Convergence to the Reynolds approximation with a double effect of roughness

TL;DR

This paper investigates how boundary roughness affects lubrication flow, demonstrating that specific rugosity profiles can accelerate flow and providing a rigorous mathematical justification for the limit system with numerical illustrations.

Contribution

It introduces a novel analysis of how boundary roughness perturbs the lubrication approximation and justifies the limit system using two-scale convergence techniques.

Findings

Flow acceleration can be achieved with appropriate roughness profiles.

Abrupt changes in the boundary profile have significant effects.

Numerical results illustrate the theoretical limit system in 3D.

Abstract

We prove that the lubrication approximation is perturbed by a non-regular roughness of the boundary. We show how the flow may be accelerated using adequate rugosity profiles on the bottom. We explicit the possible effects of some abrupt changes in the profile. The limit system is mathematically justified through a variant of the notion of two-scale convergence. Finally, we present some numerical results, illustrating the limit system in the three-dimensional case.