On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single‐ and multi‐temperature models

Abstract

SUMMARYThe first rational model of homogeneous mixtures of fluids was proposed by Truesdell in the context of rational thermodynamics. Afterwards, two different theories were developed: one with a single-temperature (ST) field of the mixture and the other one with several temperatures. The two systems are from the mathematical point of view completely different and the relationship between their solutions was not clarified.In this paper, the hyperbolic multi-temperature (MT) system of a mixture of Eulerian fluids will be explained and it will be shown that the corresponding single-temperature differential system is a principal subsystem of the MT one. As a consequence, the subcharacteristic conditions for characteristic speeds hold and this gives an upper-bound esteem for pulse speeds in an ST model. Global behaviour of smooth solutions for large time for both systems will also be discussed through the application of the ShizutaKawashima condition. Finally, as an application, the particular case of a binary mixture is considered.

The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics

“…In a recent paper, Ruggeri and Simić [27] discussed the mathematical difference between the ST and the multi temperature (MT) models when the fluid components are Eulerian gases (q α = 0, σ α = 0). They proved that the differential system of the ST model is a principal sub-system of the MT model and for large time, MT solutions converge to ST ones.…”

Entropy Principle and Recent Results in Non-Equilibrium Theories

“…It was proved that the assumptions of the previous theorems are fulfilled in both classical [117] and relativistic [118], [119] RET and also in the case of mixture of gas with multi-temperatures [120].…”

Second Law Constraints on the Dynamics of a Mixture of Two Fluids at Different Temperatures

“…In the Eringen [13] aware of that addresses this in a serious fashion is Gouin and Ruggeri [9], Ruggeri and Simić [7,8], and Ruggeri and Lou [6]. Two important ideas have emerged from their studies.…”

“…As noted by Ruggeri and Lou [6], most studies consider a common temperature for both constituents even though there are many examples where this is clearly not the case. In fact the only recent papers we are aware of that address multi-temperature mixtures of fluids are the above citation along with Ruggeri and Simić [7,8] and Gouin and Ruggeri [9]. Ruggeri and Lou [6] note that one motivation for single temperature models is to avoid difficulties in measuring constituent temperatures.…”