Surface roughness and effective stick-slip motion

The effect of random surface roughness on hydrodynamics of viscous incompressible liquid is discussed. When the hydrodynamic decay length (the viscous wave penetration depth) is larger than the correlation radius (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 stick-slip length is always negative and the effective change of viscosity near the surface is positive signifying the effective average hampering of the hydrodynamic flows by the rough surface (stick rather than slip motion). A simple hydrodynamic model illustrates 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.