Lubrication Equation Research Articles

Oil film boundary analysis of spiral oil wedge sleeve bearing based on the dynamic loading conditions

The area and location of cavitation change with time in the condition of dynamic loading. The cavitation location of sleeve bearing not only depends on axis location and velocity at a certain moment but also is affected by cavitation history, so the boundary condition need to be defined in the solution of lubrication equation. Based on the balance of axial inertia force, oil film force, and dynamic loading, the motion equation, generalized Reynolds equation and oil film thickness equation of spiral oil wedge sleeve bearing are established. The results show that the cavitation location and carrying capacity vary periodically with the change in time. The effect of dynamic loading and rotational speed on the oil film cavitation and carrying capacity is studied.

Flow of a thixotropic or antithixotropic fluid in a slowly varying channel: The weakly advective regime

A general formulation of the governing equations for the slow, steady, two-dimensional flow of a thixotropic or antithixotropic fluid in a channel of slowly varying width is described. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. We demonstrate how the lubrication equations can be further simplified in the weakly advective regime in which the advective Deborah number is comparable to the aspect ratio of the flow, and present illustrative analytical and semi-analytical solutions for particular choices of the constitutive and kinetic laws, including a purely viscous Moore–Mewis–Wagner model and a regularised viscoplastic Houška model. The lubrication results also allow the calibration and validation of cross-sectionally averaged, or otherwise reduced, descriptions of thixotropic channel flow which provide a first step towards models of thixotropic flow in porous media, and we employ them to explain why such descriptions may be inadequate.

Effect of gas rarefaction on the quality factors of micro-beam resonators

The external squeeze film damping (SFD) of microelectromechanical systems (MEMS) resonators is a dominant factor to lower the quality factor (Q-factor) due to their large surface area to volume ratio and small spacing. To improve the Q-factor of MEMS resonators, the effect of gas rarefaction (low gas ambient pressure in thin gas film thickness) or operating in higher mode should be considered in SFD analysis. The modified molecular gas lubrication (MMGL) equation is applied for modeling the SFD with gas rarefaction effects taken into consideration. The effects of inverse Knudsen number, surface accommodation coefficients (ACs) and operating frequency on SFD are discussed. The combined effects of SFD, thermoelastic damping (TED) and anchor loss on the total Q-factors of MEMS resonators are considered. The contribution of SFD on the total Q-factor (weighting of SFD) is also discussed. The results show that weighting of SFD could be decrease at low inverse Knudsen number or low ACs or operating at high resonant frequencies.

Reductions of PDEs to second order ODEs and symbolic computation

A new method to obtain second-order reductions for ordinary differential equations which are polynomial in the derivatives of the dependent variable is presented. The method is applied to obtain reductions and new solutions to several well-known equations of mathematical physics: a lubrication equation, a thin-film equation, the Zoomeron equation and a family of 5th−order partial differential equations which includes the Caudrey–Dodd–Gibbon–Sawada–Kotera, Kaup–Kupershmidt, Ito and Lax equations. Some pieces of computer algebra code to derive the reductions are also included.

An isogeometric analysis approach for solving the Reynolds equation in lubricated piston dynamics

This paper discusses the approach of isogeometric analysis (IGA) for solving the Reynolds equation in lubricated piston dynamics. The simulation model consists of two parts: the differential-algebraic equations (DAE) in piston dynamics and the Reynolds equation for lubrication of piston-cylinder interface. Here, the transient positional states of piston governed by the DAE are calculated in Dymola while the friction forces together with lateral forces and their moments are computed by solving the Reynolds equation with the IGA approach in Matlab; two parts of computation are integrated through a real-time communication program. The results show that the efficiency is improved remarkably than the traditional Finite Element Method (FEM) when the IGA approach is adopted.

2D coupled HM-XFEM modeling with cohesive zone model and applications to fluid-driven fracture network

The present work focuses on a new numerical model for the fully coupled hydro-mechanical analysis of groundwater flows through poroelastic saturated media. In particular, the presence and eventual propagation of fluid-driven fractures is accounted for within a non-regularized cohesive zone model. In this paper, the fracture propagation is considered as a reactivation process: the fracture already exists and evolves (i.e. opens or closes) on a pre-defined path initially constrained. The Talon-Curnier constitutive law is considered for the fracture interfaces and its expression has been adapted to the hydro-mechanical coupling related to the fracture evolution. The fluid pressure inside the fracture is governed by the lubrication equation. The momentum-stress balance equations involving fluid flow and deformation of the solid porous matrix are derived within the framework of the generalized Biot theory. The extended finite element method (XFEM) is preferred to a standard finite element spatial discretization in order to easily handle the presence and evolution of discontinuities in the porous medium. A set of four Lagrange multipliers is introduced to prevent spurious oscillations of the numerical solution at the interface. Comparisons between numerical results and theoretical solution assess the validity of the model presented in this paper. In addition, the hydro-mechanical interactions between neighboring fractures and the effects of the permeability of the porous medium are investigated. We also demonstrate the capability of our model to handle non-planar fracture paths.

Numerical investigation of pocketed slip slider bearing with non-Newtonian lubricant

Boundary slip as well as surface texturing is an effective method to improve the tribological performance of lubricated mechanical components. This article analyzes the combined effect of single texturing (pocketing) and wall slip on pressure that strongly related to the load-carrying capacity of slider bearing. The modified Reynolds equation for lubrication with non-Newtonian power-law fluid is proposed. The equation was solved numerically using a finite difference equation obtained by means of the micro-control volume approach. Further, numerical computations for slider bearing with several power-law indexes were compared with the presence of the pocket and slip. The numerical results showed that the characteristic of non-Newtonian is similar to Newtonian fluid with respect to hydrodynamic pressure distribution. The maximum load support is achieved when the pocket depth is equal to the film thickness.

Thermo-molecular gas-film lubrication dynamics considering boundary temperature and ambient gas (2-DOF numerical analysis by t-MGL equation)

The static and dynamic flying characteristics of a slider in a hard disk drive have become an important consideration owing to recent increases in recording density. In the present paper, the characteristics of a plane inclined slider (Case 1) and a step slider flying in either air or He (Case 2) over a running boundary wall with local temperature distributions are analyzed using the thermo-molecular gas-film lubrication (t-MGL) equation. For a plane inclined slider (Case 1), the fundamental static and dynamic characteristics are analyzed numerically and are examined through two limiting approximations: the approximation for infinite bearing number and the incompressible short bearing approximation. For a step slider (Case 2), the decreases in the minimum spacing for a slider flying in He are significant because the mean free path of He, ?He, is approximately three times that of air, ?air. The increases in the minimum spacing due to laser heating are negligible in both air and He because the heat spot size is very small. Moreover, the decrease in the minimum spacing produced by thermal deformation (projection height, dmax) by laser heating in the thermal fly-height control slider is reduced by the total additional pressure of (1) MGL pressures produced by the air-film wedge effect, (2) t-MGL pressures produced by the applied temperature distribution, and (3) van der Waals attractive pressure due to the ultra-small spacing. The spacing fluctuation in He caused by a running wavy disk is smaller than that in air, because the inlet-to-outlet spacing ratio (h1/h0) in He is larger than that in air.

An Accurate Analytical Squeeze-film Model for Lateral MEMS/MOEMS Oscillators

We report on an accurate analytical model for the squeeze-film damping in laterally oscillating MEMS and MOEMS, which has not been available up to now. The models currently used are not able to correctly predict the damping and underestimate the squeeze- film contribution by a quite large factor of ∼4. This discrepancy was overcome by choosing more appropriate boundary conditions for solving Reynold's lubrication equation which governs the behavior of thin fluid films. The model was tested against numerical FVM computations with the open source software package OpenFOAM as well as measurements of MOEMS test devices with varying width of the squeeze-film gap and is in good agreement with both. This can now be used to increase the efficiency in thedesign step not only of laterally moving microstructures.

Method of investigating the elastohydrodynamic contact in slide bearings with compressible and incompressible lubrication

An algorithm of solving the problem of elastohydrodynamic contact is presented in the paper. It is used to calculate characteristics of journal and thrust bearings with oil or gas lubrication. The algorithm is based on simultaneous solution of the problem of lubrication flow in the gap created by the bearing sliding surfaces and the problem of specifying the change in the gap shape due to the shift of position and deformations of sliding surfaces caused by lubrication pressure in the gap. Modifications of Reynolds equation describing incompressible (oil) and compressible (gas) lubrication flow in journal and thrust bearing are used for the calculation of bearing characteristics. The finite-element method is used to solve the Reynolds equation. Lagrange multipliers are used to ensure closedness of the rectangular region for a journal bearing.  A step-by-step process with error self-correction at each step is applied for solving the nonlinear Reynolds equation for compressible lubrication. The shape of the gap in bearings with incompressible lubrication is defined by calculating the equilibrium pad position in a bearing taking into account the deformations of siding surfaces and bearing parts under the action of lubrication pressure. The shape of the gap in gas bearings is defined by the deformations of elastic foils. These deformations are calculated when solving the problem of deformation and contact interaction of the foils with each other and with the bearing housing under the action of lubrication pressure. The presented method of calculation of elastohydrodynamic contact in slide bearings allows taking into account the design features of physical products for the analysis of rotor supports characteristics.

A comparative study of the preload effects on the stability performance of noncircular journal bearings using linear and nonlinear dynamic approaches

In this work, the effects of preload factor on the stability performance of noncircular two and three lobe journal bearings with incompressible micropolar lubricant are presented based on the linear and nonlinear dynamic models. Assuming that the rotor is rigid, the governing Reynolds equation for the hydrodynamic lubrication of finite length lobed journal bearings has been modified using micropolar theory. Then, the linear and nonlinear dynamic models which include a certain harmonic disturbance and a time dependent trajectory of rotor center are applied to analyze the stability performance of the considered bearings. The generalized differential quadrature method and fourth-order Runge–Kutta technique have been used to solve the governing Reynolds equation and time-dependent dynamic equations of rotor motion, respectively. Finally, the numerical results for the critical mass parameter and the whirl frequency ratio of the rotor as the stability characteristics of the lobed bearings are obtained for different values of preload factor and are compared together. Results show that the dynamic stability of the noncircular bearings is enhanced by decreasing the preload factor i.e. increasing the amount of bearing noncircularity. These enhancements are in terms of increase of the critical mass parameter and decrease for the whirl frequency ratio. Also, by comparing the two dynamic analysis methods, it is seen that the results of linear model are more conservative in different investigated cases. The results of nonlinear analysis reveal that by changing the preload factor the dynamic responses of the lobed bearings appear in different manners. From the responses, it is observed that the type of dynamic trajectory of rotor center varies from stable fixed point to limit cycle periodic motions and also contact between rotor and bearing’s shell.

A robust piston ring lubrication solver: Influence of liner groove shape, depth and density

A multigrid code using Alcouffe׳s [1] ideas has been written to solve the hydrodynamic lubrication equation for the piston ring–cylinder liner contact with a textured liner. The new program shows good convergence, even for deep grooves. Results are compared with a simplified 1D analytic model proposed by Biboulet [2]. Then a 2D analytic model of cross hatched grooves with a small angle is studied and compared with the 1D analytic model.

An Investigation on the Influence of Misalignment on Micro-pitting of a Spur Gear Pair

This study investigates the impacts of the angular misalignment that is tangent to the plane of action of spur gears on the contact fatigue life of micro-pitting crack nucleation using a computational approach. The governing equations of the mixed elastohydrodynamic lubrication are solved to determine the surface traction distributions. A boundary element-based near-surface stress formulation is implemented for the computation of the multi-axial stress fields, capturing the influence of the local surface roughness topography on the stress concentrations. A fatigue criterion is then used to assess the fatigue damage. The method of circular crown in the lead direction of the gear pair is proposed to alleviate the tooth surface load and lengthen the fatigue life under the misalignment condition. It is shown there exists an optimized crown magnitude, which is dependent on both the misalignment amount and the input torque of the gear pair, for the fatigue life improvement.

A General Finite Volume Method for the Solution of the Reynolds Lubrication Equation with a Mass-Conserving Cavitation Model

This contribution presents the development of a general discretization scheme for the solution of Reynolds equation with a mass-conserving cavitation model and its application for the numerical simulation of lubricated contacts to be discretized using irregular grids. Such scheme is based on a hybrid-type formulation, here named as element-based finite volume method that combines the flexibility of the FEM to deal with unstructured grids, while preserving the local and global fluid-flow conservation aspect of the FVM throughout the discretized domain. The accuracy and robustness of the method are successfully tested using several benchmark cases proposed in the recent literature. Simulations of fully or partially textured sliding bearings are finally employed to show the advantages of being able to adopt irregular meshes both in terms of flexibility for the discretization of complex surface features and computational speed.

Symmetric drop coalescence on an under-liquid substrate.

We have derived a modified one-dimensional lubrication equation to describe the early coalescence behavior of a symmetric sessile drop coalescence for under-liquid substrates, which takes into account the viscosities of both the drop and the surrounding medium. We found a time scale, which governs the process, and there exists a crossover time between the universal scaling of the bridge height growth h^{*}∼t^{*} (valid for both under-liquid and air) and a much slower bridge growth h^{*}∼t^{*}^{0.24} occurring at a later time. It is also found that the drop coalescence bridge profile has a self-similarity, which breaks up much earlier for under-liquid substrates as opposed to symmetric coalescence in air.

Improved modified Reynolds equation for thin-film gas lubrication from an extended slip velocity boundary condition

An extended slip velocity boundary condition is derived from the regularized 13 moment equations firstly. Different from the existing slip velocity boundary condition, the slip coefficients of the extended one are not fixed, which will change with the wall accommodation coefficient and the Knudsen number of the gas flow. Using the extended slip velocity condition, an improved modified Reynolds equation for thin-film gas lubrication is established. From solving the improved modified Reynolds equation, the pressure distribution of the slider gas bearing is obtained and has a better agreement with that from the direct simulation Monte Carlo method under different pitch angles and wall velocities. It is found that the improved modified Reynolds equation can predict a more accurate pressure distribution of the slider gas bearing than the Fukui and Kaneko's lubrication model from the linearized Boltzmann equation in the near transition regime.

Can hydrodynamic contact line paradox be solved by evaporation–condensation?

We investigate a possibility to regularize the hydrodynamic contact line singularity in the configuration of partial wetting (liquid wedge on a solid substrate) via evaporation–condensation, when an inert gas is present in the atmosphere above the liquid. The no-slip condition is imposed at the solid–liquid interface and the system is assumed to be isothermal. The mass exchange dynamics is controlled by vapor diffusion in the inert gas and interfacial kinetic resistance. The coupling between the liquid meniscus curvature and mass exchange is provided by the Kelvin effect. The atmosphere is saturated and the substrate moves at a steady velocity with respect to the liquid wedge. A multi-scale analysis is performed. The liquid dynamics description in the phase-change-controlled microregion and visco-capillary intermediate region is based on the lubrication equations. The vapor diffusion is considered in the gas phase. It is shown that from the mathematical point of view, the phase exchange relieves the contact line singularity. The liquid mass is conserved: evaporation existing on a part of the meniscus and condensation occurring over another part compensate exactly each other. However, numerical estimations carried out for three common fluids (ethanol, water and glycerol) at the ambient conditions show that the characteristic length scales are tiny.

Application of distinct element methods to simulation of hydraulic fracturing in naturally fractured reservoirs

The Distinct Element Method (DEM) represents a rock mass as an assembly of blocks (polygonal or polyhedral). Contacts between blocks correspond to discontinuities (i.e., fractures or joints) that can exhibit non-linear mechanical behavior, including slip and opening. If flow in rock fracture is approximated using the lubrication equation, coupled hydro-mechanical DEM models can be used for simulation of rock mass treatment by fluid injection. However, this approach has a limited capability for simulating fracture propagation. The synthetic rock mass (SRM) concept overcomes this limitation. In SRM, the bonded particle model (BPM), which is an assembly of circular or spherical particles bonded to each other, represents deformation and damage of intact rock. If pre-existing discontinuities are represented in the BPM, the resulting model, referred to as SRM, has the capability of simulating hydraulic fracturing in naturally fractured reservoirs. The model delivers a pattern of hydraulic fractures that evolves in response to both intact rock fracturing and sliding and opening of pre-existing joints.

Non-Newtonian Inertia Squeeze Film Characteristics in Rectangular Stepped Plates

A study of non-Newtonian inertia squeeze film in rectangular stepped plates has been presented in this paper. Applying the momentum integral method incorporating the micro-continuum theory of non-Newtonian fluids, a non-Newtonian inertia lubrication equation is derived. It is found that the fluid inertia effects yield in a higher normal load capacity as well as a longer squeeze film time as compared to the non-Newtonian stepped squeeze film in the absence of fluid inertia forces.

Mechanics of edge crack growth under transient pressure and temperature conditions

The propagation of an edge fracture under compression in a semi-infinite impermeable isotropic elastic solid, subject to suddenly applied internal fluid pressure and temperature change, is studied numerically. The process involves a strong coupling between the elastic deformation, heat conduction in solids and fluid transport in fractures. Fluid flow is described by the lubrication equation, while the fracture width and fluid front are affected by thermal stresses and fluid pressure. Numerical results are provided for the cases where the cooling-induced tensile stress can overcome the compressive stress across the crack to propagate it at a thermally controlled speed, which is larger than the speed for crack growth resulting only from applied pressure. The presence of the pressurized fluids can result in crack growth starting earlier and can produce a slow early time crack speed. As time increases, the crack motion is accelerated by the pressure acting in the fracture from the viscous fluid flow toward the crack tip. The crack growth rate varies over a wider range relative to a pure thermally driven crack growth case and a stable, rapid growth period occurs, during which the calculated crack speeds are consistent with experimental results. The crack growth curves are located in between two thermally controlled ones for the fracture with and without uniform pressure, respectively. In addition, the dimensionless factors controlling the hydrothermal crack growth are derived from a dimensional analysis. Numerical results demonstrate that viscous flow which leads to the fluid front lagging behind the crack tip can stabilize crack growth under a larger temperature change.