Conformal Mapping on Rough Boundaries II: Applications to bi-harmonic problems

TL;DR

This paper applies conformal mapping techniques to analyze bi-harmonic fields near rough boundaries, specifically in Stokes flow and stress distribution problems, providing numerical and perturbative solutions for complex surface geometries.

Contribution

It introduces a comprehensive numerical and perturbative approach to bi-harmonic problems on rough surfaces, extending conformal mapping methods to practical applications.

Findings

Surface stress can be derived from the Hilbert transform of the slope at first order.

Stress distribution on self-affine surfaces exhibits a power-law tail at large stresses.

Numerical solutions are provided for sinusoidal and self-affine rough surfaces with slopes up to 2.5.

Abstract

We use a conformal mapping method introduced in a companion paper to study the properties of bi-harmonic fields in the vicinity of rough boundaries. We focus our analysis on two different situations where such bi-harmonic problems are encountered: a Stokes flow near a rough wall and the stress distribution on the rough interface of a material in uni-axial tension. We perform a complete numerical solution of these two-dimensional problems for any univalued rough surfaces. We present results for sinusoidal and self-affine surface whose slope can locally reach 2.5. Beyond the numerical solution we present perturbative solutions of these problems. We show in particular that at first order in roughness amplitude, the surface stress of a material in uni-axial tension can be directly obtained from the Hilbert transform of the local slope. In case of self-affine surfaces, we show that the stress distribution presents, for large stresses, a power law tail whose exponent continuously depends on the roughness amplitude.