Long-wavelength rotating convection between poorly conducting boundaries

SIAM Journal on Applied Mathematics

Volumn 58, Issue 4, 1998, Pages 1338-1364

  Cox, Stephen M ^a  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

Bifurcation; Long wavelength model; Rotating convection; Square planform

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; GEOMETRY; NAVIER STOKES EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PRANDTL NUMBER; ROTATIONAL FLOW; TWO DIMENSIONAL;

KUPPERS-LORTZ INSTABILITY; LONG WAVELENGTH MODEL; ROTATING CONVECTION; SQUARE PLANFORM;

HEAT CONVECTION;

EID: 0032140110     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139996314003     Document Type: Article

References (40)

1. B. M. BOUBNOV AND G. S. GOLITSYN, Convection in Rotating Fluids, Kluwer, Dordrecht, 1995.
2. F. H. BUSSE AND H. FRICK, Square-pattern convection in fluids with strongly temperature-dependent viscosity, J. Fluid Mech., 150 (1985), pp. 451-465.
3. F. H. BUSSE AND N. RIAHI, Nonlinear convection in a layer with nearly insulating boundaries, . Fluid Mech., 96 (1980), pp. 243-256.
4. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988.
5. S. CHANDRASEKHAR, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford, 1961, reprinted by Dover, New York, 1981.
6. C. J. CHAPMAN AND M. R. E. PROCTOR, Nonlinear Rayleigh-Bénard convection between poorly conducting boundaries, J. Fluid Mech., 101 (1980), pp. 759-782.
7. R. M. CLEVER AND F. H. BUSSE, Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis, J. Fluid Mech., 94 (1979), pp. 609-627.
8. T. CLUNE AND E. KNOBLOCH, Pattern selection in rotating convection with experimental boundary conditions, Phys. Rev. E, 47 (1993), pp. 2536-2550.
9. S. M. COX, The onset of thermal convection between poorly conducting horizontal boundaries in the presence of a shear flow, SIAM J. Appl. Math., 56 (1996), pp. 1317-1328.
10. M. C. DEPASSIER AND E. A. SPIEGEL, Convection with heat flux prescribed on the boundaries of the system. I: The effect of temperature dependence of material properties, Geophys. Astrophys. Fluid Dynam., 21 (1982), pp. 167-188.
11. B. DIONNE, M. SILBER, AND A. C. SKELDON, Stability results for steady, spatially-periodic planforms, Nonlinearity, 10 (1997), pp. 321-353.
12. V. L. GERTSBERG AND G. I. SIVASHINSKY, Large cells in nonlinear Rayleigh-Bénard convection, Progr. Theoret. Phys., 66 (1981), pp. 1219-1229.
13. H. F. GOLDSTEIN, E. KNOBLOCH, AND M. SILBER, Planform selection in rotating convection, Phys. Fluids, 2 (1990), pp. 625-627.
14. H. F. GOLDSTEIN, E. KNOBLOCH, AND M. SILBER, Planform selection in rotating convection: Hexagonal symmetry, Phys. Rev. A, 46 (1992), pp. 4755-4761.
15. D. GOTTLIEB AND S. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977.
16. N. E. HURLBURT, M. R. E. PROCTOR, N. O. WEISS, AND D. P. BROWNJOHN, Nonlinear compressible magnetoconvection Part 1: Travelling waves and oscillations, J. Fluid Mech., 207 (1989), pp. 587-628.
17. D. T. J. HURLE, E. JAKEMAN, AND E. R. PIKE, On the solution of the Bénard problem with boundaries of finite conductivity, Proc. Roy. Soc. London Ser. A, 296 (1967), pp. 469-475.
18. D. R. JENKINS AND M. R. E. PROCTOR, The transition from roll to square-cell solutions in Rayleigh-Bénard convection, J. Fluid Mech., 139 (1984), pp. 461-471.
19. R. E. KELLY, The onset and development of thermal convection in fully developed shear flows, Adv. Appl. Mech., 31 (1994), pp. 35-112.
20. E. KNOBLOCH, Pattern selection in long-wavelength convection, Phys. D, 41 (1990), pp. 450-479.
21. E. KNOBLOCH AND M SILBER, Oscillatory convection in a rotating layer, Phys. D, 63 (1993), pp. 213-232.
22. G. KÜPPERS AND D. LORTZ, Transition from laminar convection to thermal turbulence in a rotating fluid layer, J. Fluid Mech., 35 (1969), pp. 609-620.
23. P. C. MATTHEWS, Hexagonal patterns in finite domains, Phys. D, 116 (1998), pp. 81-94.
24. P. C. MATTHEWS, M. R. E. PROCTOR, AND N. O. WEISS, Compressible magnetoconvection in three dimensions: Planforms and nonlinear behaviour, J. Fluid Mech., 305 (1995), pp. 281-305.
25. B. A. MALOMED, A. A. NEPOMNYASHCHY, AND A. E. NUZ, Nonequilateral hexagonal patterns, Phys. D, 70 (1994), pp. 357-369.
26. M. NEUFELD AND R. FRIEDRICH, Pattern formation in rotating Bénard convection, Internat. J. Bifur. and Chaos Appl. Sci. Engrg., 4 (1994), pp. 1155-1163.
27. M. NEUFELD AND R. FRIEDRICH, Statistical properties of the heat transport in a model of rotating Bénard convection, Phys. Rev. E, 51 (1995), pp. 2033-2045.
28. M. NEUFELD, R. FRIEDRICH, AND H. HAKEN, Order parameter equation and model equation for high Prandtl number Rayleigh-Bénard convection in a rotating large aspect ratio system, Z. Phys. B, 92 (1993), pp. 243-256.
29. L. M. PISMEN, Inertial effects in long-scale thermal convection, Phys. Lett. A, 116 (1986), pp. 241-244.
30. M. R. E. PROCTOR, Planform selection by finite-amplitude thermal convection between poorly conducting slabs, J. Fluid Mech., 113 (1981), pp. 469-485.
31. M. R. E. PROCTOR AND P. C. MATTHEWS, √2 : 1 resonance in non-Boussinesq convection, Phys. D, 97 (1996), pp. 469-485.
32. M. R. E. PROCTOR, N. O. WEISS, D. P. BROWNJOHN, AND N. E. HURLBURT, Nonlinear compressible magnetoconvection Part 2: Streaming instabilities in two dimensions, J. Fluid Mech., 280 (1994), pp. 227-253.
33. D. N. RIAHI, Nonlinear convection in a rotating layer with finite conducting boundaries, Phys. Fluids A, 2 (1990), pp. 353-358.
34. J. M. RODRÍGUEZ, C. PÉREZ-GARCÍA, M. BESTEHORN, M. FANTZ, AND R. FRIEDRICH, Pattern formation in convection of rotating fluids with broken vertical symmetry, Phys. Rev. A, 46 (1992), pp. 4729-4735.
35. L. SHTILMAN AND G. SIVASHINSKY, Hexagonal structure of large-scale Marangoni convection, Phys. D, 52 (1991), pp. 477-488.
36. G. I. SIVASHINSKY, Large cells in nonlinear Marangoni convection, Phys. D, 4 (1982), pp. 227-235.
37. G. I. SIVASHINSKY, On cellular instability in the solidification of a dilute binary alloy, Phys. D, 8 (1983), pp. 243-248.
38. E. M. SPARROW, R. J. GOLDSTEIN, AND V. K. JONSSON, Thermal instability in a horizontal fluid layer: Effect of boundary conditions and non-linear temperature profile, J. Fluid Mech., 18 (1964), pp. 513-528.
39. A. THESS AND S. A. ORSZAG, Surface-tension-driven convection at infinite Prandtl number, J. Fluid Mech., 283 (1995), pp. 201-230.
40. N. O. WEISS, Magnetoconvection, Geophys. Astrophys. Fluid Dynam., 62 (1991), pp. 229-247.