We derive a very sharp nonlinear stability result for the problem of thermal convection in a layer of dielectric fluid subject to an alternating current (AC). It is particularly important to note that the size of the initial energy in which we establish global nonlinear stability is not restricted whatsoever, and the Rayleigh-Roberts number boundary coincides with that found by a formal linear instability analysis.
Otteniamo un risultato di stabilità non lineare incondizionata per il problema della convezione termica di un fluido dielettrico soggetto ad una corrente alternata (AC). È particolarmente importante osservare che la grandezza iniziale dell'energia rispetto a cui stabiliamo il risultato di stabilità non lineare globale non ha restrizioni e i numeri critici di Rayleigh-Roberts ottenuti coincidono con quelli trovati con l'analisi formale della instabilità lineare.
@article{RLIN_1996_9_7_4_241_0,
author = {Giuseppe Mulone and Salvatore Rionero and Brian Straughan},
title = {Unconditional nonlinear stability in a polarized dielectric liquid},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {7},
year = {1996},
pages = {241-252},
zbl = {0877.76028},
mrnumber = {1454418},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_1996_9_7_4_241_0}
}
Mulone, Giuseppe; Rionero, Salvatore; Straughan, Brian. Unconditional nonlinear stability in a polarized dielectric liquid. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 7 (1996) pp. 241-252. http://gdmltest.u-ga.fr/item/RLIN_1996_9_7_4_241_0/
[1] - - , Heat transfer through strongly magnetized ferrofluids. Proc. Roy. Soc. London A, vol. 433, 1991, 165-177. | Zbl 0717.76050
[2] , Hydrodynamic and hydromagnetic stability. Dover, New York1981. | Zbl 0142.44103
[3] , On the possibility of subcriticai instabilities. In: Proc. IUTAM Symp. Herrenalb, Springer-Verlag, Berlin-Heidelberg-New York1971. | Zbl 0247.76038
[4] - , Weighted energy methods in fluid dynamics and elasticity. Lecture Notes in Mathematics, vol. 1134, Springer-Verlag, Berlin 1985. | MR 801034 | Zbl 0585.76001
[5] - , Exchange of stabilities, symmetry and nonlinear stability. Arch. Rational Mech. Anal., vol. 89, 1985, 211-228. | MR 786547 | Zbl 0622.76061
[6] - , A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem. Proc. Roy. Soc. London A, vol. 402, 1985, 257-283. | MR 828220 | Zbl 0593.76049
[7] , Stability of Fluid Motions. Vols. I, II, Springer Tracts in Natural Philosophy, vol. 27-28, 1976. | MR 449147 | Zbl 0345.76023
[8] , Some remarks on the boundary conditions for magnetic fluids. Int. J. Engng. Sci., vol. 30, 1992, 1451-1457. | MR 1187108 | Zbl 0764.76075
[9] - , An explicit example of Hopf bifurcation in fluid mechanics. Proc. Roy. Soc. London A, vol. 390, 1983, 293-320. | MR 733604 | Zbl 0588.76070
[10] - - , Electrodynamics of continuous media. 2nd ed., Pergamon Press, Oxford 1984. | MR 766230
[11] , Thermodynamics. Pitman, Boston - London - Melbourne1985. | Zbl 0637.73002
[12] - , On the nonlinear stability of the magnetic Bénard problem with rotation. Z. angew. Math. Mech., vol. 73, 1, 1993, 35-45. | MR 1291193 | Zbl 0798.76028
[13] - - , Stabilità non lineare incondizionata per la convezione elettro-termica in un liquido polarizzato. Atti XII Congr. AIMETA, Napoli, vol. V, 1995, 45-50.
[14] - - , Thermal instability with radiation by the energy method. Int. J. Heat Mass Transfer, vol. 37, 1994, 2909-2915. | Zbl 0900.76151
[15] - , A nonlinear stability problem for ferromagnetic fluids in a porous medium. Appl. Math. Letters, vol. 8, 1993, 25-29. | MR 1357246 | Zbl 0826.76028
[16] - , Nonlinear stability problem of a ferromagnetic fluid with surface tension effect. Eur. J. Mech. B/Fluids, vol. 13, 1994, 305-321. | MR 1284823 | Zbl 0811.76023
[17] , Sulla stabilità asintotica in media in magnetoidrodinamica. Ann. Matem. Pura Appl., vol. 76, 1967, 75-92. | MR 220478 | Zbl 0158.45104
[18] , Metodi variazionali per la stabilità asintotica in media in magnetoidrodinamica. Ann. Matem. Pura Appl., vol. 78, 1968, 339-364. | MR 229424 | Zbl 0182.29402
[19] , Sulla stabilità magnetofluidodinamica non lineare asintotica in media in presenza di effetto Hall. Ricerche Matem., vol. 20, 1971, 285-296. | MR 343759 | Zbl 0236.76043
[20] - , A nonlinear stability analysis of the magnetic Bénard problem through the Lyapunov direct method. Arch. Rational Mech. Anal., vol. 103, 1988, 347-368. | MR 955532 | Zbl 0666.76068
[21] , Electrohydrodynamic convection. Q. Jl. Mech. Appl. Math., vol. 22, 1969, 211-220. | Zbl 0181.55701
[22] , Ferrohydrodynamics. Cambridge Univ. Press, 1985.
[23] - - , Labyrinthine instability in magnetic and dielectric liquids. J. Magnetism and Magnetic Materials, vol. 39, 1983, 127-132.
[24] , On the stability of viscous fluid motions. Arch. Rational Mech. Anal., vol. 3, 1959, 1-13. | MR 105250 | Zbl 0086.20001
[25] , Electro-thermal convection in dielectric liquids. Chemical Physics Letters, vol. 179, 1991, 311-315.
[26] - - , Convective heat transfer through polarized dielectric liquids. Physics Fluids A, vol. 5, 1993, 3273-3279. | Zbl 0811.76022
[27] , The energy method, stability, and nonlinear convection. Appl. Math. Sci. Ser., vol. 91. Springer-Verlag, New York1992. | MR 1140924 | Zbl 0743.76006
[28] , Stability problems in electrohydrodynamics, ferrohydrodynamics and thermoelectric magnetohydrodynamics. In: - (eds.), Mathematical topics in fluid mechanics. Pitman Res. Notes Math., Longman Press, vol. 274, 1992. | MR 1204927 | Zbl 0795.76041
[29] , Nonlinear stability for convection in a polarized dielectric liquid. Article in Recent Advances in Mechanics of Structured Continua, ASME, vol. 160, 1993, 145-150.
[30] - , Effects of rotation on the thermo-convective instability of a horizontal layer of ferrofluids. Int. J. Engng. Sci., vol. 32, 1994, 237-256. | Zbl 0798.76029