Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, and Trinity College, University of Cambridge, Cambridge CB2 1TQ, United Kingdom.
Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA.
Phys Rev E. 2019 Aug;100(2-1):022802. doi: 10.1103/PhysRevE.100.022802.
A prescribed, horizontal temperature gradient is imposed upon a horizontal liquid layer bounded from above by a deformable, liquid-gas interface and bounded from below by a partial-slip, rigid surface. A steady shear flow driven by thermocapillary motion emerges. This dynamic liquid layer is susceptible to the onset of oblique three-dimensional hydrothermal waves, purely two-dimensional hydrothermal waves, longitudinal traveling waves, and longitudinal rolls depending on the capillary number. A low capillary number analysis finds that surface deformations are destabilizing for all modes of instability. There is a preference for two-dimensional hydrothermal waves when there are surface deformations. Though longitudinal traveling waves are never selected as the preferred mode of instability, these waves offer a convenient way to understand the behavior of oblique hydrothermal waves, which are near-longitudinal. This is especially the case for low capillary numbers, but oblique hydrothermal waves instead tend to align themselves with the direction of flow as the capillary number increases. Surface deformations affect longitudinal waves most significantly out of all the modes of instability, especially for low Prandtl numbers. The typical length scales shorten and the critical Marangoni numbers increase with the capillary number for all types of modes. Notably, the system selects long waves near a critical Prandtl number when the interface is nondeformable and when the layer is subject to partial slip, but this is no longer the case when the upper surface is deformable.
在一个由可变形的气液界面和部分滑移的刚性表面界定的水平液体层上施加规定的水平温度梯度。由热毛细运动驱动的稳态剪切流出现。这个动态液体层容易受到斜三维热水力波、纯二维热水力波、纵向行波和纵向涡旋的产生，这取决于毛细数。低毛细数分析发现，表面变形对所有不稳定性模式都是不稳定的。当存在表面变形时，更倾向于二维热水力波。虽然纵向行波从未被选为优先不稳定性模式，但这些波提供了一种方便的方法来理解近纵向的斜热水力波的行为。对于低毛细数尤其如此，但随着毛细数的增加，斜热水力波倾向于与流动方向对齐。表面变形对所有不稳定性模式中纵向波的影响最大，尤其是对低普朗特数的情况。所有类型的模式的典型长度尺度缩短，临界马兰戈尼数随毛细数增加而增加。值得注意的是，当界面不可变形且层处于部分滑移时，系统在接近临界普朗特数时选择长波，但当上表面可变形时，情况不再如此。
