Convective stability in the Rayleigh–Bénard and directional solidification problems: High-frequency gravity modulation

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Abstract
The effect of vertical, sinusoidal, time‐dependent gravitational acceleration on the onset of solutal convection during directional solidification is analyzed in the limit of large modulation frequency Ω. When the unmodulated state is unstable, the modulation amplitude required to stabilize the system is determined by the method of averaging, and is O(Ω). Comparison of the results from the averaged equations with numerical solutions of the full linear stability equations (based on Floquet theory) show that the difference is O(Ω1/2). When the unmodulated state is stable, resonant modes of instability occur at large modulation amplitude. These are analyzed using matched asymptotic expansions to elucidate the boundary‐layer structure for both the Rayleigh–Bénard and directional solidification configurations. The leading‐order term for the modulation amplitude is of O(Ω2); the first‐order correction of O(Ω3/2) is calculated, and the results are compared with numerical solutions of the full linear stability equations. Based on these analyses, a thorough examination of the dependence of the stability criteria on the unmodulated Rayleigh number, Schmidt number, and distribution coefficient, is carried out.