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Rayleigh-Taylor instability of viscous liquid films under a temperature-controlled inclined substrate

Youchuang Chao1,*, Lailai Zhu2,†, and Hao Yuan3

  • 1Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
  • 2Department of Mechanical Engineering, National University of Singapore, 117575, Singapore
  • 3School of Life Science and Engineering, Southwest Jiaotong University, 610031 Chengdu, China

  • *ychao@connect.hku.hk
  • †lailai_zhu@nus.edu.sg
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Phys. Rev. Fluids 6, 064001 – Published 14 June, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.064001

Abstract

We study the Rayleigh-Taylor instability of gravity-driven viscous liquid films flowing under a uniformly heated or cooled inclined substrate. The long-wave assumption is adopted to derive the evolution equation of the film, which is characterized by five dimensionless parameters including Marangoni number Ma, Biot number Bi, Reynolds number Re, Weber number We, and the inclination angle α of the substrate. Based on the long-wave equation, we systematically examine the temporal and spatiotemporal stability of the system. Temporal stability analysis shows that the thermocapillary stress reinforces the Rayleigh-Taylor instability of a heated film but counteracts the instability of a cooled film, as verified by the numerical solutions of linearized Navier-Stokes equation. In particular, this instability can be completely inhibited if a composite Marangoni number Ma* is below a critical value Ma1*. We further perform a spatiotemporal stability analysis to delineate the absolute and convective nature of the temporally unstable system. We find that the thermocapillary stress in the heated film enhances the absolute instability and suppresses the convective instability. The trend reverses for a cooled film that is featured by suppressed absolute instability and enhanced convective instability. More importantly, the transition between the absolute and convective instability can be characterized by another critical value, Ma2*, beyond which the flow stability will be triggered from the convectively into the absolutely unstable. The predictions from linear stability analysis are confirmed by numerical solutions of the full long-wave evolution equation.

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Phys. Rev. Fluids 6, 064001– Published 14 June, 2021
  • Received 15 January 2021
  • Accepted 26 May 2021
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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