Discontinuity Waves, Shock Formation and Critical Temperature in Crystals

Abstract

The propagation of discontinuity waves in a rigid heat conductor at low temperatures is studied by using a generalized non-linear Maxwell-Cattaneo equation developed in the framework of extended thermodynamics. The critical time (i.e., the instant in which a shock wave formation occurs) is evaluated in both cases of infinite and finite heat conductivity. The critical temperature e, pointed out in our previous papers concerning the propagation of shock and simple waves, once more plays an important role: in fact, now it determines two different regimes for the wave propagation and this phenomenon, from a mathematical point of view, is related to the loss of the genuine non-linearity when fl = 6. In the last sections some numerical results are given and a brief analysis about the evolution of a possible initial wave profile is performed.

Second sound and multiple shocks in superfluid helium

“…We underline that the equilibrium quantities (3), from which all the constitutive functions can be determined, have been evaluated at the saturated vapour pressure. 4 Neglecting the quadratic terms in q, since as usual we are studying the phenomenon near an equilibrium state, the previous equations coincide with the ones given by (1) and already obtained by Ruggeri and co-workers for the model of second sound propagation in crystals [19], [20], [21], [22]. But now the physical meaning of the thermal inertia α and the function ν is well explained (ν is a pressure and α is the inverse of a difference of enthalpies and kinetic energies) and the system (1) can be considered helpful also in the case of Helium II, provided we concentrate our attention only to the propagation of second sound wave.…”

“…One of the principal differences of He II with respect to the crystals as NaF and Bi [19], [20], [21], [22] is that there exist three characteristic temperatures θ for which the shape function Φ reaches the extrema and dλ/dθ, evaluated at equilibrium, changes sign, as it is possible to deduce from Figure 1. At equilibrium, these temperatures are θ 1 0.47 K, θ 2 0.95 K, θ 3 1.87 K.…”

“…From an essentially macroscopic point of view, many other models [14] start, as a rule, by a modification of the Cattaneo's equation [15] to explain thermal disturbances, also in the case of finite amplitude waves [16], [17] where a linear theory is no longer sufficient. In these last years Ruggeri and co-workers [18], [19], [20], [21], [22] studied the second sound propagation in crystals, in the framework of Extended Thermodynamics [23], introducing a thermal inertia factor in Cattaneo's equation. In [1] it was observed that this rigid heat conductor model can be obtained by a binary mixture of Euler's fluids, "making rigid" the system in order to isolate the second sound wave.…”

Mixtures of Euler’s fluids and second sound propagation in superfluid helium

“…Using the universal principles of the Extended Thermodynamics [16], Ruggeri and co-workers [17], [18], [19], [20] studied the second sound propagation in crystals introducing a thermal inertia factor in Cattaneo's equation.…”

“…Then, in [19] a temperature pulse propagating in a solid is studied by the simple waves theory, proving again thatθ plays an essential role in the wave shape changes. In [20], the theory was applied to study the propagation of discontinuity waves (acceleration waves in the language of continuum mechanics) in a rigid heat conductor at low temperature showing two different regimes, depending onθ, for the wave propagation. The model which we use in all these papers, is based on a differential non linear system constituted by an energy balance conservation equation and an evolution equation for the heat flux, namely ∂(ρε) ∂t…”