Non linear hyperbolic fields and waves

Boillat, G.

doi:10.1007/bfb0093705

Cited by 28 publications

(16 citation statements)

References 89 publications

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“…It is possible to show (Boillat [7,8], Ruggeri and Strumia [9]) that, because of (7) 1 , there exist four potentials h…”

Section: Dissipative Hyperbolic Systemsmentioning

confidence: 99%

The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics

Ruggeri

2008

Entropy

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Abstract:We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous weak solutions and to symmetrize general systems of hyperbolic balance laws. A particular attention is given on the local and global well-posedness of the relative Cauchy problem for smooth solutions. Examples are given in the case of extended thermodynamics for rarefied gases and in the case of a multi-temperature mixture of fluids.

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“…It is possible to show (Boillat [7,8], Ruggeri and Strumia [9]) that, because of (7) 1 , there exist four potentials h…”

Section: Dissipative Hyperbolic Systemsmentioning

confidence: 99%

The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics

Ruggeri

2008

Entropy

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“…Let us recall the GodunovFriedrichs-Lax method of symmetrisation of quasilinear conservation laws [4][5] (see also different applications and generalizations in [11][12]). We suppose that the system of conservation laws for n variables q has the form…”

Section: The Following Vector Relation Is An Identitymentioning

confidence: 99%

A new form of governing equations of fluids arising from Hamilton's principle

Gavrilyuk

Gouin

1999

International Journal of Engineering Science

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A new form of governing equations is derived from Hamilton's principle of least action for a constrained Lagrangian, depending on conserved quantities and their derivatives with respect to the time-space. This form yields conservation laws both for non-dispersive case (Lagrangian depends only on conserved quantities) and dispersive case (Lagrangian depends also on their derivatives). For non-dispersive case the set of conservation laws allows to rewrite the governing equations in the symmetric form of Godunov-Friedrichs-Lax. The linear stability of equilibrium states for potential motions is also studied. In particular, the dispersion relation is obtained in terms of Hermitian matrices both for non-dispersive and dispersive case. Some new results are extended to the two-fluid non-dispersive case.

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“…In this paper, we see that the dependance of entropy gradient is necessary for isentropic waves of acceleration along the interfaces: the fact that the internal energy depends not only on the gradient of matter density but also on the gradient of entropy, yields a new kind of waves which does not appear in the simpler models. It is easy to see they are exceptional waves in the sense of Lax [12] and they appear only in, at least, two-dimension spaces. Recent experiments in space laboratories, for carbonic dioxide near its critical point have showed the possibility of such waves [13] and should justify the well-founded of the thermocapillary fluid model.…”

Section: Resultsmentioning

confidence: 99%

Adiabatic waves along interfacial layers near the critical point

Gouin

2004

Comptes Rendus. Mécanique

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Near the critical point, isothermal interfacial zones are investigated starting from a non-local density of energy. From the equations of motion of thermocapillary fluids [1], we point out a new kind of adiabatic waves propagating along the interfacial layers. The waves are associated with the second derivatives of densities and propagate with a celerity depending on the proximity of the critical point. RésuméPrès du point critique, les couches interfaciales sont modéliséesà l'aide d'une densité d'énergie non locale. A partir deséquations du mouvement des fluides thermocapillaires [1], nous mettons enévidence des ondes adiabatiques se propageant le long des couches interfaciales. Ces ondes associées aux dérivées secondes des densités se meuvent avec une célérité dépendant de la proximité du point critique. Version française abrégéeLe modèle le plus simple permettant de considérer les couches capillaires et les phases comme un unique milieu continu, consisteà prendre en compte uneénergie comme la somme de deux termes : le premier correspondà l'énergie du milieu supposé uniforme et de compositionégaleà la composition locale, le second est associéà la non uniformité du fluide et exprimé par un développement en gradient de la masse volumique qui est limité au second 1 ordre [2,3]. Ce modèle permet le prolongement au cas dynamique desétudes effectuées sur les systèmes enéquilibre. L'énergie interne volumique du fluide est maintenant proposée sous la forme d'une densité α dépendant non seulement de grad ρ mais aussi de grad s où ρ et s notent respectivement la masse volumique et l'entropie spécifique. Les milieux associés appelés fluides thermocapillaires [1] ont une densité d'énergie de la forme (1). Pour un fluide isotrope, l'énergie prend la forme (2) et l'équation du mouvement s'écrit sous les formeséquivalentes (5) ou (6) dans lesquelles une entropie h et une température θ dites thermocapillaires explicitées en (4) font intervenir deux nouveaux vecteurs Φ and Ψ associésà la non homogénéité des zones interfaciales et donnés par (3). Une telleénergie interne associe dans leséquations du mouvement et de l'énergie un tenseur des contraintes sphérique. Le cas desécoulements isentropiques associésà l'équation (7) correspondà la conservation de l'énergie (8). Les mouvements isothermes représentés par (9) et (10) ont comme cas particulier leséquilibres isothermes entre phases. Il est alors possible d'étudier les profils des densités dans leséquilibres unidimensionnels correspondant aux zones interfaciales planes. Le choix d'uné energie interne thermocapillaire de la forme (11) associéeà uneénergie interne α du milieu supposé homogène ramène l'étude des zones interfaciales a l'analyse d'un système dynamique (12b), (12a) qui donne deuxéquations différentielles. L'analyse asymptotique de ce système au voisinage du point critique montre que la densité de masse pilote le système en ce sens que l'on est ramenéà une décomposition dynamique représentée par le système (15b), (15a). L'intégration de ce système nous ramène ...

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Non linear hyperbolic fields and waves

Cited by 28 publications

References 89 publications

The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics

The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics

A new form of governing equations of fluids arising from Hamilton's principle

Adiabatic waves along interfacial layers near the critical point

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