We calculate the effect of a heat current on transporting 3He dissolved in
superfluid 4He 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)
3He down a long pipe. In the regime of 3He concentrations <~10−9 and temperatures ∼0.5 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 4He first
viscosity. On the other hand, the phonon wind drives the 3He out of local
equilibrium via phonon-3He scattering. For temperatures below 0.5 K, both
the phonon and 3He 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 3He
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 3He atoms is determined by the
time dependent 3He diffusion equation, which describes the competition
between advection by the phonon wind and 3He diffusion. As a consequence of
the thermal diffusivity being small compared with the 3He diffusivity, the
scale height of the final 3He distribution is much smaller than that of the
temperature gradient. We present exact solutions of the time dependent
temperature and 3He distributions in terms of a complete set of normal
modes.Comment: NORDITA PREPRINT 2015-37, 9 pages, 6 figure
