Abstract
The cornerstone of thin-film flow modeling is the Reynolds equation—a lower-dimensional representation of the Navier–Stokes equation. The derivation of the Reynolds equation is based on explicit assumptions about the constitutive behavior of the fluid that prohibit applications in multiscale scenarios based on measured or atomistically simulated data. Here, we present a method that treats the macroscopic flow evolution and the calculation of local cross-film stresses as separate yet coupled problems—the so-called macro and micro problem. The macro problem considers mass and momentum balance for compressible fluids in a height-averaged sense and is solved using a time-explicit finite-volume scheme. Analytical solutions for the micro problem are derived for common constitutive laws and implemented into the Height-averaged Navier–Stokes (HANS) solver. We demonstrate the validity of our solver on examples, including mass-conserving cavitation, inertial effects, wall slip, and non-Newtonian fluids. The presented method is not limited to these fixed-form relations and may therefore be useful for testing constitutive relations obtained from experiment or simulation.
Highlights
The fundamental equation for hydrodynamic lubrication was derived by Reynolds [1] almost 140 years ago, accurate descriptions of thin-film fluid flow are still a research topic of ongoing interest
The derivation of the Reynolds equation is based on an asymptotic analysis of the incompressible Navier–Stokes equation under the assumption that the gap height is small compared to the lateral dimensions
We incorporate mass-conserving cavitation directly through the equation of state, by either fixing the pressure to a constant value pcav at densities lower than the saturation density, which is conceptually similar to the EA algorithm or using a unique equation of state describing the behavior of vapor, liquid, and vapor–liquid mixture as in the model of Bayada and Chupin [34]
Summary
The fundamental equation for hydrodynamic lubrication was derived by Reynolds [1] almost 140 years ago, accurate descriptions of thin-film fluid flow are still a research topic of ongoing interest. Inertial and body force terms can be neglected and the fluid pressure does not depend on the gap height coordinate. This allows integration of the momentum equations in the gap height coordinate to obtain the flow velocity profiles. It is common practice to introduce constitutive relations for the pressure-dependent density and viscosity to the asymptotic analysis a posteriori. This can lead to wrong predictions for piezoviscous fluids, where the viscosity strongly depends on pressure [5–7]. Almqvist et al [8] presented a compressible Reynolds equation for pressure–density relations that take
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