Layers and Lubrication Analysis of Micro-Tapered Pad Bearing

Modified Reynolds Equation for Thin Film Lubrication with Arbitrarily Enhanced Viscosity Surface Layers and Lubrication Analysis of Micro-Tapered Pad Bearing

Mathematical Model and Theoretical Investigation of the Performance of Journal Bearing using a discretized Reynolds Lubrication Equation with Finite Width

Hydrodynamic journal bearings are widely used for a large number of applications such as rotating turbo machinery, axial flow compressors, axial flow turbines, and internal combustion engines. These types of journals are used to support high loads high speed rotating turbo machinery elements mainly shafts and attached components and also for enhancing the quality of these machines. In this work, the importance of journal bearing and an overview of the Reynolds hydrodynamic equation in three dimensions have been studied using the Boundary Value finite-difference method. The second-order nonlinear partial differential equation has been solved using numerical iterations approach with MATLAB software. The numerical solution of the transient Reynolds equation has been studied to investigate the Pressure distribution, pressure gradient as well film lubricant thickness on journal bearings. A numerical solution using ANSYS FLUENT has been applied to investigate the main parameters that have the main influence on the performance of the journal bearing and bearing performance.

Toward a continuum description of lubrication in highly pressurized nanometer-wide constrictions: The importance of accurate slip laws.

The Reynolds lubrication equation (RLE) is widely used to design sliding contacts in mechanical machinery. While providing an excellent description of hydrodynamic lubrication, friction in boundary lubrication regions is usually considered by empirical laws, because continuum theories are expected to fail for lubricant film heights h0 ≪ 10 nm, especially in highly loaded tribosystems with normal pressures pn ≫ 0.1 GPa. Here, the performance of RLEs is validated by molecular dynamics simulations of pressurized (with pn = 0.2 to 1 GPa) hexadecane in a gold converging-diverging channel with minimum gap heights h0 = 1.4 to 9.7 nm. For pn ≤ 0.4 GPa and h0 ≥ 5 nm, agreement with the RLE requires accurate constitutive laws for pressure-dependent density and viscosity. An additional nonlinear wall slip law relating wall slip velocities to local shear stresses extends the RLE's validity to even the most severe loading condition pn = 1 GPa and h0 = 1.4 nm. Our results demonstrate an innovative route for continuum modeling of highly loaded tribological contacts under boundary lubrication.

Generalized Reynolds equation for microscale lubrication between eccentric circular cylinders based on kinetic theory

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi,Phys. Rev. Fluids, vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers.

Higher order lubrication model between slip walls

A higher order lubrication model between slip walls is proposed for predicting the flow fields that cannot be described by the standard lubrication models based on the thin-gap approximation. The analysis shows that when considering the non-negligible pressure gradient in the surface-normal direction, the local pressure can be separated into (i) the base contribution by the modified Reynolds lubrication equation and (ii) the higher order component varying in both longitudinal and wall-normal directions, which takes the form proportional to the longitudinal derivative of the local velocity of the Couette–Poiseuille flow. For both (i) and (ii), the effect of the slip boundaries appears as the apparent displacements of the no-slip solid walls, and for (i) additional terms (to the no-slip case) also appear. The validity of the higher order slip-wall lubrication model is established by comparing the analytical prediction of the pressure with the fully resolved numerical results in a relatively wide region between a no-slip corrugated wall and a flat plate with varying slip length: the contribution of the higher order term is identified as the decreased lubrication pressure due to velocity slip. The model also successfully predicts the trend of pressure change between the varying slip case and a more realistic system with constant slip length for a channel, where the thin-gap approximation does not hold.

A Novel Method to Achieve Fast Multi-Objective Optimization of Hydrostatic Porous Journal Bearings Used in Hydraulic Turbomachine

AbstractThe hydrostatic journal bearing equipped with a carbon-fiber-reinforced carbon-based porous bushing is employed in the hydraulic turbomachine. The bearing exhibits high load capacity, but may unduly consume pressurized lubricant. This study aims to maximize the load capacity and minimize the feeding power. The journal radius, nominal clearance, porous bushing length, porous bushing thickness, feeding pressure, and material permeability are selected to optimize. A fast optimization method is proposed, integrating an in-house porous journal bearing solver (PBS), sampling method, surrogate model, and genetic algorithm. Behind PBS, a theoretical flow model based on the Reynolds lubrication equation and the Darcy equation is established, and a new numerical method based on the finite difference method is proposed. PBS substitutes ansysfluent by calculating bearing performances accurately and instantly, which is the first novelty to facilitate optimization. Then, artificial neural networks are trained as error-free and time-efficient surrogate models to produce bearing objectives in the evolution, which is the second acceleration highlight. The running time is reduced significantly. The load capacity is improved by 68.1%, whereas the feeding power declines by 50.5%. In the optimized case, a sharp pressure hump leads to greater load capacity, while the radial velocity decreases, resulting in reduced feeding power.

The Partial Derivative Method for Dynamic Stiffness and Damping Coefficients of Supercritical CO2 Foil Bearings

Supercritical CO2 foil bearings are promising bearing technology for supercritical CO2 high-speed turbomachinery. The partial derivative method including complete variable perturbation of the compressible turbulent lubrication Reynolds equation is effective to predict the frequency dependent dynamic stiffness and damping coefficients of supercritical CO2 bearings. In this research, the structural perturbation of foil dynamic model was introduced into this method and then the dynamic coefficients of supercritical CO2 foil bearings were calculated. The results of parametric analysis show that the structural loss factor has little influence on the trend of dynamic coefficients changing with the dimensionless support stiffness but mainly affects the value of stiffness coefficients as well as damping coefficients. Due to the turbulence effect, the bearing number is not able to directly determine the characteristics of supercritical CO2 foil bearings, which is different from air bearings. Compared to the bearing number, the influence of the average Reynolds number on the change of dynamic coefficients with dimensionless support stiffness is more obvious.

Dynamics of the capillary rise in tilted Taylor-Hauksbee cells

In this work, we theoretically and experimentally study the issue of the spontaneous capillary rise of a viscous liquid in wedge-shaped tilted cells, with very short angles of aperture, α. We provide the equilibrium profiles yse and, by means of the Reynolds lubrication equations, we find the time-dependent profiles and the dynamic evolution of the meniscus close to the edge of the wedge, as a function of time, which follow power laws of the form ys ∼ t 1/3 . Experiments performed at various inclinations are consistent with our theoretical results.

Analysis of the effect of texturing parameters on the static characteristics of radial rigid bore aerodynamic journal bearings

Surface texturing has been widely used to improve the performance of bearings. To further investigate the effect of surface morphology on rigid bore aerodynamic journal bearings and to improve the performance of bearings, this paper investigates the effect of surface roughness and surface texture on the static characteristics of radial rigid bore aerodynamic journal bearings using numerical analysis. The compressible Reynolds lubrication equation of the rigid bore aerodynamic journal bearing is solved by Successive Over Relaxation (SOR) and finite difference method. The static performance of the bearings with different parameters was investigated. The parameters that have been studied include the shape of the texture, the depth of the texture, the number of textures, the percentage of textures, and the surface roughness. The results show that radial rigid bore aerodynamic journal bearings with specific parameter textures have 36.32% higher load carrying capacity and 1.66% lower frictional torque compared to the case without texture.

A Three-Dimensional Slip Velocity Model for Water-Lubricated Hydrodynamic Journal Bearings

Hydrodynamic journal bearings, coated with polytetrafluoroethylene (PTFE) and lubricated by water, have been widely used in ships and large-scale pumps, and the function is to maintain the stability of rotor system. However, slip velocity exists on the PTFE-coated surface, whose effect is still an open question. This study aims to investigate the static characteristics of water-lubricated hydrodynamic journal bearings under three-dimensional slip velocity boundary conditions. Firstly, under the non-slip boundary condition, the CFD (computational fluid dynamics) method with ANSYS Fluent is verified based on the Reynolds lubrication equation and the open literature. Then, a three-dimensional slip velocity equation that is based on the Navier slip velocity boundary condition is proposed and embedded into Fluent. Finally, the effects of slip length on the static characteristics are analyzed. Under the same eccentricity ratio, with the increase in slip length, the load capacity decreases due to the decrease of the pressure circumferential gradient, and the friction power decreases. Under the same eccentricity ratio and the same slip length, with the increase in the attitude angle, the load capacity and friction power increase. However, under the non-slip boundary condition, the effects of attitude angle on the load capacity and friction power are insignificant. This paper could provide a reference for studying slip velocity in the hydrodynamic journal bearing.

Numerical analysis of coating thickness and pressure distribution in blade coating flows of power-law fluids

The shear stress of a power-law fluid is exponentially related to the shear rate. Thus, unlike for Newtonian fluids, the Reynolds lubrication equation in the simplified differential form is nonlinear and cannot be solved analytically. This paper investigates the flow of a power-law fluid in comma-roll coating and planar blade coating. We establish an intermediate function F(x), defined as the ratio of the shear force τ(X,0) in the X-direction on the edge of the blade coater to the pressure gradient G(x). Using various flow patterns, the signs of F and G are determined, and expressions for ∂U/∂Y are obtained in terms of F and G. Combined with the velocity boundary condition, ∂U/∂Y is integrated and the explicit relation between the coating thickness H∞ and F is established. The pressure gradient is also obtained. Finally, dP/dX is integrated and combined with the inlet and outlet pressure conditions, and the bisection method is used to determine H∞. The coating thickness and pressure given by the proposed method for both comma-roll coating and flat blade coating are highly consistent with the exact results for n = 1, 1/2, and 1/3 calculated by Dien and Elrod's method. We deduce that the limiting thickness of comma-roll coating is (n + 1)/(2n + 1), and the limit of the pressure gradient is −[n+1/n]n. Moreover, we construct a phase diagram of the relationship between the shape parameter k of the blade coater and the power-law exponent n. These results have instructive significance and application value for guiding coating operations and applicator design.

Research on static characteristics of elastic foil gas thrust bearing applied to aeronautical ACM

In air conditioning system for aircraft, air cycle machine (ACM) is the flimsy core component, which is impacted observably by the performance and reliability of elastic foil air-dynamic pressure bearing. It's necessary to have research on static characteristics of elastic foil air-dynamic pressure bearing. Based on finite element analysis (FEA) theory, it’s clearly obtained that discretized fluid lubrication Reynolds equation of foil dynamic pressure bearing by the weighted residual method. Coupled with the elastic deformation of foil calculated by beam element model, foil gas film thickness equation is used to work out static characteristics of elastic foil air-dynamic pressure bearing by iterative calculation. And then, it’s known how the basic parameters of the bearing impact its static characteristics. The result of the method in this paper, is verified that it’s correct by the experiment data in other relevant papers, and it’s appropriate to be used as guidance and reference for the further application of elastic foil bearing in ACM.

An Extended Reynolds Equation for Non-Newtonian Lubrication With the Upper Convected Maxwell Model

Abstract The upper convected Maxwell (UCM) model is one of the ways to describe lubricant’s viscoelasticity. In this paper, a new modified Reynolds equation for the UCM model is derived. This equation may be called extended as compared to its analogs because it incorporates the complete structure of the upper convected derivative without the explicit omitting of any term or using the perturbation technique. Then, a numerical scheme for solving the viscoelastic lubrication problems with the employment of the derived equation is described. A mixed Euler–Lagrange approach is used here: the constitutive rheological equations are resolved by a semi-Lagrangian technique and the extended Reynolds equation is discretized by the finite volume method. A constant surface slope problem is considered as a test case. Excellent agreement is achieved between the numerical solution at low Deborah number and one of the approximate solutions. The results of simulations with different types of time derivative used in Maxwell model both for two- and three-dimensional cases are also discussed.

Theoretical and Numerical Investigations on Static Characteristics of Aerostatic Porous Journal Bearings

To investigate the static characteristics of aerostatic journal bearings with porous bushing, the flow model—in which the compressibility of lubricating gas is considered—is established based on the Reynolds lubrication equation, Darcy equation for porous material, and continuity equation. With the finite difference method, difference schemes for non-uniform grids, relaxation method, and virtual node method, the numerical method for the governing equations of compressible flow in porous journal bearings is proposed. The effects of nominal clearance of bearings and compressibility of gas on the static characteristics are analyzed. Under the same minimum film thickness and the same gas compressibility, as the nominal clearance widens, the load capacity, mass flow rate, and power consumption increase. Under the same minimum film thickness and the same nominal clearance, with the increase in gas polytropic index, the load capacity strengthens, while the mass flow rate and power consumption decline. This study could provide a reference for the design of porous journal bearings.

Effect of texture geometry on the performance of noncircular aerodynamic journal bearings

Today, considerable uses of aerodynamic journal bearings have attracted the attention of many researchers in the field of tribology. Therefore, the theoretical and experimental analysis of their performance has been the subject of many researches. Growing application of the gas journal bearings is due to their capability in various engineering fields such as instrumentation and high-speed machines like spindles, small-scale turbomachines, and precision gyroscopes, dental milling machines, and many other rotor-bearing systems. Upgrading the performance of gas bearings by changing their geometry or using new compressible lubricants has always been one of the suggestions proposed by the tribology researchers. Recently, the development of new technologies such as laser milling and lithography increased the possibility of creating textured surfaces as a way to improve the performance of journal bearings. So, in this study, the effects of texture geometry on the steady-state performance of two-lobe noncircular aerodynamic journal bearings are presented. For this purpose, the governing Reynolds equation of hydrodynamic gas lubrication is analyzed by finite element method according to the changes of the lubricant film thickness in presence of the cubical, cylindrical, and ellipsoidal surface textures. Results show that the creation of textures on the lower lobe compared to the upper lobe or the whole surface is more effective on the performance of two-lobe aerodynamic bearings. Also, increasing both the dimple depth and the amount of bearings noncircularity, especially at shallow texture depth, cause more significant changes in the lubricant film pressure distribution and the bearing performance parameters such as load-carrying capacity, attitude angle, and frictional power loss. Further, it is obvious that the creation of cubical, cylindrical and ellipsoidal textures especially at shallow depth with an increase in the noncircularity of bearings’ geometry have a higher impact on the steady-state performance of the studied rotor-bearing system, respectively.

Lubrication pressure model in a non-negligible gap for fluid permeation through a membrane with finite permeability

The membrane permeation of a pure fluid driven by lubrication pressure is studied in the range of wall-membrane gap widths that cannot be treated by the Reynolds lubrication equation. This lubrication effect is modeled by including a higher-order pressure component described by the wall-tangential derivative of the local Couette-Poiseuille velocity. A permeate flux model is developed by renormalizing the Couette component, which shows divergent behavior, into the lowest order pressure component, and successfully predicts the permeate flux from infinitesimal to finite permeability.

Research on motion and friction of rolling piston in rotary compressor

Based on the Reynolds equation of mixed lubrication, the dynamic model of rolling piston was established, and the axis locus, rotation velocity, sliding velocity at vane tip, and friction loss of the piston were numerically solved. The analysis results indicated that the numeric results of sliding velocity at vane tip are in good agreement with the experimental results. In addition, the influences of crankshaft rotation speed, working condition, piston density and compression structure parameters on motion and friction were analyzed. Taking the ratio of piston rotation velocity and revolution velocity (λ) as evaluation index, whether λ conforms to cosine curve can judge the motion state between piston and vane. The results show that there is not only sliding state but also rolling state between piston and vane tip, and keeping rolling state is helpful to reduce friction loss. The design schemes of lighter piston, thicker vane and larger height-diameter ratio cylinder are proposed to keep rolling state in a wider rotation speed range and smaller pressure difference condition, thereby improving the annual operating efficiency of R32 rotary compressor.

Design a cooling pillow to support a high-speed supercritical CO2 turbine shaft

• High speed supercritical CO 2 turbine rotor shaft cooling zone. • Modeling a cooling pillow to cool the shaft and to provide dynamic support. • Design charts and figures for the cooling zone. • Rotor dynamic analysis through the Campbell diagram. • Trade-off analysis in designing the cooling zone. Supercritical CO 2 cycles offer a new thermal power generation paradigm but their commercial deployment has been slow due to significant turbine design challenges. Rotor bearings and seal design are identified as the critical challenges due to the high temperatures, high pressures, and high rotational speeds, a combination of extreme values not encountered in any other past turbine applications. This paper presents a new supercritical CO 2 turbine configuration that addresses these challenges by introducing a rotor shaft cooling zone to control the temperatures encountered by the seals and the bearings. This makes it possible to use proven conventional seal and bearing technology. The salient feature is referred to as a ‘cooling pillow’ that cools the rotor shaft to a tolerable seal temperature by circulating supercritical CO 2 fluid through the annulus and at the same time provides dynamic support to the rotating shaft against vibration. Turbulent heat transfer and frictional heat generation correlations for the annuli have been employed to evaluate the cooling performance and the simplified Reynolds equation for hydrodynamic fluid film lubrication has been employed to evaluate the dynamic performance. Design charts and figures are presented using non-dimensional parameters to help with the design of supercritical CO 2 turbines at different temperatures and sizes. The COMSOL rotor-bearing system simulator consisting of bearing and disks with the main shaft has been employed to produce Campbell plots and carry out a parametric study of rotor vibration frequencies varying with the cooling zone design. The cooling capacity is found to be significantly higher than the cooling load required. Moreover, the convection heat transfer coefficient and dimensionless temperature distribution are independent of radial clearance and this offers trade-offs in terms of controlling rotor vibrations. Through the pursuit of such trade-offs, it is demonstrated that dynamic performance can be improved by up to 55% and 60% for the 0.5 MW and 10 MW turbine sizes, respectively.

Effect of lubrication in the non-Reynolds regime due to the non-negligible gap on the fluid permeation through a membrane

To understand lubrication-induced membrane permeation, the effects of permeability and membrane geometry on lubrication pressure and permeate flux are studied in a range of a wall-membrane gap width wherein the effect of lubrication cannot be resolved using the Reynolds lubrication equation. The unresolvable lubrication effect (referred to as the non-Reynolds lubrication effect) is modelled by including a higher-order effect as the wall-tangential variation of the local Couette–Poiseuille velocity. Analytical prediction of the permeate flux is then validated with the fully validated numerical simulation. The result shows that, while the traditional Reynolds lubrication model underestimates the permeate flux, the permeation with the effect of the non-Reynolds lubrication is effectively improved in a small permeability range. Furthermore, the non-Reynolds lubrication model also enables reproduction of the characteristic variation in the permeate flux along the membrane. The effective range of the permeability for the non-Reynolds permeate model is discussed through the order analysis of the pressure terms in the Reynolds and non-Reynolds lubrication regimes.