Extended Reversible and Irreversible Thermodynamics: A Hamiltonian Approach with Application to Heat Waves

Georgy Lebon; David Jou; Miroslav Grmela

doi:10.1515/jnet-2016-0035

Article

Extended Reversible and Irreversible Thermodynamics: A Hamiltonian Approach with Application to Heat Waves

Georgy Lebon , David Jou and Miroslav Grmela

Published/Copyright: October 8, 2016

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 42 Issue 2

Abstract

A new version of extended irreversible thermodynamics (EIT) satisfying a Hamiltonian structure is proposed. For pedagogical purpose, the simple problem of linear heat conduction in a rigid body is investigated to illustrate the general framework. In contrast with earlier versions of EIT wherein the heat flux was upgraded to the status of state variables, we select here its conjugate dual and higher order fluxes as new independent variables. Their time–evolution equations are formed of reversible and irreversible terms but they cannot take any arbitrary form. Restrictions are placed on the reversible terms by imposing a Hamiltonian structure while the irreversible contribution is subject to the requirement to satisfy the second law of thermodynamics. Explicit expressions of the temperature and heat flux waves are also derived.

Keywords: extended thermodynamics; generic; Hamiltonian dynamics; heat and temperature waves

Funding statement: G.L. acknowledges support from the collaborative project Bruxelles-Wallonie-Québec through grant RI 15, biennium 2015–2017. D.J. acknowledges the financial supports from the Spanish Ministry of Economy and Competitiveness under grant FIS 2012-33099, and the Ministry of Education and Science under Consolided Project Nano Therm (grant CSD-2010-00044). M.G. acknowledges funding from Natural Sciences and Engineering Research Council of Canada (NSERC).

Appendix

We complete the analysis of Section 6 by considering the third-order tensor W as extra state variable besides the zero, first- and second-order tensors variables u, q and Q. In the entropy representation, the corresponding Gibbs’ equation is

(65)ds=T−1du+w.dq+Ω:dQ+Θ⊗dW,

wherein Θ is defined by Θ=∂s/∂W with designating the triple scalar product between third-order tensors. The Legendre transformation of s wherein the odd variables q and W are replaced by their duals w and Θ, respectively, is given by

(66)s∗∗∗u,w,Q,Θ=s−w.q−Θ⊗W,

and, in differential form,

(67)ds∗∗∗=T−1du−q.dw+Ω;dQ−W⊗dΘ.

The non-dissipative dynamical equations governing the set u, w, Q, Θ are obtained from

(68)(du/dtdw/dtdQ/dtdΘ/dt)=(0∇.00−∇0−β∇.00β∇0∇.00−∇0)(ds***/duds***/dwds***/dQds***/dΘ)

which implies the following time–evolution equations

(69)dudt=−∇.q,

(70)dwdt=−∇T−1−β∇.Ω,

(71)dQdt=−β∇q−∇.W,

(72)dΘdt=−∇Ω.

It is worth to stress that the result (71) confirms that the variable W is the flux of Q, in the same way that q is the flux of u.

To derive the dissipative contribution to expression (72), we will proceed in analogy with Sections 5 and 6.3, it consists in introducing a dissipation potential Φ_W of the form

(73)ΦW=12γw∂s∂Θ2,

wherein γ_w is positive definite to satisfy the positiveness of Φ_W and to add the term ∂Φ/∂(∂s/∂Θ) (72). We further assume a linear constitutive equation Θ=−ξW between the variables Θ and Ω with ξ a positive phenomenological coefficient to guarantee that s is maximum at local equilibrium. Making use of the third equality (61) between Ω and Q, the complete time–evolution equation for the third-order uneven tensor W may be written as

(74)dWdt=−ττQλλQξT2∇Q−1τwW,

with τ_w=ξ/γ_w (>0) designating the relaxation time of the variable W, a positive quantity as it should.

Acknowledgements

The authors are grateful to both referees whose comments have contributed to clarify several aspects of the theory.

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Received: 2016-4-27

Revised: 2016-8-9

Accepted: 2016-9-9

Published Online: 2016-10-8

Published in Print: 2017-4-1

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https://doi.org/10.1515/jnet-2016-0035

Keywords for this article

extended thermodynamics; generic; Hamiltonian dynamics; heat and temperature waves