Continuum approach to phonon gas and shape changes of second sound via shock waves theory

Abstract

A continuum approach, based on the principles of modern extended thermodynamics, describing the model of a phonon gas is performed. The main difference with the ideal phonon gas theory consists in the presence of a thermal inertia. We apply the shock wave theory and discuss the selection rules for physical shocks (the Lax conditions and the entropy growth). In this way the existence of two new kinds of shocks (hot and cold shocks) in rigid heat conductors at low temperature is pointed out. In particular a critical temperature, characteristic of each material, changing the structure of the previous types of shocks is analytically deduced. This characteristic temperature permits also to explain the modification of the received second sound wave form with respect to the initial wave profile. Finally, the results are applied to the case of high-purity crystals (NaF, Bi, 3He and 4He) and compared with experimental results. PACS 44.10 -Heat conduction (models, phenomenological description). PACS 66.70 -Nonelectronic thermal conduction and heat-pulse propagation in nonmetallic solids.

A Wavefront Tracking Algorithm for N×N Nongenuinely Nonlinear Conservation Laws

“…Systems of conservation laws with characteristic fields that fulfill the assumption (A) physically arise in several contexts, for instance in studying elastodynamic (e.g., see [Dp2]) or rigid heat conductors at low temperature [RMS1,RMS2]. An example is given by the generalized Cattaneo model proposed by T. Ruggeri and co-workers [RMS1,RMS2] to describe the heat propagation in high-purity crystals (He, NaF, Bi):…”

“…An example is given by the generalized Cattaneo model proposed by T. Ruggeri and co-workers [RMS1,RMS2] to describe the heat propagation in high-purity crystals (He, NaF, Bi):…”

Discontinuity Waves, Shock Formation and Critical Temperature in Crystals

“…This coefficient vanishes when where e" (e" = 3.38 K for Bi and e" = 15.36 K for NaF) is the characteristic temperature pointed out formerly in [1][2][3]. In the present case too, this temperature plays an important role: in fact, in the next section, we show that when 8 "cross through" the critical temperature e" the genuine non-linearity is lost and so the coefficient a(t) changes its sign (see (13)) determining two different regimes for the wave propagation.…”

“…In some previous papers [1][2][3] the behaviour of a temperature pulse propagating in a solid was investigated, using a generalized non-linear Cattaneo model [4, 51 developed in the framework of extended thermodynamics [6]. In particular, in [l, 21, on the ground of the shock wave theory, it was proved the existence of a critical temperature e", characteristic of the material, such that the nature of the shock changes: in fact, if the unperturbed temperature 8, is less than e" the temperature 8, behind the shock wave front must be such that 8, > 8, (hot shock) and, vice versa, if 8, > e" the contrary happens, i.e., 8, < 8, (cold shock).…”

“…A detailed discussion of the previous system is performed in [2]. We briefly observe that the system (1)- (2) is a particular case of a quasi-linear first-order system…”