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Mode interactions in large aspect ratio convection. (English) Zbl 0902.76035
Summary: Mode interaction between odd and even modes in two-dimensional Boussinesq convection in a box is revisited. It is noted that in the large aspect ratio limit the structure of the amplitude equations depends on the boundary conditions applied at the sidewalls, however distant. With no-slip sidewall boundary conditions the equations approach those for an unbounded layer with periodic boundary conditions; this is not the case for free-slip boundary conditions. Thus only in the former case the large aspect ratio system can be considered as a small perturbation of the unbounded system. The reasons for the different large aspect ratio limit are traced to the presence of “hidden” symmetries in the stress-free case. Homotopic continuation is used to extend these results to other types of boundary conditions.

MSC:
76E15 Absolute and convective instability and stability in hydrodynamic stability
76E30 Nonlinear effects in hydrodynamic stability
80A20 Heat and mass transfer, heat flow (MSC2010)
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[1] Busse, F. H.; Or, A. C., Subharmonic and asymmetric convection rolls, Z. Angew. Math. Phys., 37, 608-623, (1986) · Zbl 0615.76052
[2] Chen, Y.-Y., Boundary conditions and linear analysis of finite-cell Rayleigh-Bé nard convection, J. Fluid Mech., 241, 549-585, (1992) · Zbl 0754.76030
[3] Crawford, J. D.; Golubitsky, M.; Gomes, M. G. M.; Knobloch, E.; Stewart, I. N.; Roberts, M. (ed.); Stewart, I. (ed.), Boundary conditions as symmetry constraints, 65-79, (1991), Berlin · Zbl 0735.35007
[4] Daniels, P. G., Asymptotic sidewall effects in rotating Bé nard convection, Z. Angew. Math. Phys., 28, 577-584, (1977) · Zbl 0378.76070
[5] Drazin, P. G., On the effects of side walls on Bé nard convection, Z. Angew. Math. Phys., 26, 239-243, (1975) · Zbl 0303.76021
[6] Esipov, A. A.; Yudovich, V., Limit behaviour of eigenvalues of boundary value problems in indefinitely widening domains (in Russian), Zh. Vych. Mat. Mat. Fiz., 13, 421-432, (1973) · Zbl 0258.47036
[7] Esipov, A. A.; Yudovich, V., Asymptotics of eigenvalues of the first boundary value problem for an ordinary differential equation on a long interval (in Russian), Zh. Vych. Mat. Mat. Fiz., 14, 342-349, (1974) · Zbl 0298.34019
[8] Gomes, G.; Stewart, I.; Chossat, P. (ed.), Hopf bifurcation on generalized rectangles with Neumann boundary conditions, 139-158, (1994), Dordrecht · Zbl 0815.58026
[9] Gomes, M. G. M.; Stewart, I., Steady PDEs on generalized rectangles: A change of genericity in mode interactions, Nonlinearity, 7, 253-272, (1994) · Zbl 0836.35018
[10] D. Jacqmin and J. Heminger. Double-diffusion with Soret effect in a rectangular geometry: Linear and nonlinear traveling wave instabilities. 1993. Preprint.
[11] Julien, K. A., Strong spatial interactions with 1:1 resonance:athree-layer convection problem, Nonlinearity, 7, 1655-1693, (1994) · Zbl 0812.34034
[12] Kidachi, H., Side wall effect on the pattern formation of the Rayleigh-Bé nard convection, Prog. Theor. Phys., 68, 49-63, (1982) · Zbl 1098.76540
[13] Knobloch, E.; Guckenheimer, J., Convective transitions induced by a varying aspect ratio, Phys. Rev. A, 27, 408-417, (1983)
[14] Landsberg, A. S.; Knobloch, E., Oscillatory bifurcations with broken translation symmetry, Phys. Rev. E, 53, 3579-3600, (1996)
[15] Landsberg, A. S.; Knobloch, E., Oscillatory doubly diffusive convection in a finite container, Phys. Rev. E, 53, 3601-3609, (1996)
[16] Nagata, W., Convection in a layer with sidewalls: bifurcation with reflection symmetries, Z. Angew. Math. Phys., 41, 812-828, (1990) · Zbl 0727.76104
[17] Schaeffer, D. G., Qualitative analysis of a model for boundary effects in the Taylor problem, Math. Proc. Camb. Phil. Soc., 87, 307-337, (1980) · Zbl 0461.76016
[18] Segel, L. A., Distant side walls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38, 203-224, (1969) · Zbl 0179.57501
[19] H. R. Zangeneh. Hopf Bifurcation in Magnetoconvection in the Presence of Sidewalls. PhD thesis, The University of British Columbia, 1993.
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