Slip corrections for vibrating wire resonators

Vibrating wire resonators have long been used as probes of the helium liquids. They have been employed as viscometers or thermometers over a wide temperature range and a corresponding range of mean free paths, from the hydrodynamic regime to the ballistic regime. The scattering of helium quasi-particles from the surface of the wire may be diffuse or specular depending on the smoothness of the wire and the helium liquid through which it is moving. The scattering is characterized by a slip length which is of order the mean free path. Recent experiments by Perisanu and Vermeulen on He-3/He-4 mixtures show that the scattering appears to be mostly specular; in this case there is an enhancement of the slip length. However, they find that the existing theory of Hojgaard Jensen et al. does not fit well to their experiments. In this paper, we present new calculations of the force on the wire for a mean free path which is less than a tenth of the wire diameter. The boundary conditions have been modified from those of Hojgaard Jensen et al.: we include the effect of curvature of the wire and we no longer assume that the boundary condition is that the normal component of velocity in the rest frame of the wire is zero. Perisanu and Vermeulen find that the theory gives an impressive fit to their mixture data.