Hydrodynamic boundary conditions and mode-mode coupling theory

Abstract

Hydrodynamic boundary conditions are often described in terms of a slipping length which linearly relates the tangential velocity and its normal derivative at the boundary. In this paper a generalized wave-number and frequency-dependent slipping length is defined in terms of the eigenmodes of the nonlocal hydrodynamic equations. The ordinary slipping length is the zero-frequency, long-wavelength limit of this generalized slipping length. A scattering-theory formalism provides the connection between the eigenmodes and the generalized slipping length. The mode-mode coupling theory of a compressible fluid in the presence of a plane wall is developed. This theory indicates that the zero-frequency generalized slipping length diverges in the long-wavelength limit. The contribution due to sound-mode coupling dominates and diverges as $k^{\frac{-1\mathrm{}}{2}}$. While this result shows the nonexistence of the ordinary slipping length, it does not invalidate the use of no-slip boundary conditions in ordinary hydrodynamics. Corrections to Stokes's law are discussed in terms of the divergence of the slipping length, and the form of the divergence at low densities is indicated.