Oscillatory thermocapillary instability of a film heated by a thick substrate
Oscillatory thermocapillary instability of a film heated by a thick substrate
复制标题
DOI：
10.1017/jfm.2019.417
复制
发表时间：
2019-02
影响因子：
3.7
通讯作者：
W. Batson;L. Cummings;D. Shirokoff;L. Kondic
中科院分区：
文献类型：
--
作者：
W. Batson;L. Cummings;D. Shirokoff;L. Kondic
在这项工作中，我们考虑了一类新的振荡不稳定性，这类振荡不稳定性与固体衬底加热的液体膜的热毛细不稳定性有关。我们假设衬底厚度和衬底-膜导热系数较大，使得衬底热扩散的影响在长波近似中保持在第一阶。因此，系统动力学由薄膜厚度的非线性偏微分方程描述，该方程与全衬底热方程非局部耦合。对一个稳定的静态进行扰动，发现它的稳定性可以用一个非自伴随特征值问题来描述。我们证明，在适当的模型参数下，线性化的特征值问题允许复特征值，这些复特征值在物理上对应于薄膜高度的振荡（时间）不稳定性。作为我们工作的主要结果，我们提供了不同模型参数对振荡不稳定性的敏感性的完整图像。利用这一描述，我们得出结论，振荡不稳定性与由绝缘衬底加热的薄膜的实验更相关。此外，我们表明振荡不稳定性，其中增长最快（最不稳定）的波数是复杂的，只出现在具有足够大的衬底厚度的系统中。最后，我们讨论了我们的模型对实际环境的适应性，并对所报告的不稳定性可以观察到的条件进行了预测。
In this work we consider a new class of oscillatory instabilities that pertain to thermocapillary destabilization of a liquid film heated by a solid substrate. We assume the substrate thickness and substrate–film thermal conductivity ratio are large so that the effect of substrate thermal diffusion is retained at leading order in the long-wave approximation. As a result, the system dynamics is described by a nonlinear partial differential equation for the film thickness that is non-locally coupled to the full substrate heat equation. Perturbing about a steady quiescent state, we find that its stability is described by a non-self-adjoint eigenvalue problem. We show that, under appropriate model parameters, the linearized eigenvalue problem admits complex eigenvalues that physically correspond to oscillatory (in time) instabilities of the thin-film height. As the principal results of our work, we provide a complete picture of the susceptibility to oscillatory instabilities for different model parameters. Using this description, we conclude that oscillatory instabilities are more relevant experimentally for films heated by insulating substrates. Furthermore, we show that oscillatory instability where the fastest-growing (most unstable) wavenumber is complex, arises only for systems with sufficiently large substrate thicknesses. Finally, we discuss adaptation of our model to a practical setting and make predictions of conditions at which the reported instabilities can be observed.