Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions

Giuseppe Mulone; Salvatore Rionero

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1998)

  • Volume: 9, Issue: 3, page 221-236
  • ISSN: 1120-6330

Abstract

top
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.

How to cite

top

Mulone, Giuseppe, and 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 9.3 (1998): 221-236. <http://eudml.org/doc/252344>.

@article{Mulone1998,
abstract = {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.},
author = {Mulone, Giuseppe, Rionero, Salvatore},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Fluid mixture heated and salted; Lyapunov unconditional nonlinear stability; Natural convection; Lyapunov direct method; binary fluid mixture; Oberbeck-Boussinesq scheme; critical stability limits},
language = {eng},
month = {9},
number = {3},
pages = {221-236},
publisher = {Accademia Nazionale dei Lincei},
title = {Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions},
url = {http://eudml.org/doc/252344},
volume = {9},
year = {1998},
}

TY - JOUR
AU - Mulone, Giuseppe
AU - Rionero, Salvatore
TI - Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1998/9//
PB - Accademia Nazionale dei Lincei
VL - 9
IS - 3
SP - 221
EP - 236
AB - 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.
LA - eng
KW - Fluid mixture heated and salted; Lyapunov unconditional nonlinear stability; Natural convection; Lyapunov direct method; binary fluid mixture; Oberbeck-Boussinesq scheme; critical stability limits
UR - http://eudml.org/doc/252344
ER -

References

top
  1. Tabor, H., Large-area solar collections for power production. Solar Energy, 7, 1963, 189-196. 
  2. Tabor, H. - Maetz, R., Solar pond project. Solar Energy, 9, 1965, 177-182. 
  3. Weinberger, H., The physics of the solar pond. Solar Energy, 8, 1964, n. 2, 45-56. 
  4. Joseph, D. D., Stability of Fluid Motions. Springer Tracts in Natural Philosophy, 27-28, Springer-Verlag, New York1976. Zbl0345.76023
  5. Sani, R. L., Ph. D. thesis. Univ. Minn., Minneapolis, 1963. 
  6. Veronis, G., On finite amplitude instability in thermohaline convection. J. Marine Res., 23, 1965, 1-17. 
  7. Nield, D. A., The thermohaline Rayleigh-Jeffreys problem. J. Fluid Mech., 29, 1967, 545-558. 
  8. Baines, P. G. - Gill, A. E., On thermohaline convection with linear gradients. J. Fluid Mech., 37, 1969, 289-306. 
  9. Shir, C. C. - Joseph, D. D., Convective instability in a temperature and concentration field. Arch. Rational Mech. Anal., 30, 1968, 38-80. Zbl0172.54903MR250557
  10. Galdi, G. P. - Straughan, B., A Nonlinear Analysis of the Stabilizing Effect of Rotation in the Bénard Problem. Proc. R. Soc. London, A, 402, 1985, 257-283. Zbl0593.76049MR828220
  11. Rionero, S. - Mulone, G., A Nonlinear Stability Analysis of the Magnetic Bénard Problem through the Lyapunov Direct Method. Arch. Rational Mech. Anal., 103, 1988, 347-368. Zbl0666.76068MR955532DOI10.1007/BF00251445
  12. Rionero, S., On the Choice of the Lyapunov Functional in the Stability of Fluid Motions. In: G. P. Galdi - B. Straughan (eds.), Energy Stability and Convection. Pitman Research Notes in Mathematics, 168, Wiley, New York1988, 392-419. Zbl0689.76015MR959780
  13. Mulone, G. - Rionero, S., On the Non-linear Stability of the Rotating Bénard Problem via the Lyapunov Direct Method. J. Mat. Anal. App., 144, 1989, 109. Zbl0682.76037
  14. Straughan, B., The Energy Method, Stability, and Nonlinear Convection. Applied Mathematical Sciences, 91, Springer-Verlag, 1992, 242 pp. Zbl0743.76006MR1140924
  15. Flavin, J. - Rionero, S., Qualitative estimates for partial differential equations. An introduction. CRC Press, Boca Raton, Florida, 1996. Zbl0862.35001MR1396085
  16. Joseph, D. D., Global stability of conduction diffusion solution. Arch. Rational Mech. Anal., 36, 1970, 285-292. Zbl0202.26602MR269180
  17. Sani, R. L., 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. Veronis, G., Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech., 34, 1968, 315-336. Zbl0175.52305
  19. Rionero, S. - Mulone, G., On the non-linear stability of parallel shear flows. Continuum Mech. Thermodyn., 3, 1991, 1-11. Zbl0760.76034MR1098352DOI10.1007/BF01128961
  20. Mulone, G., On the stability of plane parallel convective flow. Acta Mechanica, 87, 1991, 153-162. Zbl0737.76030MR1108025DOI10.1007/BF01299792
  21. Mulone, G., On the Lyapunov stability of a plane parallel convective flow of a binary mixture. Le Matematiche, 46, 1991, 283-294. Zbl0756.76027MR1228719
  22. Mulone, G., 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. Zbl0749.76025MR1158921
  23. Mulone, G., On the nonlinear stability of a fluid layer of a mixture heated and salted from below. Continuum Mech. Thermodyn., 6, 1994, 161-184. Zbl0809.76034MR1285920DOI10.1007/BF01135252
  24. Mikhlin, S. G., The problem of the minimum of a quadratic functional. Holden-Day, San Francisco1965. Zbl0121.32801MR171196
  25. Chandrasekhar, S., Hydrodynamic and hydromagnetic stability. Clarendon Press, Oxford1961. Zbl0142.44103MR128226
  26. Sattinger, D. H., The mathematical problem of hydrodynamic stability. J. Math. Mech., 19, n. 9, 1970, 797-817. Zbl0198.30401MR261182
  27. Mulone, G. - Rionero, S., On the stability of the rotating Bénard problem. Bull. Tech. Univ. Istanbul, 47, 1994, 181-202. Zbl0864.76030MR1321950

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.