Lubrication Equation Research Articles

Investigation of nonlinear dynamic behaviors of vertical rotor system supported by aerostatic bearings

Abstract A common issue associated with gas bearing-rotor systems is the tendency to generate self-excited vibrations, leading to instability. To address this problem, the fluid-structure coupling model of an aerostatic bearing-rotor system is established in this paper. Then a hybrid method combining the finite difference method and direct integration method, is employed to solve the bearing lubrication equation and rotor motion equation simultaneously. Furthermore, based on the orbit of rotor center,frequency spectrum diagram, Poincaré map, waterfall diagram and bifurcation diagram, the effects of rotational speed, rotor mass, orifice diameter and nominal clearance on the nonlinear dynamic behaviors of the bearing-rotor system are investigated. The results indicate that the system exhibits rich nonlinear behaviors with increasing rotational speed and rotor mass, including the occurrence of typical half-speed whirl. However, the nonlinear vibrations of the system can be restricted by selecting appropriate bearing structural parameters, providing theoretical guidance for the design of aerostatic bearing-rotor systems.

Linear stability theory and molecular simulations of nanofilm dewetting with disjoining pressure, strong liquid–solid slip and thermal fluctuations

The dewetting of thin nanofilms is significantly affected by thermal fluctuations, liquid–solid slip and disjoining pressure, which can be described by lubrication equations augmented by appropriately scaled noise terms, known as stochastic lubrication equations. Here molecular dynamics simulations along with a newly proposed slip-generating method are adopted to study the instability of nanofilms with arbitrary slip. These simulations show that strong-slip dewetting is distinct from weak-slip dewetting by faster growth of perturbations and fewer droplets after dewetting, which cannot be predicted by the existing stochastic lubrication equation. A new stochastic lubrication equation considering the strong-slip boundary condition is thus derived using a long-wave approximation to the equations of fluctuating hydrodynamics. The linear stability analysis of this equation, i.e. surface spectrum, agrees well with molecular simulations. Interestingly, strong slip can break down the usual Stokes limits adopted in weak-slip dewetting and bring the inertia into effect. The evolution of the standard deviation of the film height $W^2(t)={\overline {h^2}-{\bar {h}}^2}$ at the initial stage of the strong-slip dewetting is found to be $W\sim t^{1/4}$ in contrast to $W\sim t^{1/8}$ for the weak-slip dewetting.

An exact solution of the lubrication equations for the Oldroyd-B model in a hyperbolic pipe

An exact analytical solution of the lubrication equations for the steady, isothermal, incompressible flow of a viscoelastic Oldroyd-B fluid in a hyperbolic cylindrical contracting pipe is derived. The solution is valid for values of the Deborah number, De, up to order unity (De is defined as the ratio of the longest relaxation time of the polymer to the characteristic residence time of the fluid in the pipe), all values of the ratio of the polymer viscosity to the total viscosity of the fluid, η, and typical values of the contraction ratio, Λ, encountered in experiments and practical applications. It is provided in terms of the streamfunction only and is used in the momentum balance to derive a strongly non-linear ordinary differential equation of second order with unknown a function which corresponds to a modified fluid velocity along the main flow direction. The final equation is solved semi-numerically using a fully spectral (Legendre)-Galerkin approach to resolve the unknown function almost down to machine accuracy. The exact solution for the polymer extra-stresses, which is emphasized is not the full solution of the complete lubrication equations, allows for the derivation of a variety of theoretical expressions for the average pressure-drop along the pipe. In all cases, a decrease in the pressure drop compared to the Newtonian value with increasing De, η and/or Λ is predicted. The differences between the corresponding analytical solution for the planar geometrical configuration are also identified and discussed.

Fluctuation-driven dynamics of liquid nano-threads with external hydrodynamic perturbations

Instability and rupture dynamics of a liquid nano-thread, subjected to external hydrodynamic perturbations, are captured by a stochastic lubrication equation (SLE) incorporating thermal fluctuations via Gaussian white noise. Linear instability analysis of the SLE is conducted to derive the spectra and distribution functions of thermal capillary waves influenced by external perturbations and thermal fluctuations. The SLE is also solved numerically using a second-order finite difference method with a correlated noise model. Both theoretical and numerical solutions, validated through molecular dynamics, indicate that surface tension forces due to specific external perturbations overcome the random effects of thermal fluctuations, determining both the thermal capillary waves and the evolution of perturbation growth. The results also show two distinct regimes: (i) the hydrodynamic regime, where external perturbations dominate, leading to uniform ruptures, and (ii) the thermal-fluctuation regime, where external perturbations are surpassed by thermal fluctuations, resulting in non-uniform ruptures. The transition between these regimes, modelled by a criterion developed from linear instability theory, exhibits a strong dependence on the amplitudes and wavenumbers of the external perturbations.

Finite domain solution of a hydraulic fracture in a permeable rock

In this work, we present a domain-based algorithm to simulate the propagation of a plane-strain hydraulic fracture in a zero-toughness permeable elastic medium. The algorithm utilizes a domain-based method to solve the elasticity equations and integrates a multi-scale tip asymptote, which is particular to hydraulic fractures, into this framework. This integration is key to accurately model the energy dissipation and the fluid leak-off in the fracture tip region. The algorithm combines a 2D finite volume method (FVM) for solving the elasticity equation with a 1D FVM for solving the nonlinear lubrication equation. Incorporating the far-field asymptotics and using a moving-mesh scheme reduces the computational burden while improving the accuracy of the scheme. The paper concludes with an analysis of the numerical results. This study demonstrates the potential of this domain-based approach for modeling hydraulic fractures in poroelastic media.

Optimization of lubrication characteristics of wind turbine’s transmission system based on Newton Raphson method

To improve the reliability and lifespan of wind turbines, this paper takes the two-stage fixed shaft gearbox experimental platform of wind turbines as the research object. Based on Hertz contact theory, the oil film pressure and thickness in the contact area are solved by combining the equations of elastohydrodynamic lubrication and the Newton Raphson method; And the lubrication characteristics of the transmission system were analyzed to verify the correctness of the method; At the same time, in response to partial load phenomenon caused by system coupling deformation, genetic algorithm was selected to modify the gear teeth. The results show that the max unit load on the tooth face and the maximum stress of tooth root decreased by up to 26.48 % and up to 20.35 % respectively after modification which can improve the uneven distribution of oil film and the lubrication performance of the tooth surface.

Effects of humidity and temperature on quality factor of micro-beam resonators in atmospheric pressure and gas rarefaction

At atmospheric pressure (p=101325 Pa), the effects of humidity and temperature on moist air become important when discussing the quality factor of micro-cantilever and micro-bridge resonators. The squeeze film damping (SFD) problem, the dominant damping source for micro-beam resonators, is modelled using the modified molecular gas lubrication (MMGL) equation with finite element modelling (FEM) in the eigenvalue problem. The MMGL equation is modified with the effective viscosity of moist air (μeff) to account for the effects of humidity and temperature. Other damping sources, such as thermoelastic damping (TED) and the support loss of micro-beam resonators, are also calculated. The quality factor of micro-beam resonators is then discussed over a wide range of temperatures and relative humidity levels at atmospheric pressure and gas rarefaction. The results show that the quality factor of micro-cantilever and micro-bridge resonators increases as both humidity and temperature rise in atmospheric pressure and gas rarefaction. Furthermore, the quality factor of a micro-bridge resonator with changes in humidity and temperature is significantly higher than that of a micro-cantilever resonator in atmospheric pressure and gas rarefaction.

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

Optimal metabolite transport in hollow fibre membrane bioreactors

Hollow fibre membrane bioreactors provide a fast and efficient method for engineering functional tissue for use in medical treatments. Flow is utilised to overcome mass transport limitations by perfusing a nutrient-rich culture medium through the fibre lumen, which can then transport along the fibre lumen or across the porous membrane wall. Cells seeded at the outer membrane wall consume the nutrient and subsequently produce waste metabolites, which are transported away through an external extra-capillary space (ECS) along with excess nutrient. We present and investigate a two-dimensional axisymmetric model for fluid flow and solute transport through a single-fibre bioreactor configuration, with cells seeded to the external fibre wall. Fluid flow is modelled by steady lubrication and Darcy equations, which are coupled to the solute transport problem modelled by a system of advection–diffusion equations, supplemented with a reaction term to model the cell layer. Our model analysis reveals how spatially varying wall permeability distributions can be utilised to provide uniform nutrient delivery to a spatially uniform, homogeneous cell population. We also reveal how maximising the transmural pressure drop across the membrane wall is the dominant mechanism for waste removal rather than traditional experimental methods of flushing the ECS.

The modified lubrication equation

The modified lubrication equation

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.

Lubrication dynamics of a settling plate

If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power law: $h\sim t^{-1/2}$ . We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex. We find that, depending upon the initial position of the plate and the position of the centre of force, the plate might either make contact in finite time or settle progressively without ever making contact. Our results show an excellent agreement between analytical exact solutions, asymptotic solutions and numerical studies of the lubrication equations.

Generation of microbubbles via a tapered capillary

We propose a novel method for efficient production of microbubbles based on a tapered capillary with an interiorly attached filament. When gas–liquid displacement driven by an input pressure occurs in the capillary, the gas cone ruptures close to the orifice of the capillary. The generated microbubbles can be pushed out of the capillary and collected by a liquid tank when the pressure is appropriately selected. A liquid column is employed in the straight part of the capillary, which can sustain the liquid film near the capillary orifice and hence the bubble generation by transporting liquid along the filament. Within the working pressure range, increasing the input air pressure leads to a decrease in the microbubble diameter. The minimum diameter of the microbubbles is approximately equal to the orifice diameter of the tapered capillary. In our experiments, microbubbles with a minimum diameter of 1.56 μm can be realized. Theoretically, we derive a one-dimensional unsteady lubrication equation describing the evolution of the gas–liquid interface in a tapered tube. The bubble pinch-off is justified by the numerical solution of the lubrication equation. In particular, the predicted bubble diameters are in agreement with the experimental measurements.

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.

Non-axisymmetric modes in ultrathin annular liquid films coating a cylindrical fiber

This paper presents a detailed analysis of the three-dimensional stability properties of an annular liquid film coating a cylindrical fiber in the presence of van der Waals (vdW) interactions, whose influence depends on the wettability of the solid by the liquid. Under wetting conditions, vdW interactions can stabilize a uniform annular film when its thickness is smaller than a critical value that depends only on the fiber radius, the Hamaker constant, and the surface tension coefficient [Quéré et al., Science 249(4974), 1256–1260 (1990)]. In contrast, under non-wetting conditions, both surface tension and vdW forces contribute to destabilize the interface, and non-axisymmetric modes may become dominant depending on the thickness of the film and the relative strength of the surface tension and vdW forces. We perform temporal stability analyses of both the Stokes and lubrication equations of motion, allowing us to reveal the dominant azimuthal mode, as well as the optimal axial wavenumber and the corresponding temporal growth rate, as a function of the relevant governing parameters.

Research on the Secondary Motion of Engine Piston Considering the Transport of Lubricating Oil

<div>At present, it is generally considered in the analysis of the secondary motion of engine piston that the piston skirt–cylinder liner friction pair is fully lubricated in an engine operating cycle. However, in practice, when the piston moves upward, the amount of lubricating oil at the inlet may not ensure that the friction pair is fully lubricated. In this article, the secondary motion of piston is studied when the transport of lubricating oil is considered to determine the lubrication condition of piston skirt–cylinder liner friction pair. The secondary motion of piston is solved based on the combined piston motion model, hydrodynamic lubrication model, asperity contact model, and lubricating oil flow model. The secondary motion equation of piston is solved by the Broyden method. The hydrodynamic lubrication equation is solved by the finite difference method. The asperity contact between piston skirt and cylinder liner is calculated by the Greenwood model. The flow of lubricating oil is analyzed based on the theory of fluid mechanics. The results indicate that, when the actual transport of lubricating oil is considered to determine the lubrication condition of piston skirt–cylinder liner friction pair, the secondary motion of piston is remarkably different from that in which the flooded lubrication is assumed in an engine operating cycle. Therefore, it is helpful to improve the accuracy and make the analysis closer to the actual engine operating situation that the transport of lubricating oil is considered in the analysis of the secondary motion of engine piston.</div>

Finite domain solution of a KGD hydraulic fracture in the viscosity-dominated regime

This paper describes a numerical algorithm for solving the classic problem of a plane strain (KGD) fracture propagating in an impermeable elastic medium with zero toughness. The method, which takes advantage of the self-similar nature of the solution, combines a domain-based scheme to solve the elasticity equations and a finite volume method to solve the nonlinear lubrication equation. This work represents a first step towards developing a model able to account for pore pressure diffusion in the medium and corresponding poroelastic effects, noting that these processes are more efficiently solved using a domain-based rather than a boundary integral method. To enhance the efficiency and accuracy of the numerical scheme, the far-field crack asymptotics is embedded in the discretized elastic relationship between the fluid pressure and the crack opening, while the coupled fluid-solid tip asymptote is enforced in a weak form when solving the nonlinear lubrication equation. The proposed technique yields results that closely match the analytical solution, even with a coarse mesh. This approach offers potential for addressing more complex hydraulic fracturing problems in the future.

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.

Numerical solution of dry friction in point contact using the unified Reynolds method combined with rarefied gas effect

Dry friction problems are widespread in engineering applications. However, the current research on the contact pressure between interfaces mainly relies on static contact mechanics solutions, neglecting the influence of the contact pressure generated by the gases in the environment. In this paper, the ambient gas is considered as a lubricant with lubricating effects. Drawing inspiration from hydrodynamic lubrication methods, a unified mixed lubrication equation for dry friction problems is developed, taking into account the influence of gas rarefaction effects. This study computes the dry friction contact pressure under different winding speeds and loads. The advantage of this method lies in its ability to automatically distinguish between the contact and noncontact regions during the calculation process, enabling the determination of the contact pressure over the entire contact area. The computational results demonstrate that at low entrainment velocities, there is minimal deviation in contact pressure and contact area compared to Hertzian contact. However, as the entrainment velocity increases, the actual pressure-bearing area enlarges compared to static contact, and there is a smooth transition of pressure at the contact edge, which cannot be obtained from static contact analysis. Finally, the numerical solution of the contact pressure when the sliding speed spans several orders of magnitude is given, and the calculation results show that the numerical model has good robustness. This numerical approach offers valuable insights for guiding the design of air bearings in practical applications.

Knudsen Maximum Effect in Micro-Scale Gas Lubrication

Abstract Micro-scale gas lubrication has been studied for several decades, with extensive research on non-equilibrium flow effects like velocity slip and thermal creep. However, the Knudsen maximum effect in micro-scale gas lubrication has not been reported yet. To address this, we analyzed the load capacity characteristics of slider bearings with ultra-thin film gas lubrication equations derived from the linearized Boltzmann-Bathnagar–Gross–Krook model equation (Fukui–Kaneko lubrication model) under the condition of constant bearing number. Our study reveals that there exists a maximum value of load capacity for slider bearings when the reference Knudsen number is about unity. This happens because the dimensionless mass flowrate of micro-scale gas flows has a minimum value when the reference Knudsen number approaches unity. Understanding the Knudsen maximum effect is crucial when designing micro-nano devices related to gas lubrication, as it implies that there exists an optimum clearance for maximum load capacity.