The Rotation of Liquid Helium II. II. The Theory of Mutual Friction in Uniformly Rotating Helium II

H. E. Hall, W. F. Vinen

Abstract

A discussion is given of models for the rotation of helium II involving regions of concentrated vorticity, and it is shown thermodynamically that an arrangement of vortex lines is energetically preferable to an arrangement of vortex sheets. It is suggested that such models exhibit the property of mutual friction, owing to the possibility of collisions between normal fluid excitations and the regions of concentrated superfluid vorticity; the observed anisotropy of this mutual friction (part I of this paper) is consistent only with a vortex-line model, so that the theoretical decision in favour of this model is confirmed by experiment. A detailed calculation of the magnitude and temperature-dependence of this mutual friction is given for the quantized vortex-line model of Onsager (1949) and Feynman (1955). The vortex lines are treated as classical vortex lines belonging entirely to the superfluid. The force of mutual friction arising from the collision of rotons with these lines is calculated in terms of the roton-line collision diameter $\overline{\sigma}$, taking into account a tendency for the lines to drag the gas of excitations (i.e. the normal fluid) in their vicinity, and a transverse motion of the lines due to the Magnus effect. The calculated mutual friction contains two components: one parallel to, and one perpendicular to, (v$_{s}$-v$_{n}$). The magnitude of the former component agrees well with the experimental results if $\overline{\sigma}$ is taken to be about 10 angstrom. The agreement between theory and experiment confirms that the normal fluid is dragged by the lines, and shows that the spacing of the lines must be close to the theoretical value given by Feynman; but it provides no evidence for or against a motion of the lines due to the Magnus effect. A rough value for $\overline{\sigma}$ is calculated in an appendix, and shown to agree as well as can be expected with the value derived from experiment.