Phonon heat transport and Knudsen's minimum in liquid helium at low temperatures

The flow of heat in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at temperatures below 0.6°K is considered. Calculations are carried out for the specific cases of heat flow along a tube and flow down the channel between two parallel plates. It is assumed that the mean free path ${\Lambda }_{\parallel }$ for small-angle collisions between phonons is much less than the tube diameter or the spacing between the plates, but the calculations are valid for arbitrary values of the mean free path ${\Lambda }_{\perp }$ for large-angle collisions. For small ${\Lambda }_{\perp }$ the results are equivalent to the predictions of the Navier-Stokes equation with corrections for first- and second-order slip at the boundaries. For large ${\Lambda }_{\perp }$ the heat flow is close to the value given by Casimir's formula. At intermediate values of ${\Lambda }_{\perp }$ an effect similar to Knudsen's minimum is predicted by the theory. The calculations are compared briefly with the experimental results of Whitworth.