This paper investigates the Rayleigh Benard instability of a viscous, Newtonian, Boussinesq fluid with time-periodic boundary temperature modulation using the framework of weakly nonlinear theory. Critical Rayleigh number is computed for asymptotic stability criterion using the energy method accompanied by variational algorithm. Subcritical instability is found to occur under two conditions: when modulation is in anti-phase and when modulation is imposed only on the lower boundary. Supercritical stability is witnessed during in-phase modulation. In all the three cases of the relative phase of two boundary temperatures, the effect of modulation is found to be weaker for infinitesimal disturbances. The findings of the present study could be referred in several applications where appropriate temperature modulation is of prime concern.

本文利用弱非线性理论的框架研究了具有时间周期边界温度调制的粘性牛顿布森涅克流体的瑞利贝纳德不稳定性。使用能量方法和变分算法来计算渐近稳定性准则的临界瑞利数。发现在两个条件下发生亚临界不稳定性：当调制处于反相状态且仅在下边界施加调制时。在同相调制过程中可以看到超临界稳定性。在两个边界温度的相对相位的所有三种情况下，对于极小的扰动，调制的作用都较弱。本研究的发现可在一些主要考虑适当温度调制的应用中提供参考。