Fast Solvers for the Reynolds Equation on Piecewise Linear Geometries

TL;DR

The paper presents two Schur-complement-based solvers for the Reynolds equation on piecewise-height geometries: a piecewise constant-height (PWC) method and a more general piecewise linear-height (PWL) method. The PWL approach achieves linear-time complexity in the number of height pieces, outperforming PWC and standard finite-difference methods, while maintaining second-order accuracy for non-linear heights. Through four textured-slider examples, the work benchmarks Reynolds-equation solutions against full Stokes flows, highlighting the limits of lubrication theory in regions with large surface gradients or discontinuities. Overall, the methods provide robust, scalable tools for fast Reynolds solutions in lubrication problems and clarify the regime where lubrication theory remains valid for slider bearings.

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.

Figures (6)

• Figure 1: The pressure and velocity solutions for the backward facing step with the Reynolds equation (left) and the Stokes equation (right). The solution to the Reynolds equation underestimates the pressure drop $\Delta P$, and does not capture corner flow recirculation.
• Figure 1: The PWL method is linear time and faster than the PWC method (quadratic time) and the FD method (cubic time).
• Figure 1: Convergence of solutions to the Reynolds equation for the given sinusoidal slider.
• Figure 2: The pressure and velocity solutions for the wedge slider with the Reynolds equation (left) and the Stokes equation (right). The solutions to the Reynolds and the Stokes equations are similar in the case of a moderately sloped wedge.
• Figure 3: The pressure and velocity solutions for the logistic step with the Reynolds equation (left) and the Stokes equation (right). As with the backward facing step, the Reynolds equation underestimates the pressure drop $\Delta P$, and does not capture cross film pressure variation or corner flow recirculation as seen with the Stokes equation.