The stability of a nonisothermal system consisting of two superimposed fluid layers: a thin liquid film layer and a gas layer sandwiched between differentially heated horizontal solid plates in the gravity field, is investigated. The system is assumed to be subjected to the Rayleigh–Taylor instability (RTI) with the Marangoni effect that either enhances the RTI or opposes it and to the tangential harmonic vibration of the upper substrate. A set of reduced evolution equations is derived based on the weighted-residual integral boundary layer approach, and the investigation is carried out in the framework of this set. The base state of the system represents a time-periodic flow, and its linear stability analysis is carried out using the Floquet theory in the large-time limit. The nonlinear dynamics of the system is investigated numerically in the case of either a static or vibrating substrate. Among the possible outcomes of the nonlinear dynamics, there is the emergence of ruptured states of the liquid film with rupture taking place at either the upper or lower substrate and also the emergence of saturated continuous flows of the liquid film. We also find that the nonlinear dynamics of the system is consistent with the results of the linear stability analysis in terms of enhancement or attenuation of interfacial distortion.

中文翻译：

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瑞利-泰勒不稳定性条件下受谐波切向强迫的非等温两薄流体层系统动力学

研究了由两个叠加流体层组成的非等温系统的稳定性：薄液膜层和夹在重力场中不同加热水平固体板之间的气体层。假设该系统受到瑞利-泰勒不稳定性 (RTI) 和 Marangoni 效应的影响，该效应要么增强 RTI，要么反对它以及上基板的切向谐波振动。基于加权残差积分边界层方法推导了一组简化的演化方程，并在该方程组的框架内进行了研究。系统的基态代表一个时间周期流，在大时限下利用Floquet理论对其进行线性稳定性分析。在静态或振动基板的情况下，系统的非线性动力学被数值研究。在非线性动力学的可能结果中，出现了液膜破裂状态，破裂发生在上基底或下基底，也出现了饱和的液膜连续流动。我们还发现系统的非线性动力学在界面畸变的增强或衰减方面与线性稳定性分析的结果一致。