Search

This paper constitutes the third part of a series of works on lower-dimensional models in lubrication. In Part A, it was shown that implicit constitutive theory must be used in the modelling of incompressible fluids with pressure-dependent viscosity and that it is not possible to obtain a lower-dimensional model for the pressure just by letting the film thickness go to zero, as in the proof of the classical Reynolds equation. In Part B, a new method for deriving lower-dimensional models of thin-film flow of fluids with pressure-dependent viscosity was presented. Here, in Part C, we also incorporate the energy equation so as to include fluids with both temperature and pressure dependent viscosity. By asymptotic analysis of this system, as the film thickness goes to zero, we derive a simplified model of the flow. We also carry out an asymptotic analysis of the boundary condition, in the case where the normal stress is specified on one part of the boundary and the velocity on the remaining part. [ABSTRACT FROM AUTHOR]

Most of the problems in lubrication are studied within the context of Reynolds' equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero. [ABSTRACT FROM AUTHOR]

The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure. [ABSTRACT FROM AUTHOR]

The bearing geometry has a big impact on the performance of a hydrodynamic thrust bearing. For this reason, shape optimisation of the bearing surface has been carried out for some time, with Lord Rayleigh's early publication dated back to 1918. There are several recent results e.g. optimal bearing geometries that maximise the load carrying capacity for hydrodynamic thrust bearings. Currently, many engineers are making an effort to include sustainability in their work, which increases the need for bearings with lower friction and higher load carrying capacity. Improving these two qualities will result in lower energy consumption and increase the lifetime of applications, which are outcomes that will contribute to a sustainable future. For this reason, there is a need to find geometries that have performance characteristics of as low coefficient of friction torque as possible. In this work, the topological optimisation method of moving asymptotes is employed to optimise bearing geometries with the objective of minimising the coefficient of friction torque. The results are both optimised bearing geometries that minimise the coefficient of friction torque and bearing geometries that maximise the load carrying capacity. The bearing geometries are of comparable aspect ratios to the ones uses in recent publications. The present article also covers minimisation of friction torque on ring bearing geometries, also known as thrust washers. The results are thrust washers with periodical geometries, where the number of periodical segments has a high impact on the geometrical outcome. [ABSTRACT FROM AUTHOR]

The search for the optimal bearing geometry has been on for over a century. In a publication from 1918, Lord Rayleigh revealed the infinitely wide bearing geometry that maximises the load carrying capacity under incompressible flow, i.e. the Rayleigh step bearing. Four decades ago, Rohde, who continued on the same path, revealed the finitely wide bearing geometry that maximises the load carrying capacity, referred to as the Rayleigh-pocket bearing. Since then, the numerical results have been perfected with highly refined meshes, all converging to the same Rayleigh-pocket bearing. During recent years new methods for performing topology optimisations have been developed and one of those is the method of moving asymptotes, frequently used in the area of structural mechanics. In this work, the method of moving asymptotes is employed to find optimal bearing geometries under incompressible flow, for three different objectives. Among the results obtained are (i) show new bearing geometries that maximise the load carrying capacity, which performs better than the ones available, (ii) new bearing geometries minimising the coefficient of friction and (iii) new bearing geometries minimising the friction force for a given load carrying capacity are presented as well. [ABSTRACT FROM AUTHOR]

It has been over a century since the interest in inventing the optimal topology for bearings arose. A significant achievement was published by Lord Rayleigh, who found the step-bearing geometry which maximise the load-carrying capacity when the classical Reynolds equation is used to model thin film flow of an iso-viscous and incompressible fluid. Since then, new optimisation methods considering some variants of governing equations for finding the best possible bearings have surfaced, one of which will be presented in this paper. Here, two different formulations for compressible flow, i.e. ideal gas and constant bulk modulus compressibility, as well as the classical Reynolds formulation will be used in combination with the method of moving asymptotes for topological optimisation. All three of these problem formulations provide us with unique geometries, which either maximise the load-carrying capacity or minimise friction, for fluids with a wide variety of compressibility. [ABSTRACT FROM AUTHOR]

During operation, the mating surfaces of a metal-to-metal seal typically undergo significant plastic deformation, which in turn can have beneficial effect on its performance. In previous studies, it has, for instance, been shown that plastic deformation can provide for better sealing during unloading. Those studies did, however, only consider flow through unrealistically small domains. Therefore, it is possible that this might be a size effect, which would not be apparent in a real situation with a much larger domain. In this paper, we develop a model which can handle real-sized seal domains at the same time as fine details of the surface topography. More precisely, we construct a two-scale model, in which the global scale represents the seal domain and where the influence of the fine details at the local scale are represented by a stochastic element. By means of this stochastic two-scale model, we show that the beneficial effect associated with the plastic deformation persists also when real-sized seal domains are considered. [ABSTRACT FROM AUTHOR]

This study considers flow through the gap left between two surfaces during unloading, in other words, when an applied load is gradually reduced after loading to a state where plastic deformation occurs. In particular, the permeability of the gap is studied. It was found that a substantial reduction of the applied load is required before the permeability starts to increase significantly. The explanation for this phenomenon is given by the combination of components with different wavelengths present in the surface. Components with long wavelengths deform elastically and those with shorter wavelengths may also deform plastically. We found that plastic deformation acts to keep the permeability nearly constant at the beginning of the unloading and elastic spring-back is responsible for the rapid increase at lower loads. This principle constitutes a basis for the strategy that was developed in order to predict the load at which the rapid increase of permeability starts. [ABSTRACT FROM AUTHOR]

Abstract: We homogenize a Reynolds equation with rapidly oscillating film thickness function , assuming a constant compressibility factor in the pressure–density relation. The oscillations are due to roughness on the bounding surfaces of the fluid film. As shown by previous studies, homogenization is an effective approach for analyzing the effects of surface roughness in hydrodynamic lubrication. By two-scale convergence theory we obtain the limit problem (homogenized equation) and strong convergence in for the unknown density . By adding a small corrector term, convergence is obtained also in the Sobolev norm. [Copyright &y& Elsevier]

Different averaging techniques have proved to be useful for analyzing the effects of surface roughness in hydrodynamic lubrication. This paper compares two of these averaging techniques, namely the flow factor method by Patir and Cheng (P&C) and homogenization. It has been rigorously proved by many authors that the homogenization method provides a correct solution for arbitrary roughness. In this work it is shown that the two methods coincide if and only if the roughness exhibits certain symmetries. Hence, homogenization is always the preferred method. [ABSTRACT FROM AUTHOR]

This paper presents a complete derivation and design of a physics-informed neural network (PINN) applicable to solve initial and boundary value problems described by linear ordinary differential equations. The objective with this technical note is not to develop a numerical solution procedure which is more accurate and efficient than standard finite element- or finite difference-based methods, but to give a fully explicit mathematical description of a PINN and to present an application example in the context of hydrodynamic lubrication. It is, however, worth noticing that the PINN developed herein, contrary to FEM and FDM, is a meshless method and that training does not require big data which is typical in machine learning. [ABSTRACT FROM AUTHOR]

Leakage in static metal-to-metal seals is predominantly determined by the topography of the contacting surfaces. The topography consists of features that span the entire range from its carefully engineered geometry down to micro-sized surface asperities. The mesh density necessary to fully resolve all the features, in this large span of length scales, generates too many degrees of freedom for a direct numerical approach to be applicable. Some kind of sophistication, either incorporated in the mathematical model or in the numerical solution procedure or even a combination of both is therefore required. For instance, in a two-scale model, the geometrical features can be addressed in the global-scale model, while the features belonging to length scales smaller than a given cut-off value are addressed in the local-scale model. However, the classical two-scale approaches do not explicitly address the stochastic nature of the surfaces, and this has turned out to be a requirement in order to obtain quantitative predictions of leakage in metal-to-metal seals. In this work, we present a continued development of an already existing two-scale model, which incorporates a stochastic element. The novelty lies in the way we characterise the permeability at the local scale and how this is used to build a more efficient and useful approach. [ABSTRACT FROM AUTHOR]