Surface Roughness and Effective Stick-Slip Motion

TL;DR

This paper investigates how random surface roughness affects viscous fluid flow, introducing effective boundary conditions characterized by stick-slip length and viscosity renormalization, which depend on surface inhomogeneity correlations.

Contribution

It derives explicit formulas for effective boundary parameters based on surface roughness correlation functions, enabling simplified hydrodynamic modeling of rough surfaces.

Findings

Effective stick-slip length is negative, indicating flow slowdown.

Maximum dissipation frequency reveals surface inhomogeneity size.

Explicit relations connect surface roughness correlations to hydrodynamic boundary conditions.

Abstract

The effect of random surface roughness on hydrodynamics of viscous incompressible liquid is discussed. Roughness-driven contributions to hydrodynamic flows, energy dissipation, and friction force are calculated in a wide range of parameters. When the hydrodynamic decay length (the viscous wave penetration depth) is larger than the size of random surface inhomogeneities, it is possible to replace a random rough surface by effective stick-slip boundary conditions on a flat surface with two constants: the stick-slip length and the renormalization of viscosity near the boundary. The stick-slip length and the renormalization coefficient are expressed explicitly via the correlation function of random surface inhomogeneities. The effective stick-slip length is always negative signifying the effective slow-down of the hydrodynamic flows by the rough surface (stick rather than slip motion). A simple hydrodynamic model is presented as an illustration of these general hydrodynamic results. The effective boundary parameters are analyzed numerically for Gaussian, power-law and exponentially decaying correlators with various indices. The maximum on the frequency dependence of the dissipation allows one to extract the correlation radius (characteristic size) of the surface inhomogeneities directly from, for example, experiments with torsional quartz oscillators.