  The size-dependent and flux-dependent effective thermal conductivity of
narrow capillaries filled with superfluid helium is analyzed from a
thermodynamic continuum perspective. The classical Landau evaluation of the
effective thermal conductivity of quiescent superfluid, or the Gorter-Mellinck
regime of turbulent superfluids, are extended to describe the transition to
ballistic regime in narrow channels wherein the radius $R$ is comparable to (or
smaller than) the phonon mean-free path $\ell$ in superfluid helium. To do so
we start from an extended equation for the heat flux incorporating non-local
terms, and take into consideration a heat slip flow along the walls of the
tube. This leads from an effective thermal conductivity proportional to $R^2$
(Landau regime) to another one proportional to $R\ell$ (ballistic regime). We
consider two kinds of flows: along cylindrical pipes and along two infinite
parallel plates.

Superconductivity;Mesoscale and Nanoscale Physics