This compendium comprises models and numerical solution procedure for tribological interfaces. It describes the tribological contact and the classical lubrication regimes. A thorough derivation of the Reynolds equation, governing the fluid pressure, from the Navier-Stokes momentum equations and the continuity equation for conservation of mass, is presented along with its analytical solution for the infinitely wide linear slider bearing.

The compilation of the compendium was conducted by the first author during his tenure as Professor at the Division of Machine Elements, Department of Engineering Sciences and Mathematics, Luleå University of Technology and by the second author during his tenure as a postdoctoral researcher at the same division.

Although the compilation of this text is the work solely of the authors, the models and solution procedure presented herein is joint development of many good colleagues and co-authors. Our sincere gratitude is extended towards them all.

Predicting the contact mechanical response for various types of surfaces is and has long been a subject, where many researchers have made valuable contributions. This is because the surface topography has a tremendous impact on the tribological performance of many applications. The contact mechanics problem can be solved in many ways, with less accurate but fast asperity-based models on one end to highly accurate but not as fast rigorous numerical methods on the other. A mathematical model as fast as an asperity-based, yet as accurate as a rigorous numerical method is, of course, preferred. Artificial neural network (ANN)-based models are fast and can be trained to interpret how in- and output of processes are correlated. Herein, 1,536 surface topographies are generated with different properties, corresponding to three height probability density and two power spectrum functions, for which, the areal roughness parameters are calculated. A numerical contact mechanics approach was employed to obtain the response for each of the 1,536 surface topographies, and this was done using four different values of the hardness per surface and for a range of loads. From the results, 14 in situ areal roughness parameters and six contact mechanical parameters were calculated. The load, the hardness, and the areal roughness parameters for the original surfaces were assembled as input to a training set, and the in situ areal roughness parameters and the contact mechanical parameters were used as output. A suitable architecture for the ANN was developed and the training set was used to optimize its parameters. The prediction accuracy of the ANN was validated on a test set containing specimens not seen during training. The result is a quickly executing ANN, that given a surface topography represented by areal roughness parameters, can predict the contact mechanical response with reasonable accuracy. The most important contact mechanical parameters, that is, the real area of contact, the average interfacial separation, and the contact stiffness can in fact be predicted with high accuracy. As the model is only trained on six different combinations of height probability density and power spectrum functions, one can say that an output should only be trusted if the input surface can be represented with one of these.

In this work, the stiffness, i.e., the derivative of the load-separation curve, is studied for self-affine fractal surfaces with non-Gaussian height distribution. In particular, the heights of the surfaces are assumed to follow a Weibull distribution. We find that a linear relation between stiffness and load, well established for Gaussian surfaces, is not obtained in this case. Instead, a power law, which can be motivated by dimensionality analysis, is a better descriptor. Also unlike Gaussian surfaces, we find that the stiffness curve is no longer independent of the Hurst exponent in this case. We carefully asses the possible convergence errors to ensure that our conclusions are not affected by them.

This paper presents the development of a model enabling the analysis of rarefied gas flow through highly heterogeneous porous media. To capture the characteristics associated with the global- and the local-scale topology of the permeable phase in a typical porous medium, the heterogeneous multi-scale method, which is a flexible framework for constructing two-scale models, was employed. The rapid spatial variations associated with the local-scale topology are accounted for stochastically, by treating the permeability of different local-scale domains as a random variable. The results obtained with the present model show that an increase in the spatial variability in the heterogeneous topology of the porous medium significantly reduces the relevance of rarefaction effects. This clearly shows the necessity of considering a realistic description of the pore topology and questions the applicability of the results obtained for topologies exhibiting regular pore patterns. Although the present model is developed to study low Knudsen number flows, i.e. the slip-flow regime, the same development procedure could be readily adapted for other regimes as well.

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.

This chapter introduces the reader to lubrication theory and describes the governing equations, models and methods that can be used to simulate various types of lubricated systems. It starts with an introduction to the tribological contact and to the different lubrication regimes. The basis for the classical lubrication theory is then given and thereafter follows a presentation of how to obtain the Reynolds equation by means of scaling and asymptotic analysis of the Navier–Stokes equations. After having obtained the Reynolds equation, a quite elaborate presentation of cavitation algorithms is given. It includes discretisation and presents the analytical solution for a pocket bearing as a benchmark model problem. Then, the concept of homogenisation of surface roughness is introduced. This starts from the simplest iso-viscous and incompressible case, expands to include compressibility with a constant bulk modulus constitutive relation and then also addresses the case of ideal gases. Thereafter, the relation between homogenised coefficients and the Patir and Cheng flow factors is described and finally it is shown how to incorporate the effect of mixed lubrication into the model.

In this work we present a simple method to generate surface topography. The main advantage of the presented method as compared with those available in the literature is that the power spectrum and the height probability distribution can be specified independently. In this article we present the method and show its versatility by generating surface topographies with three different height probability distributions: the Weibull distribution, a bimodal distribution and a distribution containing a delta function that represents worn surfaces. The MATLAB-code we used to generate the numerical examples are also provided to the reader.

The fact that the film is thin is in lubrication theory utilised to simplify the full Navier–Stokes system of equations. For incompressible and iso-viscous fluids, it turns out that the inertial terms are small enough to be neglected. However, for a compressible fluid, we show that the influence of inertia depends on the (constitutive) density-pressure relationship and may not always be neglected. We consider a class of iso-viscous fluids obeying a power-law type of compressibility, which in particular includes both incompressible fluids and ideal gases. We show by scaling and asymptotic analysis, that the degree of compressibility determines whether the terms governing inertia may or may not be neglected. For instance, for an ideal gas, the inertial terms remain regardless of the film height-to-length ratio. However, by means of a specific modified Reynolds number that we define we show that the magnitudes of the inertial terms rarely are large enough to be influential. In addition, we consider fluids obeying the well-known Dowson and Higginson density-pressure relationship and show that the inertial terms can be neglected, which allows for obtaining a Reynolds type of equation. Finally, some numerical examples are presented in order to illustrate our theoretical results.

Metal-to-metal seals are used in connections of casing and tubing in oil and gas wells. This paper describes the mechanisms of sealing metal-to-metal seals as investigated using an experimental setup and a stochastic numerical sealing model. Experiments were conducted for a variety of thread compounds and applied pin/box surface coatings. The results were used to validate a stochastic numerical sealing model for sealability. The model couples a contact-mechanics model with a flow model and takes into account the influence of all the surface-topography features by introducing the concept of seal permeability. Once validated, the model was used together with the experimental results to better understand the sealing mechanisms of metal-to-metal seals.

The sealing configuration is a face seal with an 80-mm roundoff radius on one face pressing against a flat on the other face. The face-seal specimens were manufactured from P110 tubing to ensure material properties that are representative for casing or tubing. The test setup used is designed for investigating only the metal-to-metal seal of the connection. The setup can perform rotary sliding under constant load to simulate surface changes during makeup and subsequently perform a leakage test. The sealing limit is determined by applying 700-bar fluid pressure and then gradually reducing the normal force until leakage is observed. The data are subsequently used to validate the previously published stochastic numerical sealing model.

The results indicate a strong dependence on the type of thread compound used for the onset of leakage. The thread compound affects the amount of wear and thus changes the surface topography of the interacting surfaces. It is shown that the stochastic numerical sealing model is capable of predicting the onset of leakage within the experimental accuracy. The model shows further that certain surface topographical features improve the sealing performance. In particular, a surface manufactured by turning on a lathe that is in contact with, for instance, a smooth shot-blasted surface topography leads to highly localized contact areas, which in turn yield the best sealing performance.

Metal-to-metal seals are used in connections of casing in oil and gas wells. This paper describes the mechanisms of sealing of metal-to-metal seals as investigated using an experimental set-up and a sealability model. Experiments were conducted for a variety of thread compounds and applied pin/box surface coatings. The results were used to validate a numerical model for sealability. The stochastic model couples a contact mechanics model with a flow model and takes the influence of all the surface topography features into account. Once validated, the model was used together with the experimental results to explain the sealing mechanisms of metal-to-metal seals. The sealing configuration is a face seal with an R=80 mm round-off radius pressing against a flat. The face seal specimens were manufactured from P110 tubing. The used test set-up is designed for investigating only the metal-to-metal seal of the connection. The set-up can carry out rotary sliding under constant load to simulate surface evolution during make-up and subsequently perform a leakage test. The sealing limit is determined by applying 700 bar fluid pressure and then gradually reducing the normal force until leakage is observed. The data is subsequently used to validate a previously published model. The results indicate a strong dependence of the type of thread compound used on the onset of leakage. The thread compound affects the amount of wear and thus changes the surface topography of the interacting surfaces. It is shown that the sealability model is capable to predict the onset of leakage within the experimental accuracy. The model shows further that certain surface topographical features improve the sealing performance. Namely, a turned against a flat surface topography leads to highly localized contact areas, which in turn yields the best sealing performance. The combination of experimental data with the validated model leads to much deeper insights for the sealing mechanisms than what could be obtained using either on their own.