Second sound and multiple shocks in superfluid helium

Abstract.

A temperature pulse propagating in superfluid helium is studied through the simple waves theory. Our aim is to determine the shape change of this pulse, initially represented by a gaussian profile, using a generalized non-linear Cattaneo model proposed, in the framework of Extended Thermodynamics, by Ruggeri and co-workers in the case of a rigid conductor. The theoretical basis of our arguments is given in a paper [1] where the differential system of a binary mixture of Euler’s fluids is written as a system for a single heat conducting fluid. We prove that there exist three characteristic temperatures $\tilde{\theta}$, playing an essential role in the shape change of the propagating second sound wave; in particular, several families of multiple shocks (i.e. usual double shocks, double shocks only ahead or behind the wave profile, and very strange quadri-shocks) can appear, depending on the relation among the unperturbed temperature of Helium II and the characteristic temperatures and, in some cases, on the wave’s amplitude. Both the cases of a hot wave and a cold wave are discussed, proving that this last process is not symmetric with respect to the previous one. Finally, suitable choices of some parameters are suggested in order to better point out the changes of the wave profile and, in particular, the formation of multiple shocks.