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