Eur. Phys. J. E (2010) 32: 147-164
https://doi.org/10.1140/epje/i2010-10610-7

Regular Article

Viscous compressible hydrodynamics at planes, spheres and cylinders with finite surface slip

¹ Physics Department, Technical University Munich, 85748, Garching, Germany
² IJS, Jamova 39, SI-1000, Ljubljana, Slovenia

^* e-mail: aerbas@ph.tum.de

Received: 27 November 2009
Accepted: 2 June 2010
Published online: 25 June 2010

Abstract

We consider the linearized time-dependent Navier-Stokes equation including finite compressibility and viscosity. We first constitute the Green's function, from which we derive the flow profiles and response functions for a plane, a sphere and a cylinder for arbitrary surface slip length. For high driving frequency the flow pattern is dominated by the diffusion of vorticity and compression, for low frequency compression propagates in the form of sound waves which are exponentially damped at a screening length larger than the sound wave length. The crossover between the diffusive and propagative compression regimes occurs at the fluid's intrinsic frequency ∼ c ² / , with c the speed of sound, the fluid density and the viscosity. In the propagative regime the hydrodynamic response function of spheres and cylinders exhibits a high-frequency resonance when the particle size is of the order of the sound wave length. A distinct low-frequency resonance occurs at the boundary between the propagative and diffusive regimes. Those resonant features should be detectable experimentally by tracking the diffusion of particles, as well as by measuring the fluctuation spectrum or the response spectrum of trapped particles. Since the response function depends sensitively on the slip length, in principle the slip length can be deduced from an experimentally measured response function.