Effect of parametric modulation on the onset of convection inHe3-He4mixtures near theλline

The onset of convection in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures is considered with the temperature difference sinusoidally modulated between the plates in the Rayleigh-Bénard geometry. The system can show stabilization or destabilization depending on the mean temperature. For oscillatory instability, parametric resonances should occur. As a preliminary step, we obtain analytic results under idealized boundary conditions for the onset of convection in the unmodulated case. The results there are in qualitative agreement with the experiments of Lee, Lucas, and Tyler.