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The interaction of two spatially resonant patterns in thermal convection. I. Exact 1:2 resonance. (English) Zbl 0649.76018
A two-layer model in Cartesian geometry is defined which allows the study of exact 2:1 resonance where two distinct convective modes with preferred horizontal wavenumbers in the ratio 1:2 bifurcate simultaneously. The proposed model consists of two superimposed fluid layers which are separated by a thin conducting plate and are heated from below. If the ratio of the depths of the two layers is close to 1:2 two distinct linear modes of convection can occur with a preferred horizontal wavenumber ratio of 1:2. Using weakly nonlinear theory the nonlinear interaction of these two convective modes is studied, displaying a rich variety of behavior. The mathematics should be applicable in many other problems. A full discussion of the evolution equations is presented demonstrating the occurrence of travelling waves for a wide range of parameters. In addition modulated waves and an attracting homoclinic orbit are discussed and the importance of imperfections in the system is stressed.
Reviewer: W.Koch

MSC:
76E15 Absolute and convective instability and stability in hydrodynamic stability
76V05 Reaction effects in flows
76M99 Basic methods in fluid mechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
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