Abstract We consider the linear stability of evaporating thin films falling down an inclined plate. The one sided-model presented first by “Burelbach, J.P., Bankoff, S.G., Davis, S.H., 1988, Nonlinear stability of evaporating/condensing liquid films, Journal of Fluid Mechanics 195, 463–494. ” was implemented to decouple the dynamics of the liquid than those of the vapor at the interface, at which the evaporation is modeled based on a thermal equilibrium approach. We consider the base state solution derived by “Joo, S., Davis, S.H., Bankoff, S., 1991, Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers, Journal of Fluid Mechanics 230, 117–146. ” which is based on the slow evaporation assumption. In previous works, only low dimensional models. i.e. the long wave theory, have been analysed for evaporating liquid films. Conversely in this paper, we extend the Orr-Sommerfeld eigenvalue problem for a film falling down a heated wall to include evaporation effects namely, vapor recoil and mass loss. As expected, we observe that the long wave theory fails in predicting the correct behavior when the inertia is strong or the wavenumber k is large. We confirm that the instability induced by vapor recoil (E-mode) behaves in a similar fashion to the instability due to the thermocapillary effect (S-mode). Both the S-mode and the E-mode can enhance each other, specially, at low Reynolds numbers Re. Moreover, we examine the perturbation energy budget to have an insight into the instabilities mechanism. We show that the presence of evaporation adds a new term corresponding to the work done by vapor recoil at the interface (VRE). We also find that the main contributor to the perturbation kinetic energy in the unstable E-mode is the work done by shear stress while VRE is negligible, unless Re

中文翻译：

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蒸发降液膜的线性稳定性分析

摘要 我们考虑蒸发薄膜从斜板上落下的线性稳定性。“Burelbach, JP, Bankoff, SG, Davis, SH, 1988, 蒸发/冷凝液膜的非线性稳定性, Journal of Fluid Mechanics 195, 463–494 首次提出了单边模型。”的实施是为了将界面处的液体动力学与蒸汽动力学解耦，在该界面处，基于热平衡方法对蒸发进行建模。我们考虑由“Joo, S., Davis, SH, Bankoff, S., 1991，加热降膜的长波不稳定性：均匀层的二维理论，流体力学杂志 230, 117– 146. ”这是基于缓慢蒸发假设。在以前的工作中，只有低维模型。即长波理论，已经分析了蒸发液膜。相反，在本文中，我们扩展了薄膜从加热壁上落下的 Orr-Sommerfeld 特征值问题，以包括蒸发效应，即蒸汽反冲和质量损失。正如预期的那样，我们观察到当惯性强或波数 k 大时，长波理论无法预测正确的行为。我们确认蒸汽反冲（E 模式）引起的不稳定性与热毛细管效应（S 模式）引起的不稳定性表现相似。S 模式和 E 模式都可以相互增强，特别是在低雷诺数 Re 下。此外，我们检查扰动能量预算以深入了解不稳定性机制。我们表明，蒸发的存在增加了一个新术语，对应于界面处蒸汽反冲 (VRE) 所做的功。