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Sliding instability of draining fluid films

Published online by Cambridge University Press:  19 October 2018

Georg F. Dietze Open the ORCID record for Georg F. Dietze [Opens in a new window] ,
Jason R. Picardo Open the ORCID record for Jason R. Picardo [Opens in a new window]  and
R. Narayanan
Georg F. Dietze*
Affiliation:
Laboratoire FAST, Univ. Paris-Sud, CNRS, Université Paris-Saclay, F-91405, Orsay, France
Jason R. Picardo
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, 560089, India
R. Narayanan
Affiliation:
Department of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA
*
†Email address for correspondence: dietze@fast.u-psud.fr
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Abstract

The aim of this paper is to show that the spontaneous sliding of drops forming from an interfacial instability on the surface of a wall-bounded fluid film is caused by a symmetry-breaking secondary instability. As an example, we consider a water film suspended from a ceiling that drains into drops due to the Rayleigh–Taylor instability. Loss of symmetry is observed after the film has attained a quasi-steady state, following the buckling of the thin residual film separating two drops, whereby two extremely thin secondary troughs are generated. Instability emanates from these secondary troughs, which are very sensitive to surface curvature perturbations because drainage there is dominated by capillary pressure gradients. We have performed two types of linear stability analysis. Firstly, applying the frozen-time approximation to the quasi-steady base state and assuming exponential temporal growth, we have identified a single, asymmetric, unstable eigenmode, constituting a concerted sliding motion of the large drops and secondary troughs. Secondly, applying transient stability analysis to the time-dependent base state, we have found that the latter is unstable at all times after the residual film has buckled, and that localized pulses at the secondary troughs are most effective in triggering the aforementioned sliding eigenmode. The onset of sliding is controlled by the level of ambient noise, but, in the range studied, always occurs in the quasi-steady regime of the base state. The sliding instability is also observed in a very thin gas film underneath a liquid layer, which we have checked for physical properties encountered underneath Leidenfrost drops. In contrast, adding Marangoni stresses to the problem substantially modifies the draining mechanism and can suppress the sliding instability.

JFM classification

Interfacial Flows (free surface): Capillary flows Instability: Instability Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , pp. 111 - 141
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Copyright
© 2018 Cambridge University Press 

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References

Alexeev, A. & Oron, A. 2007 Suppression of the Rayleigh–Taylor instability of thin liquid films by the Marangoni effect. Phys. Fluids 19 (8), 082101.Google Scholar
Balestra, G., Brun, P.-T. & Gallaire, F. 2016 Rayleigh–Taylor instability under curved substrates: an optimal transient growth analysis. Phys. Rev. Fluids 1, 083902.Google Scholar
Bonn, D. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739–805.Google Scholar
Boos, W. & Thess, A. 1999 Cascade of structures in long-wavelength marangoni instability. Phys. Fluids 11 (6), 1484–1494.Google Scholar
Boyd, J. P. 1989 Chebyshev and Fourier Spectral Methods. Springer.Google Scholar
Burton, J. C., Sharpe, A. L., van der Veen, R. C. A., Franco, A. & Nagel, S. R. 2012 Geometry of the vapor layer under a Leidenfrost drop. Phys. Rev. Lett. 109, 074301.Google Scholar
Chang, H. C., Demekhin, E. A. & Kalaidin, E. 1996 Simulation of noise-driven wave dynamics on a falling film. AIChE J. 42 (6), 1553–1568.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2013 Wavy liquid films in interaction with a confined laminar gas flow. J. Fluid Mech. 722, 348–393.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2015 Films in narrow tubes. J. Fluid Mech. 762, 68–109.Google Scholar
Duruk, S. & Oron, A. 2016 Nonlinear dynamics of a thin liquid film deposited on a laterally oscillating corrugated surface in the high-frequency limit. Phys. Fluids 28 (11), 112101.Google Scholar
Frumkin, V. & Oron, A. 2016 Liquid film flow along a substrate with an asymmetric topography sustained by the thermocapillary effect. Phys. Fluids 28 (8), 082107.Google Scholar
Glasner, K. 2007 The dynamics of pendant droplets on a one-dimensional surface. Phys. Fluids 19 (10), 102104.Google Scholar
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384.Google Scholar
Israelachvili, J. 2011 Intermolecular and Surface Forces. Academic Press.Google Scholar
Landau, L. D. & Levich, B. 1942 Dragging of liquid by a moving plate. Acta Physicochim. USSR 17, 42–54.Google Scholar
Linke, H., Alemán, B. J., Melling, L. D., Taormina, M. J., Francis, M. J., Dow-Hygelund, C. C., Narayanan, V., Taylor, R. P. & Stout, A. 2006 Self-propelled Leidenfrost droplets. Phys. Rev. Lett. 96, 154502.Google Scholar
Lister, J. R., Morrison, N. F. & Rallison, J. M. 2006a Sedimentation of a two-dimensional drop towards a rigid horizontal plate. J. Fluid Mech. 552, 345–351.Google Scholar
Lister, J. R., Rallison, J. M., King, A. A., Cummings, L. J. & Jensen, O. E. 2006b Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.Google Scholar
Ma, X., Liétor-Santos, J.-J. & Burton, J. C. 2015 The many faces of a Leidenfrost drop. Phys. Fluids 27, 091109.Google Scholar
Ma, X., Liétor-Santos, J.-J. & Burton, J. C. 2017 Star-shaped oscillations of Leidenfrost drops. Phys. Rev. Fluids 2, 031602(R).Google Scholar
Oron, A. 2000 Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films. Phys. Fluids 12 (7), 1633–1645.Google Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.Google Scholar
Quéré, D. 2013 Leidenfrost dynamics. Annu. Rev. Fluid Mech. 45, 197–215.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14 (1), 170–183.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids 1, 1484–1501.Google Scholar

Dietze et al. supplementary movie 1

Evolution of the suspended water film toward a quasi-steady state (h0=1 mm, Bo=0.134). The movie shows the evolution from panel 2a to panel 2f in slow motion, allowing to observe the three stages detailed in panels 3c, 3d, and 3e. The ordinate has been rescaled logarithmically to better highlight the secondary troughs.

Download Dietze et al. supplementary movie 1(Video)
Video 622.9 KB

Dietze et al. supplementary movie 2

Loss of symmetry and sliding of the suspended water film (h0=1 mm, Bo=0.134). The movie shows the evolution between panels 2a and 2i, whereby the framerate has been increased with respect to movie 1 in order to focus on the loss of symmetry and sliding detailed in panel 3f and figure 4. The ordinate has been rescaled logarithmically to better highlight the secondary troughs.

Download Dietze et al. supplementary movie 2(Video)
Video 853.8 KB

Dietze et al. supplementary movie 3

Evolution of the suspended water film with additional Marangoni stresses (h0=1 mm, Bo=0.134, Ma=-0.2). The movie shows the buckling cascade of figure 11 in action. The ordinate has been rescaled logarithmically to better highlight the evolution of the troughs.

Download Dietze et al. supplementary movie 3(Video)
Video 2.6 MB
Supplementary material: File

Dietze et al. supplementary data

Supplementary data

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