Container geometry and the transition to unsteady Bénard convection in porous media

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Abstract
For convection in three-dimensional boxes of fluid-saturated porous media, an analog of the Rayleigh–Bénard problem at high Prandtl number, it is shown that cascades to time-dependent motion cannot occur within the weakly nonlinear regime for boxes up to moderate size. The convective nonlinearity forces the phase space that describes the interaction of finite-amplitude convection patterns to be globally attracting to large disturbances. In addition, the fastest growing patterns inhibit others in a way that precludes transitions to time-periodic motions. The analysis exploits a closed-form dependence on geometry for the infinite set of coefficients describing the nonlinear interactions in a representation of the governing equations by eigenfunction expansion.