Linear stability analysis of evaporating falling liquid films
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In order to improve our understanding of the wavy dynamics of evaporating falling liquid films, we perform a linear stability analysis using the Orr Sommerfeld (OS) eigenvalue problem. In the present work, the OS problem is extended to include two effects due to evaporation, namely, vapor recoil and mass loss. We present a numerical scheme based on a Chebyshev spectral method to solve the eigenvalue problem. Moreover, we validate our model by comparing the results against the long wave theory in the region of small wavenumber and weak inertia. We also demonstrate how the long wave theory completely fails in predicting the correct behavior when the inertia is strong or the wavenumber is large. By performing a perturbation energy analysis, we confirm that the instability induced by vapor recoil (E-mode) behaves in a similar fashion to the instability due ot Marangoni effect (S-mode). Through the same analysis, we demonstrate that both the Smode and the E-mode can enhance each other.