The rotation of liquid helium II II. The theory of mutual friction in uniformly rotating helium II

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 σ̅, 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 σ̅ is taken to be about 10 Å. 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 σ̅ is calculated in an appendix, and shown to agree as well as can be expected with the value derived from experiment.

Vortex phase transitions in 21/2 dimensions

“…9). This con-clusion agrees well with the visual observation of superfluid vortices, which look smooth, and with standard results on their stretching behavior; for example, the rate of change ~~ of the line length L per unit volume is ~~ '" L in a classical fluid [4], while it is '" L 3 / 2 w in a superfluid, where w is a "counterflow" velocity that vanishes at Tx [37].…”

“…The Euler equations for incompressible flow are not compatible with constant T, and we have already pointed out above that the quantization of circulation is not a major effect. The important qualitative difference between classical and superfluid vortices is that the latter do not have to move at the velocity of the fluid that surrounds them [37]. Schwarz [38] has proposed that superfluid vortex motion can be described by the local self-induction approximation with Hall-Vinen friction and reconnection added.…”

Numerical Studies of Quantum Turbulence

“…The mutual friction was later found to come from the interaction between quantized vortices and normal fluid [11,12]. The original idea of quantized circulation had already been suggested by Onsager [13], but Feynman proposed the concept of a quantized vortex filament and a turbulent superfluid state consisting of a tangle of quantized vortices [14].…”

“…The idea of the mutual friction is developed to another macroscopic model of two fluids, namely the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model [12,21]. The HVBK model is a continuous coarse-grained model which assumes that superfluid contains a large number of parallel quantized vortices in each fluid parcel and ignores the details of individual vortices and their fast dynamics.…”

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“…A considerable amount is known about the properties [12][13][14][15] of vortices, and about how they are created at very low temperatures [16][17][18] but the mechanism responsible for the vorticity that appears 19,8 as a result of passing through the λ-transition (which separates the normal helium-I and the superfluid helium-II phases) is not understood. One possibility is that preexisting rotational flow caused by e.g.…”