Transport in ultradilute solutions of He in superfluid He
arXiv:1505.01468
Abstract
We calculate the effect of a heat current on transporting He dissolved in superfluid He at ultralow concentration, as will be utilized in a proposed experimental search for the electric dipole moment of the neutron (nEDM). In this experiment, a phonon wind will generated to drive (partly depolarized) He down a long pipe. In the regime of He concentrations and temperatures K, the phonons comprising the heat current are kept in a flowing local equilibrium by small angle phonon-phonon scattering, while they transfer momentum to the walls via the He first viscosity. On the other hand, the phonon wind drives the He out of local equilibrium via phonon-He scattering. For temperatures below K, both the phonon and He mean free paths can reach the centimeter scale, and we calculate the effects on the transport coefficients. We derive the relevant transport coefficients, the phonon thermal conductivity and the He diffusion constants from the Boltzmann equation. We calculate the effect of scattering from the walls of the pipe and show that it may be characterized by the average distance from points inside the pipe to the walls. The temporal evolution of the spatial distribution of the He atoms is determined by the time dependent He diffusion equation, which describes the competition between advection by the phonon wind and He diffusion. As a consequence of the thermal diffusivity being small compared with the He diffusivity, the scale height of the final He distribution is much smaller than that of the temperature gradient. We present exact solutions of the time dependent temperature and He distributions in terms of a complete set of normal modes.
NORDITA PREPRINT 2015-37, 9 pages, 6 figures