Lubrication Equation Research Articles

Research on Turbulent Lubrication of Misaligned Journal Bearing Considering the Axial Motion of Journal

Abstract Based on the coordinate transformation method, the generalized turbulent lubrication equation considering the axial motion of the journal is derived. The finite-difference method is used to solve the generalized turbulent lubrication equation. The variations of turbulent lubrication performance with axial displacement for different axial movement velocity of the journal, journal misalignment angle, and initial mid-plane eccentricity ratio are obtained. The results show that when the axial movement velocity of the journal increases, the mid-plane eccentricity ratio of the bearing and the minimum film thickness remain unchanged, the average Reynolds number, maximum film pressure, load capacity, frictional power, and side leakage increases. As the axial displacement of the journal increases, the greater the misalignment angle of the journal, the greater the degree of misalignment, maximum film pressure, load capacity, and misalignment moment of the bearing. The greater the initial mid-plane eccentricity ratio, the greater the degree of journal misalignment, maximum film pressure, load capacity, frictional power, and misalignment moment.

Three dimensional simulation of flow development of triple-layer lubricated pipeline transport

In heavy oil pipelining, a major challenge is to reliably achieve high rates of pressure drop reduction in stable annular flow patterns. Sarmadi et al. (2017), introduced a novel methodology for efficient transportation of heavy oil via a triple-layer core-annular flow. The lubricating outer layer is separated from the core by a shaped skin comprising a yield stress fluid. The lubricant reduces the pressure drop, the unyielded skin eliminates interfacial instabilities and the shaped interface produces lift force in the lubrication layer to balance buoyancy of the core. Sarmadi et al. (2018), studied how to sculpt the interface in stable controlled way and how to establish the flow. Here we present three dimensional (3D) triple-layer computations which capture the buoyant motion of the core to reach its equilibrium position. The 3D computations are performed using a finite element method and adaptively aligned meshes to track dynamically the interfaces, benchmarked against axisymmetric computations from Sarmadi et al. (2018). The study shows that these flows may stably become established with control over interface shape, but development lengths (times) for the core to attain equilibrium are relatively long, meaning extensive 3D computation. We also present a simplified analytical model using the lubrication approximation and equations of motion for the lubricant and skin layers. This model allows us to quickly estimate motion to the balanced configuration for a given shape and initial conditions.

Thermo-molecular gas-film lubrication (t-MGL) considering temperature distributions and accommodation coefficients: analyses by quasi-free-molecular t-MGL equation

In the present paper, the air-bearing characteristics of a fixed 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 and imperfect surface accommodation coefficients, the molecular reflections at the surface of which are a mixture of diffuse reflection (α) and specular reflection (1−α) (Maxwell-type reflection), are analyzed using the thermo-molecular gas-film lubrication (t-MGL) equation. In particular, since the minimum spacing is ultra-small (~ 1 nm), the quasi-free-molecular t-MGL equation, in which the gas temperature, τG, is assumed to be that in the free molecular limit, τGfm, defined by temperatures and accommodation coefficients at the disk, τW0 and α0, respectively, and those at the slider, τW1 and α1, respectively, is used. For a plane inclined slider (Case 1), the fundamental static and dynamic characteristics are analyzed numerically and are compared with the approximation results for an infinite bearing number (t-MGLqfm, Λ→∞). When the spacing is ultra-small and only the running boundary (disk) has a local temperature distribution, the additional static pressure produced by the local boundary temperature distribution is approximately proportional to the equivalent accommodation coefficient $$ \gamma_{\alpha } \equiv \alpha_{0} (2 - \alpha_{1} )/[2\{ 1 - (1 - \alpha_{0} )(1 - \alpha_{1} )\} ] $$. For a step slider (Case 2), the following results are revealed: (1) the spacing decreases occur as the accommodation coefficient of the disk, α0, decrease i.e., the ratio of specular reflection at the disk increases, (2) the increases in the minimum spacing due to local temperature distribution are negligible in both air and He because the heat spot size is very small, (3) the decreases in the minimum spacing for a slider flying in He are significant, and (4) the spacing fluctuation caused by a running wavy disk varies according to both the surface accommodation coefficients and the ambient gas (air/He) and can be estimated precisely.

Effect of geometric shape of micro-grooves on the performance of textured hybrid thrust pad bearing

A numerical simulation is performed to investigate the synergizing influence of surface texture (micro-grooves) and electrically conducting lubricant on static and dynamic performance of hybrid thrust pad bearings operating under transverse magnetic field. Linear momentum and mass conservation equation are solved simultaneously to derive modified Reynolds equation for magnetohydrodynamic lubrication. Finite element approach has been used to obtain couple solution of modified Reynolds equation and restrictor (orifice) flow equation. A mass conserving algorithm (Jakobsson–Floberg–Olsson boundary condition) has been implemented to simulate cavitation phenomenon within micro-grooves. A parametric investigation (based on load-carrying capacity/fluid film pressure) is carried out to obtain optimum attributes for various micro-grooves shapes. Providing micro-groove on bearings surfaces results in a significant increase in load-carrying capacity, stiffness coefficient and reduction in frictional power loss of the bearing. Employing of electrically conducting lubricant is seen to be enhancing the load-carrying capacity and damping characteristics of the bearing.

Probabilistic Sensitivity Analysis of Wear Property for MEMS Gas Bearing

In this study, the sensitivity of MEMS gas bearing’ performance to the wear in different axial and circumferential positions is investigated in detail. Rarefaction effect is introduced into the transient and steady lubrication equation, and then the finite element method (FEM) is employed to solve the equations. The stochastic process is adopted to simulate wear distribution and view of probability is proposed to describe the change laws of the static and dynamic performance of the bearing. Then, the static and dynamic characteristics of the bearing in 50 wear conditions are calculated for each case. Furthermore, the standard deviations and correlation coefficients of the bearing performance sample points are analyzed to demonstrate the influence degree of wear in different positions.

A Semi-Infinite Hydraulic Fracture Driven by a Herschel–Bulkley Fluid

Abstract Hydraulic fracturing is an industrial process often applied to enhance oil and gas recovery. Under this process, fractures are generated by the injection of highly pressurized fluids, which often exhibit shear-thinning rheology and yield stress. The global fracture propagation is influenced by various processes occurring near the fracture tip. To gain an insight into fracture propagation, the problem of a semi-infinite hydraulic fracture propagating in a permeable linear elastic rock is solved. To investigate the effect of fluid yield stress, we focus on a fracture driven by Herschel–Bulkley fluid. The mathematical model consists of the elasticity equation, the lubrication equation, and the propagation criterion for the semi-infinite plane strain fracture to obtain the fracture opening. The non-linear system of governing equations is represented in the non-singular form and solved numerically using Newton’s method. The solution is influenced by the competing processes related to rock toughness, fluid properties, and leak-off. The effects of these phenomena prevail at different length scales, and the corresponding limits can be described via analytical solutions. For a Herschel-Bulkley fluid, an additional limiting solution related to the fluid yield stress is obtained, and the regions of the dominance of limiting solutions affected by the yield stress are investigated. Finally, a faster approximate solution for the problem is proposed and its accuracy against a numerical solution is evaluated. The obtained result can be applied in hydraulic fracturing simulators to account for the effect of Herschel–Bulkley fluid rheology on the near-tip region.

A 2D explicit numerical scheme–based pore pressure cohesive zone model for simulating hydraulic fracture propagation in naturally fractured formation

AbstractIn this work, a 2D pore pressure cohesive zone model is presented to simulate the hydraulic fracture propagation in naturally fractured formation, in which the fracturing process is governed by a bilinear cohesive zone model equipped with Coulomb's friction law and the fluid flow within the hydraulic fracture is described by the lubrication equation. In this model, fluid leak‐off into rock matrix is ignored and a fully explicit temporal integration scheme is adopted to overcome the convergence issue of conventional implicit scheme (eg, Newton‐Raphson method). The advantage of the proposed model is that it does not require any special crossing criterion to determine the interaction behavior when the hydraulic fracture hits the natural fracture. Implementation of the model was described in detail, and then, the model was verified with well‐known analytical solution of KGD problem and criteria of hydraulic fracture crossing natural fracture. The capability of the proposed model to capture the interaction behavior between hydraulic fracture and natural fracture was demonstrated by the good agreement of the modeling result and analytical solution. Several numerical cases were performed to investigate the impact of key factors on fracture network evolution during hydraulic fracturing treatment. Results show that fracture network is not only governed by the spatial distribution of natural fracture, but also greatly affected by the initial hydraulic aperture of natural fracture and in situ stress contrast.

Immersed boundary/solid method for the numerical simulation of particle-laden flows

An outlook of computational fluid dynamics methods based on the fixed Cartesian grid, especially the immersed boundary (IB) method and related techniques, is provided. The basic idea of IB methods and the improvements to maintain consistency of boundary condition and mass conservation, that is the consistent projection method, are discussed. Then, recent examples for particle-laden flows are introduced. To deal with many particles, the immersed solid method is picked up as an efficient alternative to the IB method. In particular, we place focus on the flow and heat transfer of dense mixture of particles and liquid. In such a condition, the flow in narrow gap in between solid interfaces is considered by the lubrication equation. Finally, for further development of application in large-scale turbulent flows laden with particles, the importance of explicitly-defined volume averaging is pointed out based on our recent progress.

Static characteristics analysis of spherical hybrid sliding bearings

PurposeThe spherical hybrid sliding bearings (SHSBs) can be used in ultra-precision and heavy-duty machine tools. However, there is little related research for these bearings. The purpose of this study is to investigate the static characteristics and effect factors affecting SHSBs by fluid lubrication.Design/methodology/approachBased on the theories of fluid lubrication, the Reynolds equation of general Newtonian fluid is derived to obtain the steady-state lubrication equation. The system is solved by the finite difference method and the relaxation iterative method on the staggered grid to obtain the thickness and the pressure distribution of the oil film. The radial and axial load capacities of SHSBs are determined by the pressure field integration over the spherical surface.FindingsThe results show that the parameters such as oil supply pressure, bearing clearance, eccentricity ratio, rotating speed and orifices’ number affecting the static characteristics of bearings are significant and the cross-coupling effect exists.Originality/valueThe lubrication model of SHSB is established to analyze the pressure distribution with a variety of oil film thickness. The laws of oil supply pressure, bearing clearance, eccentricity ratio, rotating speed and orifices’ number on the load capacities are researched.

Effect of inertia on the cavitation phenomena of hydrodynamic textured bearings considering slip

Surface modification of a lubricated bearing, such as hydrophobic coating inducing slip situation and texturing, is proved to enhance hydrodynamic performance. As widely known, in textured surface lubricant inertia and cavitation can significantly affect the hydrodynamic pressure profile. However, a brief literature review indicates that studies related to the correlation between cavitation and inertia, especially in the presence of slip, are considerably limited. The present study examines the effect of inertia on cavitation phenomena by considering the slip boundary using two approaches, namely computational fluid dynamics based on full Navier–Stokes equations and analytical lubrication equation based on the Reynolds equation. The modified Reynolds equation with slip concept is used with respect to the slip effect applied on the surface of the bearing. The results indicate that the inertia as well as the slip condition significantly affects the cavitation area. It is also highlighted that the cavitation area reduces by increasing the inertia effect, and it becomes smaller when the slip is introduced.

An experimental and numerical study on the wall lubrication force in dispersed liquid-liquid flow

Many analytical, experimental and numerical works have been devoted to the accurate prediction of volumetric fraction distribution in the wall region, which is important for the numerical simulation of dispersed flow. Within the two-fluid-model framework, it depends upon the correct modeling of interfacial forces, and among them there is the wall lubrication force. In this prospective paper, we try to contribute with the discussion by studying the effect of wall lubrication forces in a dispersed liquid-liquid flow. Instead of bubbles, oil droplets dispersed in turbulent water flow are moved away from the wall. This phenomenon has been recently observed in a dispersed oil-water flow in a horizontal pipe and one of the findings is that it is related to drag reduction. One of the industries that have interest in this kind of problem is the energy industry. The pipe flow of oil-in-water dispersion is very common in the Brazilian offshore production scenario. An experimental campaign was carried out to measure the wall water thickness as a function of mixture velocity. The frictional pressure-drop was also measured to estimate the oil droplet average diameter. The Reynolds lubrication equation is solved using the finite element method to compute the lubrication force in the gap between a typical oil droplet and the pipe wall. Two methods were used to solve it, the Gauss-Seidel and Successive Overrelaxation. The ratio of lubrication to buoyancy forces is computed for the flow condition studied in this work, and the results suggest that the occurrence of wall lubrication force is a reasonable explanation for the observed concentration of dispersed oil droplets in the turbulent core of the pipe.

Film thickness distribution in gravity-driven pancake-shaped droplets rising in a Hele-Shaw cell.

We study here experimentally, numerically and using a lubrication approach, the shape, velocity and lubrication film thickness distribution of a droplet rising in a vertical Hele-Shaw cell. The droplet is surrounded by a stationary immiscible fluid and moves purely due to buoyancy. A low density difference between the two media helps to operate in a regime with capillary number Ca lying between 0.03 and 0.35, where Ca =μo Ud /γ is built with the surrounding oil viscosity γ0 , the droplet velocity Ud and surface tension γ. The experimental data show that in this regime the droplet velocity is not influenced by the thickness of the thin lubricating film and the dynamic meniscus. For iso-viscous cases, experimental and three-dimensional numerical results of the film thickness distribution agree well with each other. The mean film thickness is well captured by the Aussillous & Quéré (Phys. Fluids, vol. 12 (10), 2000, pp. 2367-2371) model with fitting parameters. The droplet also exhibits the 'catamaran' shape that has been identified experimentally for a pressure-driven counterpart (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501). This pattern has been rationalized using a two-dimensional lubrication equation. In particular, we show that this peculiar film thickness distribution is intrinsically related to the anisotropy of the fluxes induced by the droplet's motion.

A comprehensive mathematical model for an accurate calculation of the oil film thickness of strip cold rolling

The surface quality of cold rolled strip is related to a greater extent on the rolling oil film thickness, and there are many factors that affect the oil film thickness. Considering the various factors comprehensively, an integrated mathematical model is established, such as roughness of rolls and strips, elastohydrodynamic lubrication, friction heat and plastic deformation heat in the rolling zone, viscosity varying with temperature and pressure, etc. A series of equations are developed, such as the Reynolds equation of partial membrane hydrodynamic lubrication based on average flow theory, equation of oil film thickness on rough elastic surface, the thermal interface equations between strip, oil film and roller surface, surface asperity contact pressure equation, lubricant viscosity and density equations, motion equation of the oil film, etc. This model is solved by finite difference method to get the film pressure, oil film thickness, and temperature distribution in the rolling zone. The average rolling pressure, the roll, and strip temperature calculated by the model are very close to the field test results. Comparing the minimum film thickness calculated by the model with the regression formula of other literature test, the error is less than 10%. The minimum oil film thickness is analyzed. It increases with the decrease of the rolling force and is approximately proportional to the rolling speed and lubricant viscosity.

Effects of rotor surface texture on rotary vane actuator end sealing performance

Abstract The dynamic pressure oil film is difficult to form between the end seal ring and the rotor during the hydraulic rotary vane actuator (RVA) operates, which results in the sealing surfaces sustain severe friction and wear. Thus, a series of circular micro-dimples was processed on the rotor surface using laser surface texturing technology to reduce these. The mathematical model of the end seal was established based on the fluid dynamic lubrication equation. In the meanwhile, the friction torque and leakage between the sealing clearances were calculated. In addition, verification experiments were performed. The results showed that the rotor surface texture can effectively improve the RVA end sealing performance under lubricated conditions.

Viscoelastic Lubricant Deformation and Disk-to-Head Transfer During Heat-Assisted Magnetic Recording

One of the challenges in heat-assisted magnetic recording (HAMR) is the formation of write-induced head contamination at the near-field transducer. A possible mechanism that has been proposed for this contamination is the transfer of lubricant from the disk to the head due to temperature driven evaporation/condensation. Most previous studies on lubricant depletion due to laser heating have assumed the lubricant to be a viscous fluid and have modeled its behavior using the traditional lubrication theory. However, perfluoropolyether lubricants are viscoelastic fluids and are expected to exhibit a combination of viscous and elastic behavior at the time and length scales of HAMR conditions. In this paper, we use a modified Reynolds lubrication equation for the viscoelastic fluid that employs the linear Maxwell constitutive model. We use this modified lubrication equation to develop a model that predicts the disk-to-head lubricant transfer during HAMR writing. This model simultaneously determines the thermocapillary stress driven deformation and evaporation of the viscoelastic lubricant film on the disk, the diffusion of the vapor phase lubricant in the air bearing, and the evolution of the condensed lubricant film on the head. We investigate the effects of lubricant type (Zdol versus Ztetraol), head/disk temperature, initial lubricant thickness, and laser spot size on the lubricant transfer process. Simulation results show a significant difference between the rates of transfer for Zdol (timescale of nanoseconds) versus Ztetraol (timescale of microseconds). The amount of transfer increases with the disk temperature and the initial lubricant thickness.

Effect of misalignment on the dynamic characteristic of MEMS gas bearing considering rarefaction effect

Abstract In this paper, the dynamic characteristics of MEMS gas bearing are investigated. Angular misalignment and rarefaction effect are introduced into the lubrication equation based on FK model. The partial derivative method is adopted to solve the lubrication equation. The comparison of the calculating results between continuous model and rarefied model shows that the rarefaction effect has obvious impact on the dynamic performance of MEMS gas bearing. Then the influence of misalignment on the dynamic coefficients of MEMS gas bearing is studied and the influence laws on 8 dynamic characteristic coefficients are obtained. Furthermore, the limit dynamic performances considering misalignment and rarefaction effect are discussed in detail.

INVESTIGATION OF COUPLED EFFECTS OF TEMPERATURE AND RELATIVE HUMIDITY ON DYNAMIC PERFORMANCE OF MEMS CANTILEVER RESONATORS IN GAS RAREFACTION

The modified molecular gas lubrication (MMGL) equation with the effective viscosity of moist air is utilized to solve for the squeeze film damping (SFD) problem on the dynamic performance of MEMS cantilever resonators. Thus, the coupled effects of temperature and relative humidity are discussed on the Q-factors of MEMS cantilever resonators in a wide range of gas rarefaction (pressure, p and accommodation coefficients, ACs) and resonant mode of vibration. The results showed that the Q-factor of moist air decreases more significantly as temperature and relative humidity increase at higher gas rarefaction (lower p, and ACs) conditions.

Pore pressure cohesive zone modelling of complex hydraulic fracture propagation in a permeable medium

A pore pressure cohesive zone method (PCZM) is proposed to simulate the complex hydraulic fracturing process in a permeable poroelastic medium. A shared node scheme of the cohesive elements is developed to consider the arbitrary fracture propagation and fluid flow at the intersection. The fracture propagation is based on a bilinear cohesive law, while the tangential flow within the fracture is controlled by the lubrication equation. The fluid leak-off within the cohesive element is calculated according to the distribution of pressure. Three models are given to verify the capability of the PCZM in simulating the complex fracturing process. The effects of the approaching angle, cementing strength, in situ stress and natural fracture on the fracture geometry are investigated. These studies indicate that the proposed PCZM can accurately predict complex hydraulic fractures (HFs) and provide a useful way to study HF mechanisms.

Load and loss for high-speed lubrication flows of pressurized gases between non-concentric cylinders

We examine the high-speed flow of pressurized gases between non-concentric cylinders where the inner cylinder rotates at constant speed while the outer cylinder is stationary. The flow is taken to be steady, two-dimensional, compressible, laminar, single phase and governed by a Reynolds lubrication equation. Approximations for the lubricating force and friction loss are derived using a perturbation expansion for large speed numbers. The present theory is valid for general Navier–Stokes fluids at nearly all states corresponding to ideal, dense and supercritical gases. Results of interest include the observation that pressurization gives rise to large increases in the lubricating force and decreases in the fluid friction. The lubrication force is found to scale with the bulk modulus. Within the context of the Reynolds equation an exact relation between total heat transfer and power loss is developed.

A local discontinuous Galerkin method for the compressible Reynolds lubrication equation

We present an extension of the local discontinuous Galerkin (LDG) method introduced in [9] for nonlinear diffusion problems to nonlinear stationary convection–diffusion problems. We develop a numerical study of the convergence properties of the new method and solve the stationary compressible Reynolds lubrication equation under some realistic conditions.