Phonon hydrodynamics of short channels with entrance-length effects: solids and superfluids

Abstract In this paper we propose an expression for the effective thermal conductivity of thin and short channels in the context of non-linear phonon hydrodynamics with entrance-length effects. The channels communicate two thermal reservoirs of the same material than the channel, so that there are no interfaces at the longitudinal boundaries of the channel, but only an abrupt change of the radius. The channels are so short that the fully-developed Poiseuille flow usually assumed in phonon hydrodynamics cannot be reached in the length of the system. The corresponding expression is analogous to that arising in narrow and short channels of superfluid helium between two helium reservoirs at slightly different temperatures.