Range of validity of weakly nonlinear theory in the rayleigh-bénard problem

Journal of the Physical Society of Japan

Volumn 78, Issue 8, 2009, Pages

  Generalis, Sotos Constantinou ^a   Fujimura, Kaoru ^a,b  

^a ASTON UNIVERSITY   (United Kingdom)

^b TOTTORI UNIVERSITY   (Japan)

Author keywords

Homogeneous heating; Nonlinear steady solutions; Stability; Stuart landau equation; Thermal convection

Indexed keywords

EID: 69149086901     PISSN: 00319015     EISSN: 13474073     Source Type: Journal    
DOI: 10.1143/JPSJ.78.084401     Document Type: Article

References (33)

1. J. T. Stuart: J. Fluid Mech. 9 (1960) 353.
2. J. Watson: J. Fluid Mech. 9 (1960) 371.
3. W. V. R. Malkus and G. Veronis: J. Fluid Mech. 4 (1958) 225.
4. W. Eckhaus: Studies in Nonlinear Stability Theory (Springer, New York, 1965).
5. A. C. Newell and J. A. Whitehead: J. Fluid Mech. 38 (1969) 279.
6. K. Stewartson and J. T. Stuart: J. Fluid Mech. 48 (1971) 529.
7. R. C. DiPrima, W. Eckhaus, and L. A. Segel: J. Fluid Mech. 49 (1971) 705.
8. F. H. Busse: Dr. Thesis, University of Munich (1962) [Translation: Rand Rep. LT-66-19 Rand Corp., Santa Monica, California].
9. F. H. Busse: J. Fluid Mech. 30 (1967) 625.
10. F. H. Busse: Rep. Prog. Phys. 41 (1978) 1929.
11. A. Schlüter, D. Lortz, and F. H. Busse: J. Fluid Mech. 23 (1965) 129.
12. J. Carr: Applications of Centre Manifold Theory (Springer, New York, 1981).
13. M. Cheng and H.-C. Chang: Phys. Fluids A 2 (1990) 1364.
14. K. Fujimura: Proc. R. Soc. London, Ser. A 434 (1991) 719.
15. K. Fujimura: Proc. R. Soc. London, Ser. A 453 (1997) 181.
16. H. L. Kuo: J. Fluid Mech. 10 (1961) 611.
17. T. Herbert: AIAA J. 18 (1980) 243.
18. T. Herbert: J. Fluid Mech. 126 (1983) 167.
19. A. Kelley: J. Differ. Equations 3 (1967) 546.
20. A. J. Roberts: SIAM J. Math. Anal. 20 (1989) 1447.
21. K. Fujimura: Proc. R. Soc. London, Ser. A 424 (1989) 373.
22. 1415, 21)
23. A. H. Nayfeh: Perturbation Methods (Wiley, New York, 1973).
24. M. Golubitsky and D. G. Schaeffer: Singularity and Groups in Bifurcation Theory (Springer, New York, 1985) Vol. I.
25. L.-Y. Chen, N. Goldenfeld, and Y. Oono: Phys. Rev. E 54 (1996) 376.
26. jn = 0 at z = 0;1.
27. To avoid an occurrence of the zero divisor in the center manifold, we assume that Re σ > 0.
28. F. H. Busse: private communication (2009).
29. M. Nagata and S. Generalis: J. Heat Transfer 124 (2002) 635.
30. S. Generalis and M. Nagata: J. Heat Transfer 125 (2003) 795.
31. M. Nagata and S. Generalis: C. R. Mec. 332 (2004) 9.
32. For P < 1 the values M = 91, N = 10 were used in the fully nonlinear solutions of (3.1, 3.2) in order to establish whether the two methods agreed.
33. At the critical wavenumber 1.02055, the maximum growth rate is achieved at Re = 12099.