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

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

Conformal Mapping on Rough Boundaries I: Applications to harmonic problems

The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this boundary. We introduce a conformal mapping technique that is tailored to this problem in two dimensions. An efficient algorithm is introduced to compute the conformal map for arbitrarily chosen boundaries. Harmonic fields can then simply be read from the conformal map. We discuss applications to "equivalent" smooth interfaces. We study the correlations between the topography and the field at the surface. Finally we apply the conformal map to the computation of inhomogeneous harmonic fields such as the derivation of Green function for localized flux on the surface of a rough boundary