For binary mixtures of fluids without chemical reactions, but with components having different temperatures, the Hamilton principle of least action is able to produce the equation of motion for each component and a balance equation of the total heat exchange between components. In this nonconservative case, a Gibbs dynamical identity connecting the equations of momenta, masses, energy and heat exchange allows to deduce the balance equation of energy of the mixture. Due to the unknown exchange of heat between components, the number of obtained equations is less than the number of field variables. The second law of thermodynamics constrains the possible expression of a supplementary constitutive equation closing the system of equations. The exchange of energy between components produces an increasing rate of entropy and creates a dynamical pressure term associated with the difference of temperature between components. This new dynamical pressure term fits with the results obtained by classical thermodynamical arguments in [1] and confirms that the Hamilton principle can afford to obtain the equations of motions for multi-temperature mixtures of fluids.
Referenze Bibliografiche
[1]
H. GOUIN -
T. RUGGERI,
Identification of an average temperature and a dynamical pressure in multi-temperature mixture of fluids,
Physical Review E,
78 (
2008), 016303. |
fulltext (doi) |
MR 2595596[2]
M. ISHII,
Thermo-fluid dynamic theory of two-phase flows, Paris,
Eyrolles,
1975. |
Zbl 0325.76135[3] R. I. NIGMATULIN, Dynamics of multiphase media, Hemisphere Publ. Corp., New York, 1991.
[4] D. A. DREW, Mathematical modeling of two-phase flow, Annual Rev. Fluid Mech., 15 (1983), 261-291.
[5]
A. BIESHEUVEL -
L. VAN WIJNGAARDEN,
Two-phase flow equations for a dilute dispersion of gas bubbles in liquid,
J. Fluid Mech.,
148 (
1984), 301-318. |
Zbl 0585.76153[6]
R. M. BOWEN,
Theory of mixtures, in
Continuum Physics, Vol
III,
A. C. Eringen Ed.,
Academic Press (London,
1976), 1-127. |
MR 468445[7]
L. D. LANDAU -
E. LIFSHITS,
Fluid mechanics,
Pergamon Press (London,
1989). |
MR 108121[8] S. PUTTERMAN, Superfluid hydrodynamics , Elsevier (New York, 1974).
[9]
I. M. KHALATNIKOV,
Theory of superfluidity,
Nauka (Moscow,
1971). |
MR 202431[10]
C. TRUESDELL,
Rational thermodynamics,
Series in Modern Applied Mathematics,
MacGraw-Hill (New York,
1969). |
MR 366236[11] I. MÜLLER, Thermodynamics, Interaction of Mechanics and Mathematics Series, Pitman (London, 1985).
[13]
T. RUGGERI -
S. SIMIĆ,
On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single and multi-temperature models,
Math. Meth. Appl. Sci.,
30 (
2007), 827-849. |
fulltext (doi) |
MR 2310555 |
Zbl 1136.76428[15] J. SERRIN, Mathematical principles of classical fluid mechanics, Encyclopedia of Physics, VIII/1, S. Flügge Ed., Springer (Berlin, 1960), 125-263.
[16]
H. GOUIN,
Thermodynamic form of the equation of motion for perfect fluids of grade n,
Comptes Rendus Acad. Sci. Paris,
305, II (
1987), 833-838. |
MR 979611 |
Zbl 0621.76003[17]
H. GOUIN -
J. F. DEBIEVE,
Variational principle involving the stress tensor in elastodynamics,
Int. J. Engng. Sci.,
24 (
1986), 1057-1069. |
Zbl 0587.73024[18]
A. BEDFORD -
D. S. DRUMHELLER,
Recent advances. Theories of immiscible and structured mixtures,
Int. J. Engng. Sci.,
21 (
1983), 863-960. |
fulltext (doi) |
MR 711700 |
Zbl 0534.76105[19]
V. L. BERDICHEVSKY,
Variational principles of continuum mechanics,
Nauka (Moscow,
1983). |
MR 734171 |
Zbl 1189.49002[20]
J. A. GEURST,
Variational principles and two-fluid hydrodynamics of bubbly liquid/ gas mixtures,
Physica A,
135 (
1986), 455-486. |
fulltext (doi) |
MR 870328[21]
H. GOUIN,
Variational theory of mixtures in continuum mechanics,
Eur. J. Mech. B/ Fluids,
9 (
1990), 469-491. |
MR 1075270 |
Zbl 0704.76001[22]
S. GAVRILYUK -
H. GOUIN -
YU. PEREPECHKO,
A variational principle for two-fluid models,
Comptes Rendus Acad. Sci. Paris,
324, II (
1997), 483-490. |
Zbl 0877.76057[23]
H. GOUIN -
T. RUGGERI,
Mixture of fluids involving entropy gradients and acceleration waves in interfacial layers,
Eur. J. Mech. B/Fluids,
24 (
2005), 596-613. |
fulltext (doi) |
MR 2149382 |
Zbl 1069.76054[24]
H. GOUIN -
S. GAVRILYUK,
Hamilton's principle and Rankine-Hugoniot conditions for general motions of mixtures,
Meccanica,
34 (
1999), 39-47. |
fulltext (doi) |
MR 1683706 |
Zbl 0947.76090[26] D. LHUILLIER, From molecular mixtures to suspensions of particles, J. Physique II, 5 (1995), 19-36.
[27]
S. R. DE GROOT -
W. A. VAN LEEUWEN -
CH. G. VAN WEERT,
Relativistic kinetic theory.
North-Holland (Amsterdam,
1980). |
MR 635279[29] T. RUGGERI - S. SIMIĆ, Mixture of gases with multi-temperature: Identification of a macroscopic average temperature, in: Proceedings of Workshop Mathematical Physics Models and Engineering Sciences, Liguori Editore (Napoli, 2008), 455-462.
[30]
T. RUGGERI -
S. SIMIĆ,
Mixture of gases with multi-temperature: Maxwellian iteration, in:
Asymptotic Methods in Non Linear Wave Phenomena,
T. Ruggeri and
M. Sammartino Eds.,
World Scientific (Singapore,
2007), 186-194. |
fulltext (doi) |
MR 2370503 |
Zbl 1127.80002[31]
S. GAVRILYUK -
H. GOUIN,
A new form of governing equations of fluids arising from Hamilton's principle,
Int. J. Engng. Sci.,
37 (
1999), 1495-1520. |
fulltext (doi) |
MR 1709736 |
Zbl 1210.76009[32] S. WEINBERG, Entropy generation and the survival of protogalaxies in an expanding universe, Astrophys. J., 168 (1971), 175-194.
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