Mutual friction in a heat current in liquid helium II III. Theory of the mutual friction

Abstract As was shown in part II, the Gorter-Mellink mutual friction force in a heat current is probably associated with turbulence in the superfluid. Following Feynman, it is suggested that this turbulence takes the form of a tangled mass of quantized vortex lines, so that the mutual friction probably arises from collisions between thermal excitations and these vortex lines. From the observed properties of the mutual friction it is deduced that the walls of the channel carrying the heat current play no essential role in the generation, maintenance or decay of the turbulence, but merely introduce a number of incidental complications; the present paper ignores these complications and deals therefore with the idealized case of a homogeneous heat current in an unbounded volume of helium. The turbulence in this idealized case must be homogeneous, and it is shown from experimental evidence that it is probably also isotropic. Values of the force exerted on unit length of a vortex line, which have been derived from the study of the attenuation of second sound in uniformly rotating helium, are used to calculate the Gorter-Mellink force per unit volume in terms of the length of line per unit volume; then by a simple dimensional argument it is shown that the force must depend on (vs — vn) in a manner agreeing with experiment. An attempt is made to produce a detailed theory of the generation and decay of superfluid turbulence: it is shown first that owing to the Magnus effect the turbulence can probably be built up by the action of the mutual friction force exerted on the individual lines, although the way in which turbulence can be initiated in undisturbed helium is not known, and secondly that the turbulence can probably decay in a manner closely analogous to the decay of homogeneous turbulence in an ordinary fluid. Equations for the rate of generation and decay of turbulence are obtained by dimensional arguments, and by analogy with formulae known to apply to turbulence in an ordinary fluid. Comparison of the equations with the experimental results described in parts I and II reveals good agreement, and makes it possible to deduce the form and magnitude of a term describing the effect of the unknown initiation process.