Emergence of Non-Fourier Hierarchies

Fülöp, Tamás; Kovács, Róbert; Lovas, Á.; Rieth, Ágnes; Fodor, Tamás; Szücs, Mátyás; Ván, P.; Gróf, Gyula

doi:10.3390/e20110832

Cited by 35 publications

(13 citation statements)

References 45 publications

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“…Even if quantities known from mechanics form a closed system of equations to solve numerically, monitoring temperature (or other thermodynamical quantities) for a nonreversible system can give insight on the processes and phenomena, for example, pointing out the presence of viscoelasticity/rheology, and displaying when plastic changes start [10]. In addition, temperature can also react, in the form of thermal expansion and heat conduction, even in situations where one is not prepared for this 'surprise' [11]. arXiv:1908.07975v3 [physics.class-ph] 24 Oct 2019 Furthermore, in a sense, thermodynamics is a stability theory.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Fülöp

Kovács

Szücs

et al. 2020

Entropy

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On the example of the Poynting-Thomson-Zener rheological model for solids, which exhibits both dissipation and wave propagation -with nonlinear dispersion relation -, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error and dispersive error are analysed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest -elastic -level of modelling.

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Section: Introductionmentioning

confidence: 99%

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Fülöp

Kovács

Szücs

et al. 2020

Entropy

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“…The fourth paper [ 17 ], by Tamás Fülöp, Róbert Kovács, Ádám Lovas, Ágnes Rieth, Tamás Fodor, Mátyás Szücs, Péter Ván, and Gyula Gróf, analyzes the non-Fourier heat conduction phenomenon on room temperature and proposes to use the Guyer-Krumhansl equation to replace classical Fourier’s law for room-temperature phenomena in the modeling of heterogeneous materials. Then, generalized heat conduction equations are introduced where Fourier heat conduction is coupled to elasticity via thermal expansion, resulting in a particular generalized heat equation for the temperature field.…”

Section: Modern Fourier Heat Theorymentioning

confidence: 99%

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Barbaresco¹,

Gazeau

2019

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For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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“…A principal limitation of this model is that Fourier heat conduction leads to a parabolic differential equation for temperature, where the signal propagation speed is infinite. Moreover, several experiments justify deviation from the Fourier-like behaviour, e.g., low-temperature measurements of heat conduction in superfluid helium [9,10,11] and in NaF crystals [12,13,14], and room-temperature measurements on heterogeneous samples like rocks and layered structures [15,16,17].…”

Section: Introductionmentioning

confidence: 99%

A case study of non-Fourier heat conduction using Internal Variables and GENERIC

Szücs,

Pavelka,

Kovács

et al. 2021

Preprint

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Applying simultaneously the methodology of Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and the framework of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC), we demonstrate that, in heat conduction theories, entropy current multipliers can be interpreted as relaxed state variables. Fourier's law and its various extensionsthe Maxwell-Cattaneo-Vernotte, Guyer-Krumhansl, Jeffrey's type, Ginzburg-Landau (Allen-Cahn) type and ballistic-diffusive-heat conduction equations are derived in both formulations. Along these lines, a comparison of NET-IV and GENERIC is also performed. Our results may pave the way for microscopic/multiscale understanding of beyond-Fourier heat conduction, and open new ways for numerical simulations of heat-conduction problems.1 When heat current density is treated as a state variable then the obtained equation is a Ginzburg-Landau one ). In parallel, when a vectorial internal variable is assumed and heat current density is looked for as a function of the state variables then the obtained equation is Cahn-Hilliard-like. For details, see [27,28].

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Emergence of Non-Fourier Hierarchies

Cited by 35 publications

References 45 publications

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

A case study of non-Fourier heat conduction using Internal Variables and GENERIC

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