Elsevier

Physics Letters A

Volume 373, Issue 34, 17 August 2009, Pages 3052-3055
Physics Letters A

Heat conduction in multi-temperature mixtures of fluids: the role of the average temperature

https://doi.org/10.1016/j.physleta.2009.06.037Get rights and content

Abstract

The theory of mixture of fluids in which each constituent has its own temperature is more realistic than the models in which all the components have a common temperature and it is necessary in several physical situations, such as plasma physics. Nevertheless, from the theoretical point of view, the main problem regards the measurement of the temperature of each constituent. In this Letter we consider the simple case of the one-dimensional steady heat conduction between two walls and we adopt the concept of average temperature recently proposed by Ruggeri and co-workers such that the whole internal energy of the mixture depends only on the average temperature. Assuming that the two walls have given temperatures, we can first verify that the temperature behavior is different from the linear one predicted in the mixture theories with a single temperature. Moreover, we can also prove that, for a mixture of n constituents, the measure of the average temperature at 2(n1) points allows to know the temperature behavior of each constituent everywhere.

Section snippets

The multi-temperature model of homogeneous mixtures

The macroscopic theory of homogeneous mixtures is developed, within the framework of rational thermodynamics, on the assumption that each constituent obeys the same balance laws as a single fluid [1], [2]:ραt+div(ραvα)=τα,(ραvα)t+div(ραvαvαtα)=mα,(12ραvα2+ραεα)t+div{(12ραvα2+ραεα)vαtαvα+qα}=eα, where vα=vα and ρα,vα,εα,tα,qα, respectively denote the mass density, the velocity, the internal energy, the stress tensor and the heat flux of the constituent α of a mixture of n species (α=1,

Average temperature

The (MT) theory is of course the most realistic one and, also in agreement with the kinetic theory [7], it is a necessary theory in several physical situations, in particular in plasma physics [8]. Nevertheless, from the theoretical point of view, the main problem remains how it is possible to measure the temperatures of each constituent. Therefore, a question of definition of a macroscopic average temperature has to be posed. In this Letter we reconsider the definition of average temperature

Heat conduction in a mixture

We consider a one-dimensional mixture of gas at rest (vα=0), without chemical reactions (τα=0) between two walls 0xL, maintained at two different temperatures T(0)=T0, T(L)=TL. In both (CT) and (ST), the static field equation reduces to the global energy equation (2) that reads divq=0. In the one-dimensional case, this equation combined with the Fourier law with constant heat conductivity, yields the classical result of a linear behavior temperature profile as for a single fluid:T=0T=(TLT0)

Acknowledgements

This Letter was developed during a stay of J.L. as visiting professor of GNFM at the CIRAM – University of Bologna. It was supported in part (T.R.) by the GNFM-INdAM and in part (J.L.) by the Natural Science item of China under grant No.10701053 and Shanghai Leading Academic Discipline Project S30104.

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