Fast Solver for the Reynolds Equation on Piecewise Linear Geometries

TL;DR

This paper introduces a fast, linear-time solver for the Reynolds equation on piecewise linear geometries, enabling efficient analysis of lubrication problems with complex surface textures.

Contribution

It develops a novel exact solution formulation for piecewise linear geometries and demonstrates second-order accuracy with efficient linear-time computation.

Findings

The piecewise linear method operates in linear time relative to the number of components.

The methods achieve second-order accuracy for non-linear height approximations.

Application to textured geometries reveals limits of lubrication theory validity.

Abstract

The Reynolds equation is derived from the incompressible Navier Stokes equations under the lubrication assumptions of a long and thin domain geometry and a small scaled Reynolds number. The Reynolds equation is an elliptic differential equation and a dramatic simplification from the governing equations. When the fluid domain is piecewise linear, the Reynolds equation has an exact solution that we formulate by coupling the exact solutions of each piecewise component. We consider a formulation specifically for piecewise constant heights, and a more general formulation for piecewise linear heights; in both cases the linear system is inverted using the Schur complement. These methods can also be applied in the case of non-linear heights by approximating the height as piecewise constant or piecewise linear, in which case the methods achieve second order accuracy. We assess the time complexity of the two methods, and determine that the method for piecewise linear heights is linear time for the number of piecewise components. As an application of these methods, we explore the limits of validity for lubrication theory by comparing the solutions of the Reynolds and the Stokes equations for a variety of linear and non-linear textured slider geometries.