Mutual friction in a heat current in liquid helium II I. Experiments on steady heat currents

Abstract

Experiments have been carried out on the conduction of heat through helium II in channels of large rectangular cross-section ($\sim $ 2 $\times $ 6 mm) for small heat current densities. The observed relationship between temperature gradient and heat-current density can be interpreted phenomenologically in terms of the Gorter-Mellink (1949) mutual friction force, F$_{sn}\approx $ A$\rho _{s}\rho _{n}$(v$_{s}$-v$_{n}$)$^{3}$ per unit volume, in the two-fluid model, and observed values of A have been found to agree fairly well with those deduced from earlier measurements. Evidence is presented to show that the magnitude of the mutual friction is determined entirely by the value of (v$_{s}$ - v$_{n}$), independently of the boundary conditions imposed on the flow. A study of the propagation of second sound across the heat currents has shown that, while the presence of the heat current leads to no observable change in the velocity of the second sound, it does lead to an attenuation; the attenuation is linear and approximately proportional to the square of the heat-current density. This behaviour can be described phenomenologically in terms of the two-fluid model, if it is assumed that, in the presence of both a steady heat current and a second sound wave, the Gorter-Mellink mutual friction must be generalized to the form F$_{sn}$ = A$\rho _{s}\rho _{n}$ U$^{2}$u, where u is the instantaneous relative velocity between the two fluids and U is the time-average of this relative velocity. This result shows that in the presence of a steady heat current one or both of the fluids must become modified in some way, and that an essentially linear mutual friction is associated with this modification. Observation of changes in the attenuation of second sound provides a more sensitive method of measuring mutual friction than does the observation of temperature gradients, and it has been shown by the former technique that in the channels used in the present work there is a critical heat current below which the mutual friction is either absent or very small.