Height-Averaged Navier–Stokes Solver for Hydrodynamic Lubrication

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.