The Lyapunov direct method is applied to study nonlinear exponential stability of a basic motionless state to imposed linear temperature and concentration fields of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme. Stress-free and rigid surfaces are considered and absence of Hopf bifurcation is assumed. We prove the coincidence of the linear and (unconditional) nonlinear critical stability limits, when the ratio between the Schmidt and the Prandtl numbers is less or equal to 1. Precisely, we obtain necessary and sufficient conditions of unconditional nonlinear exponential stability of the basic motionless state.
Si applica il metodo diretto di Lyapunov allo studio della stabilità non lineare esponenziale della soluzione di conduzione-diffusione di una miscela fluida binaria riscaldata e salata da sotto, nello schema di Oberbeck-Boussinesq. Si considerano superfici rigide e stress-free ; si suppone che non ci sia biforcazione di Hopf. Supposto che il rapporto fra i numeri di Schmidt e di Prandtl è minore o uguale a 1, proviamo la coincidenza fra i parametri critici della stabilità lineare e non lineare. Si ottengono condizioni necessarie e sufficienti di stabilità non lineare esponenziale del moto base.
@article{RLIN_1998_9_9_3_221_0,
author = {Giuseppe Mulone and Salvatore Rionero},
title = {Unconditional nonlinear exponential stability in the B\'enard problem for a mixture: necessary and sufficient conditions},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {9},
year = {1998},
pages = {221-236},
zbl = {0922.76171},
mrnumber = {1683010},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_1998_9_9_3_221_0}
}
Mulone, Giuseppe; Rionero, Salvatore. Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 9 (1998) pp. 221-236. http://gdmltest.u-ga.fr/item/RLIN_1998_9_9_3_221_0/
[1] , Large-area solar collections for power production. Solar Energy, 7, 1963, 189-196.
[2] - , Solar pond project. Solar Energy, 9, 1965, 177-182.
[3] , The physics of the solar pond. Solar Energy, 8, 1964, n. 2, 45-56.
[4] , Stability of Fluid Motions. Springer Tracts in Natural Philosophy, 27-28, Springer-Verlag, New York1976. | Zbl 0345.76023
[5] , Ph. D. thesis. Univ. Minn., Minneapolis, 1963.
[6] , On finite amplitude instability in thermohaline convection. J. Marine Res., 23, 1965, 1-17.
[7] , The thermohaline Rayleigh-Jeffreys problem. J. Fluid Mech., 29, 1967, 545-558.
[8] - , On thermohaline convection with linear gradients. J. Fluid Mech., 37, 1969, 289-306.
[9] - , Convective instability in a temperature and concentration field. Arch. Rational Mech. Anal., 30, 1968, 38-80. | MR 250557 | Zbl 0172.54903
[10] - , A Nonlinear Analysis of the Stabilizing Effect of Rotation in the Bénard Problem. Proc. R. Soc. London, A, 402, 1985, 257-283. | MR 828220 | Zbl 0593.76049
[11] - , A Nonlinear Stability Analysis of the Magnetic Bénard Problem through the Lyapunov Direct Method. Arch. Rational Mech. Anal., 103, 1988, 347-368. | MR 955532 | Zbl 0666.76068
[12] , On the Choice of the Lyapunov Functional in the Stability of Fluid Motions. In: - (eds.), Energy Stability and Convection. Pitman Research Notes in Mathematics, 168, Wiley, New York 1988, 392-419. | MR 959780 | Zbl 0689.76015
[13] - , On the Non-linear Stability of the Rotating Bénard Problem via the Lyapunov Direct Method. J. Mat. Anal. App., 144, 1989, 109. | Zbl 0682.76037
[14] , The Energy Method, Stability, and Nonlinear Convection. Applied Mathematical Sciences, 91, Springer-Verlag, 1992, 242 pp. | MR 1140924 | Zbl 0743.76006
[15] - , Qualitative estimates for partial differential equations. An introduction. CRC Press, Boca Raton, Florida, 1996. | MR 1396085 | Zbl 0862.35001
[16] , Global stability of conduction diffusion solution. Arch. Rational Mech. Anal., 36, 1970, 285-292. | MR 269180 | Zbl 0202.26602
[17] , On finite amplitude roll cell disturbances in a fluid layer subject to heat and mass transfer. A. I. Ch. E. Journal, 11, 1965, 971-980.
[18] , Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech., 34, 1968, 315-336. | Zbl 0175.52305
[19] - , On the non-linear stability of parallel shear flows. Continuum Mech. Thermodyn., 3, 1991, 1-11. | MR 1098352 | Zbl 0760.76034
[20] , On the stability of plane parallel convective flow. Acta Mechanica, 87, 1991, 153-162. | MR 1108025 | Zbl 0737.76030
[21] , On the Lyapunov stability of a plane parallel convective flow of a binary mixture. Le Matematiche, 46, 1991, 283-294. | MR 1228719 | Zbl 0756.76027
[22] , On the stability of plane parallel convective mixture through the Lyapunov second method. Atti Acc. Peloritana Pericolanti Cl. Sci Fis. Natur., 68, 1991, 491-516. | MR 1158921 | Zbl 0749.76025
[23] , On the nonlinear stability of a fluid layer of a mixture heated and salted from below. Continuum Mech. Thermodyn., 6, 1994, 161-184. | MR 1285920 | Zbl 0809.76034
[24] , The problem of the minimum of a quadratic functional. Holden-Day, San Francisco1965. | MR 171196 | Zbl 0121.32801
[25] , Hydrodynamic and hydromagnetic stability. Clarendon Press, Oxford1961. | MR 128226 | Zbl 0142.44103
[26] , The mathematical problem of hydrodynamic stability. J. Math. Mech., 19, n. 9, 1970, 797-817. | MR 261182 | Zbl 0198.30401
[27] - , On the stability of the rotating Bénard problem. Bull. Tech. Univ. Istanbul, 47, 1994, 181-202. | MR 1321950 | Zbl 0864.76030