Practical Error Estimates for Reynolds' Lubrication Approximation and its Higher Order Corrections

TL;DR

This paper develops a method to compute higher-order corrections to Reynolds' lubrication approximation, providing rigorous error bounds that depend explicitly on the geometry, validated through numerical comparisons.

Contribution

It introduces a systematic way to calculate arbitrary order terms in the lubrication expansion with explicit, geometry-dependent error bounds.

Findings

Error bounds are independent of the geometry function h(x)

The expansion is asymptotic, not convergent, even for analytic h(x)

Numerical validation confirms the theoretical error estimates

Abstract

Reynolds' lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds' equation may be thought of as the zeroth order term in an expansion of the solution of the Stokes equations in powers of the aspect ratio $\epsilon$ of the domain. In this paper, we show how to compute the terms in this expansion to arbitrary order on a two-dimensional, $x$-periodic domain and derive rigorous, a-priori error bounds for the difference between the exact solution and the truncated expansion solution. Unlike previous studies of this sort, the constants in our error bounds are either independent of the function $h(x)$ describing the geometry, or depend on $h$ and its derivatives in an explicit, intuitive way. Specifically, if the expansion is truncated at order $2k$, the error is $O(\epsilon^{2k+2})$ and $h$ enters into the error bound only through its first and third inverse moments $\int_0^1 h(x)^{-m} dx$, $m=1,3$ and via the max norms $\big\|\frac{1}{\ell!} h^{\ell-1} \partial_x^\ell h\big\|_\infty$, $1\le\ell\le2k+2$. We validate our estimates by comparing with finite element solutions and present numerical evidence that suggests that even when $h$ is real analytic and periodic, the expansion solution forms an asymptotic series rather than a convergent series.