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Some Recent Results on Multi-temperature Mixture of Fluids

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Continuous Media with Microstructure

Abstract

We present a survey on some recent results concerning the different models of a mixture of compressible fluids. In particular we discuss the most realistic case of a mixture when each constituent has its own temperature (MT) and we first compare the solutions of this model with the one whit a unique common temperature (ST). In the case of Eulerian fluids it will be shown that the corresponding (ST) differential system is a principal subsystem of the (MT) one. Global behavior of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta-Kawashima condition. Than we introduce the concept of the average temperature of mixture based upon the consideration that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. Through the procedure of Maxwellian iteration a new constitutive equation for non-equilibrium temperatures of constituents is obtained in a classical limit, together with the Fick’s law for the diffusion flux. To justify the Maxwellian iteration, we present for dissipative fluids a possible approach of a classical theory of mixture with multi-temperature and we prove that the differences of temperatures between the constituents imply the existence of a new dynamical pressure even if the fluids have a zero bulk viscosity. in the case of the one-dimensional steady heat conduction between two walls, we have verified that the main effect of multi-temperature is that the average temperature is not a linear function of the distance as in the case of the ST theory.

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Authors and Affiliations

  1. Department of Mathematics and Research Center of Applied Mathematics (C.I.R.A.M.), University of Bologna, Via Saragozza 8, 40123, Bologna, Italy

    Tommaso Ruggeri

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  1. Tommaso Ruggeri
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Editors and Affiliations

  1. FG Grundbau und Bodenmechanik, Technische Universität Berlin, Gustav-Meyer-Allee 25, 13355, Berlin, Germany

    Bettina Albers

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Ruggeri, T. (2010). Some Recent Results on Multi-temperature Mixture of Fluids. In: Albers, B. (eds) Continuous Media with Microstructure. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11445-8_5

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  • DOI: https://doi.org/10.1007/978-3-642-11445-8_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11444-1

  • Online ISBN: 978-3-642-11445-8

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