Geodesic
Unconditional nonlinear stability in a polarized dielectric liquid
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 7 (1996) no. 4, pp. 241-252.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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_a4,
     author = {Mulone, Giuseppe and Rionero, Salvatore and Straughan, Brian},
     title = {Unconditional nonlinear stability in a	polarized dielectric liquid},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {241--252},
     publisher = {mathdoc},
     volume = {Ser. 9, 7},
     number = {4},
     year = {1996},
     zbl = {0877.76028},
     mrnumber = {801034},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/RLIN_1996_9_7_4_a4/}
}
TY  - JOUR
AU  - Mulone, Giuseppe
AU  - Rionero, Salvatore
AU  - Straughan, Brian
TI  - Unconditional nonlinear stability in a	polarized dielectric liquid
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 1996
SP  - 241
EP  - 252
VL  - 7
IS  - 4
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/RLIN_1996_9_7_4_a4/
LA  - en
ID  - RLIN_1996_9_7_4_a4
ER  - 
%0 Journal Article
%A Mulone, Giuseppe
%A Rionero, Salvatore
%A Straughan, Brian
%T Unconditional nonlinear stability in a	polarized dielectric liquid
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 1996
%P 241-252
%V 7
%N 4
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/RLIN_1996_9_7_4_a4/
%G en
%F RLIN_1996_9_7_4_a4
Mulone, Giuseppe; Rionero, Salvatore; Straughan, Brian. Unconditional nonlinear stability in a	polarized dielectric liquid. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 7 (1996) no. 4, pp. 241-252. https://geodesic-test.mathdoc.fr/item/RLIN_1996_9_7_4_a4/

[1] P. J. Blennerhassett - F. Lin - P. J. Stiles, Heat transfer through strongly magnetized ferrofluids. Proc. Roy. Soc. London A, vol. 433, 1991, 165-177. | Zbl

[2] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability. Dover, New York 1981. | Zbl

[3] S. H. Davis, On the possibility of subcriticai instabilities. In: Proc. IUTAM Symp. Herrenalb, Springer-Verlag, Berlin-Heidelberg-New York 1971. | Zbl

[4] G. P. Galdi - S. Rionero, Weighted energy methods in fluid dynamics and elasticity. Lecture Notes in Mathematics, vol. 1134, Springer-Verlag, Berlin 1985. | MR | Zbl

[5] G. P. Galdi - B. Straughan, Exchange of stabilities, symmetry and nonlinear stability. Arch. Rational Mech. Anal., vol. 89, 1985, 211-228. | DOI | MR | Zbl

[6] G. P. Galdi - B. Straughan, A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem. Proc. Roy. Soc. London A, vol. 402, 1985, 257-283. | MR | Zbl

[7] D. D. Joseph, Stability of Fluid Motions. Vols. I, II, Springer Tracts in Natural Philosophy, vol. 27-28, 1976. | MR | Zbl

[8] P. N. Kaloni, Some remarks on the boundary conditions for magnetic fluids. Int. J. Engng. Sci., vol. 30, 1992, 1451-1457. | DOI | MR | Zbl

[9] P. Kloeden - R. Wells, An explicit example of Hopf bifurcation in fluid mechanics. Proc. Roy. Soc. London A, vol. 390, 1983, 293-320. | MR | Zbl

[10] L. D. Landau - E. M. Lifshitz - L. P. Pitaevskii, Electrodynamics of continuous media. 2nd ed., Pergamon Press, Oxford 1984. | MR

[11] I. Muller, Thermodynamics. Pitman, Boston - London - Melbourne 1985. | Zbl

[12] G. Mulone - S. Rionero, On the nonlinear stability of the magnetic Bénard problem with rotation. Z. angew. Math. Mech., vol. 73, 1, 1993, 35-45. | DOI | MR | Zbl

[13] G. Mulone - S. Rionero - B. Straughan, Stabilità non lineare incondizionata per la convezione elettro-termica in un liquido polarizzato. Atti XII Congr. AIMETA, Napoli, vol. V, 1995, 45-50.

[14] G. P. Neitzel - M. K. Smith - M. J. Bolander, Thermal instability with radiation by the energy method. Int. J. Heat Mass Transfer, vol. 37, 1994, 2909-2915. | Zbl

[15] Y. Qin - J. Chadam, A nonlinear stability problem for ferromagnetic fluids in a porous medium. Appl. Math. Letters, vol. 8, 1993, 25-29. | DOI | MR | Zbl

[16] Y. Qin - P. N. Kaloni, Nonlinear stability problem of a ferromagnetic fluid with surface tension effect. Eur. J. Mech. B/Fluids, vol. 13, 1994, 305-321. | MR | Zbl

[17] S. Rionero, Sulla stabilità asintotica in media in magnetoidrodinamica. Ann. Matem. Pura Appl., vol. 76, 1967, 75-92. | MR | Zbl

[18] S. Rionero, Metodi variazionali per la stabilità asintotica in media in magnetoidrodinamica. Ann. Matem. Pura Appl., vol. 78, 1968, 339-364. | MR | Zbl

[19] S. Rionero, Sulla stabilità magnetofluidodinamica non lineare asintotica in media in presenza di effetto Hall. Ricerche Matem., vol. 20, 1971, 285-296. | MR | Zbl

[20] S. Rionero - G. Mulone, A nonlinear stability analysis of the magnetic Bénard problem through the Lyapunov direct method. Arch. Rational Mech. Anal., vol. 103, 1988, 347-368. | DOI | MR | Zbl

[21] P. H. Roberts, Electrohydrodynamic convection. Q. Jl. Mech. Appl. Math., vol. 22, 1969, 211-220. | Zbl

[22] R. H. Rosenweig, Ferrohydrodynamics. Cambridge Univ. Press, 1985.

[23] R. H. Rosenweig - M. Zahn - R. Shumovich, Labyrinthine instability in magnetic and dielectric liquids. J. Magnetism and Magnetic Materials, vol. 39, 1983, 127-132.

[24] J. Serrin, On the stability of viscous fluid motions. Arch. Rational Mech. Anal., vol. 3, 1959, 1-13. | MR | Zbl

[25] P. J. Stiles, Electro-thermal convection in dielectric liquids. Chemical Physics Letters, vol. 179, 1991, 311-315.

[26] P. J. Stiles - F. Lin - P. J. Blennerhassett, Convective heat transfer through polarized dielectric liquids. Physics Fluids A, vol. 5, 1993, 3273-3279. | Zbl

[27] B. Straughan, The energy method, stability, and nonlinear convection. Appl. Math. Sci. Ser., vol. 91. Springer-Verlag, New York 1992. | MR | Zbl

[28] B. Straughan, Stability problems in electrohydrodynamics, ferrohydrodynamics and thermoelectric magnetohydrodynamics. In: J. F. RODRIGUEZ - A. SEQUEIRA (eds.), Mathematical topics in fluid mechanics. Pitman Res. Notes Math., Longman Press, vol. 274, 1992. | MR | Zbl

[29] B. Straughan, 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] S. Venkatasubramanian - P. N. Kaloni, Effects of rotation on the thermo-convective instability of a horizontal layer of ferrofluids. Int. J. Engng. Sci., vol. 32, 1994, 237-256. | Zbl