Spatial distribution of vortices and anisotropy of mutual friction in rotating He II

Abstract

Accurate measurements of the attenuation of second sound in a rotating parallelepipedic cavity are reported. The cavity, which is used as a half-wave resonator, can be tilted about one of its symmetry axes so that the two large faces make an arbitrary angle $\ensuremath{\theta}$ with the angular velocity $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}$. A continuum model is developed predicting that vortices, at equilibrium, must bend near the inclined plane walls, and terminate perpendicular to the boundary. The curvature of the vortices, otherwise parallel to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}$ and uniformly distributed, takes place over a characteristic depth ${d}_{0}=0.29{\ensuremath{\Omega}}^{\frac{\ensuremath{-}1}{2}}$ mm which is small compared with the dimensions of the cavity. The resulting changes in density and orientation of vortex lines, though localized near the walls, measurably affect the detailed dependence on both $\ensuremath{\Omega}$ and $\ensuremath{\theta}$ of the quality factors of resonant modes. Experimental results taken with two fundamental modes of the cavity support our model. As a by-product of these measurements we obtain information about the angular dependence of mutual friction. A small but clearly nonzero attenuation was observed for a second sound propagated in the direction of vortex lines. This axial attenuation can be described by introducing, besides the first mutual-friction parameter $B$, another dissipative coefficient ${B}^{\ensuremath{'}\ensuremath{'}}$ which is about 2.5% of $B$ (at 1.9 K). In Hall and Vinen's equations of motion of He II this ${B}^{\ensuremath{'}\ensuremath{'}}$ should represent the axial component (along $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}$) of the mutual-friction force. In this work we have deliberately ignored the intricate problem of metastable states, trying to get rid of them in experiments; thus reported results all refer to states of (or near) thermodynamic equilibrium.