Lubrication Equation Research Articles

Influence of Bearing Wear on the Stability and Modal Characteristics of a Flexible Rotor Supported on Powder-Lubricated Journal Bearings

This study has investigated the influence of journal bearing wear on the dynamic behaviour of a flexible rotor with a central disc. Rotors supported on journal bearings are susceptible to self-excited whirling, leading to unstable conditions. Prior knowledge of the stability limit speed is important to avoid the excessive vibration of rotating machines. For the study in this paper, journal bearings were lubricated with powder owing to high-temperature applications where conventional oil lubricants would fail to perform. The governing equations for lubrication were derived using a simplified grain theory based on the theory of dense gases. The rotor shaft was discretized considering Timoshenko beam elements. Modal analysis was conducted to obtain the system’s natural frequencies, mode shapes, damping factors, stability limit speed, and unbalance response. This study has also evaluated the influence of wear depth on the dynamic behaviour of the rotor shaft system and found that bearing wear significantly affects the stiffness and damping characteristics of lubricating film. Consequently, free and forced vibration behaviour is also affected. It has been found that increased wear depth improves stability limit speed but has little influence on the unbalance response.

Capillary Imbibition in a Diverging Flexible Channel.

We study the imbibition of a wetting liquid between flexible sheets that are fixed at both ends. Assuming a narrow gap between the sheets, we solve the lubrication equation coupled with a slender body deformation. When the sheets are parallel, we find that the deformation initially speeds up the flow, as shown in previous studies, but only up to the middle of the channel. As the channel contracts, the hydrodynamic resistance increases and ultimately slows down the filling process. Below a threshold stiffness, the channel collapses and imbibition stops. We propose a scaling of the filling duration near this threshold. Next, we show that if the sheets are initially tilted with a minimal angle, the channel avoids collapse. The liquid front pulls the diverging sheets and spreads in a nearly parallel portion, which maintains the capillary propulsion and enhances the wicking. Therefore, while it is established that diverging rigid plates imbibe liquids slower than parallel ones do, we show that elasticity reverses this principle: diverging flexible sheets imbibe liquids faster than parallel ones. We find an optimal tilt angle that gives the shortest filling time.

Asymptotic analysis of contact line dynamics of spreading/retracting drops on spherical surfaces

We investigate the spreading and retraction of a small sessile drop on a sphere governed by capillary and viscous forces. The lubrication equation established in spherical coordinates is solved analytically and numerically. The Navier slip model is adopted to overcome the singularity at the contact line. An asymptotic matching method is employed to study the contact line movement. The results show that the spreading process is always faster than the retraction process for a given drop volume. The position and speed of the contact line can be well-predicted using the asymptotic theory during the whole process of spreading and the late stage of retraction, while the theory becomes invalid at the early time of retraction because the macroscopic interface is significantly perturbed by the moving contact line.

Fluctuation-driven instability of nanoscale liquid films on chemically heterogeneous substrates

The instability and dewetting of bounded liquid films at the nanoscale are shown to strongly depend on thermal fluctuations. In this work, this fluctuation-driven instability of films on a chemically heterogeneous substrate is investigated both by a stochastic lubrication equation (SLE), which models thermal fluctuations with white noise, and by molecular dynamics (MD). Linear instability analysis for the SLE, considering slip and intermolecular forces of the heterogeneous substrates, is employed to derive a spectrum of the thermal capillary waves and their corresponding interface roughness, which are then quantitatively confirmed by numerical solutions for the SLE and MD. The fluctuation-driven instability is found to be enhanced by the slip and intermolecular forces, with the latter becoming dominant toward the final rupture. Moreover, the results suggest that a heterogeneous substrate can be approximated as a homogeneous one with effective (averaged) slip lengths and Hamaker constants for the intermolecular forces. The gradients of the slip and intermolecular forces due to the heterogeneities are shown not to affect the instability significantly.

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 hydro‐mechanically‐coupled XFEM model for the injection‐induced evolution of multiple fractures

AbstractIn this paper, a two‐dimensional eXtended Finite Element Method (XFEM) solution is presented for the hydro‐mechanically coupled hydro‐frac‐induced propagation of multiple fractures in rocks. Fractures are considered one‐dimensional objects, and the rock matrix is regarded as a two‐dimensional medium. Rules for improving the accuracy of the solution are included to deal with the transfer of variables between fractures and matrix in the coupling process. The main variables include fluid pressure and fracture aperture. The following assumptions are made: (1) the fluid pressure inside a fracture is applied as a net pressure; (2) fluid leak‐off is ignored; (3) the fluid front is regarded as a fracture front provided that the area of fluid lag has a small effect on fracture propagation compared to the area at which the fluid pressure is applied. The fluid flow is governed by the lubrication equation, while the XFEM is used to describe the behaviour of the elastic medium. The simulation results are compared with analytical solutions for a single hydraulic fracture model to verify and validate the proposed algorithm. A simulation with multiple hydraulic fractures is also performed to investigate the interference effect of multiple fractures and fracture propagation. The shadow effect caused by multiple fractures is analysed to manifest the stress variation along the propagation path.

Transient analysis for offset-elliptical journal bearings under misalignment and dynamic loading

Although the offset-elliptical journal bearings (OEJBs) are widely used in industry, there are few researches on the OEJBs, especially under the conditions of journal misalignment and sudden load. To overcome such deficiency, the dynamic response of misaligned OEJBs subjected to the dynamic load has been studied in this paper. Based on the Reynolds equation, Euler equation, and kinetic equations, a hydrodynamic model is developed and validated to conduct the transient analysis for the OEJBs. Moreover, the modified film thickness of the misaligned OEJB is given. The partial derivative method and the finite difference method are adopted to solve the lubrication equations. The correlations of bearing characteristics such as the minimum film thickness, maximum film pressure, journal orbits, and time-varying dynamic coefficients with the misalignment and dynamic load are studied. In addition, the effects of the misalignment, rotating speed, and clearance ratio on the transient performance of OEJBs under the dynamic load are discussed in detail. The results indicate that the journal misalignment and dynamic load have a significant impact on the transient performance of OEJBs. The decrease of the film thickness caused by the journal misalignment can be aggravated by the suddenly applied load. The proposed model can provide theoretical fundament in the design and monitoring of the dynamically loaded bearing-rotor system.

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.

Nonlinear dynamics of unstably stratified two-layer shear flow in a horizontal channel

The Rayleigh–Taylor instability at the interface of two sheared fluid layers in a horizontal channel is investigated in the absence of inertia. The dynamics of the flow is described by a nonlinear lubrication equation which is solved numerically for adverse density stratifications. The early-time dynamics features a number of finger-like protrusions of different heights at the interface. The fingers travel at different speeds leading to a sequence of merging events after which the interface eventually settles to a near-saturated state, comprising only one finger that includes most of the lower fluid. For sufficiently large density stratifications, the final state spans the height of the channel and includes two thin fluid films at each wall, both of which undergo chaotic dynamics, but finite-time touch-down/touch-up is shown to be precluded by the shear flow. An asymptotic analysis in the large-Bond-number limit (intense density stratification) reveals the finer structure of the final state including Landau–Levich-type connection regions. The asymptotic solutions are compared with numerical results of the lubrication model as well as direct numerical simulations, and excellent agreement is observed between the three in terms of interfacial structure, wave speed and film thicknesses.

Part I - Universal equation of non-Newtonian hydrodynamic lubrication

Part I - Universal equation of non-Newtonian hydrodynamic lubrication

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.

A faster optimal solver for thin film flows

A new, multigrid-based, algorithm is proposed for the solution of the discretized lubrication equations which are widely used to model a broad class of thin film flow phenomena. The approach is based upon the use of a Newton-Krylov solver for the nonlinear algebraic systems of equations that arise following mesh-based spatial discretizations and implicit time discretizations. The novel contribution is to propose a block-based preconditioner that includes two applications of algebraic multigrid (AMG) as a key component: thus allowing AMG techniques to be applied in the solution of thin film flow problems for the first time. An implementation of this preconditioned solver is demonstrated for a typical thin film scenario of free-surface flow down a non-smooth inclined plane, considering both steady-state and time-dependent configurations. Systematic computational testing is undertaken in comparison with two other state-of-the-art multigrid methods in order to demonstrate the substantial computational advantages of the proposed algorithm.

Modified Reynolds Equations for Thin Film Lubrication with Saturated High-Viscosity Surface Layer and Lubrication Analysis of Tapered Pad Bearing

Modified Reynolds Equations for Thin Film Lubrication with Saturated High-Viscosity Surface Layer and Lubrication Analysis of Tapered Pad Bearing

Experimental investigation for novel hybrid journal bearing with hydrostatic squeeze film and metal mesh damper in series

PurposeThis study aims to present a novel hydrostatic squeeze film-metal mesh journal bearing (HS-MMJB), which uses both hydrostatic squeeze film damper (HSFD) and metal mesh damper (MMD), to suppress the vibration of rotor-bearing systems.Design/methodology/approachThe lubrication equations were introduced to calculate the dynamic characteristics of HS-MMJB, and the response analyses of rotor systems were carried out. Experiments were conducted to study the vibration reduction of a rotor system with HS-MMJB. In addition, experiments for different oil supply pressures in the HS-MMJB were conducted.FindingsThe theoretical and experimental results show that the HS-MMJB exhibits excellent damping and vibration attenuation characteristics. Moreover, the stability of the rotor system can be improved by controlling the oil supply pressure.Originality/valueThere is a dearth of research on vibration characteristics of rotor system support by journal bearing combining HSFD and MMD. Moreover, the active oil pressure control is implemented to improve the stability of rotor system.

Elasticity can affect droplet coalescence

Coalescence of two droplets on a solid substrate is an interfacial phenomenon that imposes the challenges of capturing the complex contact line motion and energy interaction between the solid–liquid interface. Recent investigations on the coalescence of polymeric droplets on a solid substrate have reported strong disagreements; the heart of the issue is whether coalescence of polymeric drops is similar to that of Newtonian fluid and is independent of molecular relaxation, or whether the role of entanglement of polymeric chains leads to a transition kinetics different from that of Newtonian fluid. Via this article, we resolve the disagreements through a discussion on the effects of merging method on the dominant forces governing the coalescence process, i.e., inertia, dissipation, and relaxation. In this regard, two methods of merging have been identified, namely, the droplet spreading method and the volume filling method. Our study unveils that the coalescence dynamics of polymeric drops is not universal and, in fact, is contingent of the method by which the coalescence is triggered. Additionally, we demonstrate the spatial features of the bridge at different time instants by a similarity analysis. We also theoretically obtain a universal bridge profile by employing the similarity parameter in a modified thin film lubrication equation for polymeric fluids.

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.

Effects of temperature and relative humidity on resonant frequency of mems cantilever resonators under atmospheric pressure

In this study, the effects of temperature and relative humidity on the resonant frequency of a micro-electro-mechanical system (MEMS) cantilever resonator under atmospheric pressure (p=101325 Pa) are discussed. The squeeze film damping (SFD) problem of MEMS cantilever resonators is modeled by solving the modified molecular gas lubrication (MMGL) equation, the equation of motion of micro-cantilever, and their appropriate boundary conditions, simultaneously in the eigen-value problem. The effective viscosity (µeff(RH, T)) of moist air is utilized to modify the MMGL equation to consider the effects of temperature and relative humidity under atmospheric pressure. Thus, the effects of temperature (T) and relative humidity (RH) on the resonant frequency of MEMS cantilever resonators over a wide range of gap thicknesses and under atmospheric pressure are discussed. The results showed that the frequency shift increases as the relative humidity and temperature increase. The influence of relative humidity on the resonant frequency becomes more significant under conditions of higher temperature and smaller gap thickness.

Analysis for Hydrodynamic Wedge-Platform Thrust Slider Bearing with Ultralow Surface Separation

For the case of ultralow surface separation, in a hydrodynamic wedge-platform thrust slider bearing, the outlet zone and a portion of the inlet zone are in boundary lubrication, while most of the inlet zone is in the multiscale lubrication contributed by both the adsorbed boundary layer and the intermediate continuum fluid film. The present paper first presents the mathematical derivations for the generated pressure and carried load of this bearing based on the governing equation for boundary lubrication and the multiscale flow equation. Then, the full numerical calculation is carried out to verify the analytical derivations. It was found that the mathematical derivations normally have considerable errors when calculating the hydrodynamic pressure distribution in the bearing, owing to introducing the equivalent parameter λ bf , e which is constant in the inlet zone; however they can be used to calculate the carried load of the bearing when the surface separation in the outlet zone is sufficiently high. The study suggests the necessity of the numerical calculation of the hydrodynamic pressure and even the carried load of this bearing. It is also shown that owing to the fluid-bearing surface interaction, the pressure and carried load of this bearing are significantly greater than those calculated from the classical hydrodynamic theory.

Research on wear detection mechanism of cylinder liner-piston ring based on energy dissipation and AE

In the field of cylinder liner-piston ring (CL-PR) wear calculations, the traditional calculation models are difficult to use as a guide for wear detection. Therefore, a new analysis model of CL-PR wear based on energy dissipation theory is established in this paper. By using a mixed lubrication equation and simulation modification method, a new characterization formula of CL-PR wear is theoretically established. The maximum error between its calculated results and the simulation results is only 1.167%. The qualitative relationship between the wear volume and the frequency band energy of the acoustic emission (AE) signal is established based on the energy conversion relationship. The AE signal acquisition platform of CL-PR wear is built on the EV80 combustion engine, and the Bayes Shrink method is used to set the threshold of large-amplitude disturbance AE events for noise reduction. The wear signal is processed by wavelet packet energy spectrum decomposition, and the resulting wear energy spectrum can better characterize the change of wear characteristics. The BP (Back Propagation) neural network is trained with the characteristic parameters of the energy spectrum, and the classification accuracy of training and finally experimental results reach 98.6% and 100%, respectively, proving that AE technology can be applied to CL-PR wear monitoring combined with the neural network.

Numerical Simulation of Fracture Propagation during Refracturing

Hydraulic fracturing is repeated in some unconventional wells after production since the initial fracturing treatment. Due to prior production, the stress field around the existing fractures possibly rotates, and this impacts the refracturing operation. In this study, an extended finite element model (XFEM) including junction enrichments of intersecting fractures was proposed to simulate fracture propagation during refracturing in the cemented fractured reservoirs. In the XFEM model, a lubrication equation coupling both tangential and normal flow in hydraulic fractures (HFs) was used to describe the fluid flow behavior within the fractured elements, and the Newton-Raphson method was used to solve the nonlinear fluid–solid coupling system of the refracturing model. The effects of approaching angle, stress anisotropy, and production time were discussed. The results showed that the effects of these factors on improvement of fracture complexity during refracturing depend on the reservoir parameters and the stress field. The characteristics of the injection pressure curves during refracturing were analyzed.