  We have studied the thermal conductivity of confined superfluids on a
bar-like geometry. We use the planar magnet lattice model on a lattice $H\times
H\times L$ with $L \gg H$. We have applied open boundary conditions on the bar
sides (the confined directions of length $H$) and periodic along the long
direction. We have adopted a hybrid Monte Carlo algorithm to efficiently deal
with the critical slowing down and in order to solve the dynamical equations of
motion we use a discretization technique which introduces errors only
$O((\delta t)^6)$ in the time step $\delta t$. Our results demonstrate the
validity of scaling using known values of the critical exponents and we
obtained the scaling function of the thermal resistivity. We find that our
results for the thermal resistivity scaling function are in very good agreement
with the available experimental results for pores using the temp
