Heat conduction in multi-temperature mixtures of fluids: the role of the average temperature
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]: where and , 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 (
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 (), without chemical reactions () between two walls , maintained at two different temperatures , . In both (CT) and (ST), the static field equation reduces to the global energy equation (2) that reads . 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:
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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