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Height-averaged Navier-Stokes solver for hydrodynamic lubrication

Modelling hydrodynamic lubrication is crucial in the design of engineering components as well as for a fundamental understanding of friction mechanisms. The cornerstone of thin-film flow modelling is the Reynolds equation -- a lower-dimensional representation of the Stokes equation. However, the derivation of the Reynolds equation is based on assumptions and fixed form constitutive relations, that may not generally be valid, especially when studying systems under extreme conditions. Furthermore, these explicit assumptions about the constitutive behaviour of the fluid prohibit applications in a multiscale scenario based on measured or atomistically simulated data. Here, we present a method that considers the full compressible Navier-Stokes equation in a height-averaged sense for arbitrary constitutive relations. We perform numerical tests by using a reformulation of the viscous stress tensor for laminar flow to validate the presented method comparing to results from conventional Reynolds solutions. The versatility of the method is shown by incorporating models for mass-conserving cavitation, wall slip and non-Newtonian fluids. This allows testing of new constitutive relations that not necessarily need to take a fixed form, and may be obtained from experimental or simulation data.