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Propagation of capillary waves and ejection of small droplets in rapid droplet spreading

Published online by Cambridge University Press:  12 March 2012

H. Ding ,
E. Q. Li ,
F. H. Zhang ,
Y. Sui ,
P. D. M. Spelt  and
S. T. Thoroddsen
H. Ding
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, 230027 Hefei, China Department of Chemical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106-5080, USA
E. Q. Li
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 119260 Singapore Division of Physical Sciences and Engineering & CCRC, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
F. H. Zhang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 119260 Singapore Singapore-MIT Alliance, National University of Singapore, 117576 Singapore
Y. Sui
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
P. D. M. Spelt*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK Département Mécanique, Université de Lyon 1, and Laboratoire de la Mécanique des Fluides & d’Acoustique, CNRS, 69134 Ecully, France
S. T. Thoroddsen
Affiliation:
Division of Physical Sciences and Engineering & CCRC, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
*
†Email address for correspondence: peter.spelt@univ-lyon1.fr
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Abstract

A new regime of droplet ejection following the slow deposition of drops onto a near-complete wetting solid substrate is identified in experiments and direct numerical simulations; a coalescence cascade subsequent to pinch-off is also observed for the first time. Results of numerical simulations indicate that the propagation of capillary waves that lead to pinch-off is closely related to the self-similar behaviour observed in the inviscid recoil of droplets, and that motions of the crests and troughs of capillary waves along the interface do not depend on the wettability and surface tension (or Ohnesorge number). The simulations also show that a self-similar theory for universal pinch-off can be used for the time evolution of the pinching neck. However, although good agreement is also found with the double-cone shape of the pinching neck for droplet ejection in drop deposition on a pool of the same liquid, substantial deviations are observed in such a comparison for droplet ejection in rapid drop spreading (including the newly identified regime). This deviation is shown to result from interference by the solid substrate, a rapid downwards acceleration of the top of the drop surface and the rapid spreading process. The experiments also confirm non-monotonic spreading behaviour observed previously only in numerical simulations, and suggest substantial inertial effects on the relation between an apparent contact angle and the dimensionless contact-line speed.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Capillary waves Interfacial Flows (free surface): Contact lines

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 697 , 25 April 2012 , pp. 92 - 114
Copyright
Copyright © Cambridge University Press 2012

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Ding et al. supplementary movie

Movie 1. Typical sequence of a second-stage pinchoff for a drop of water and glycerin mixture. Oh=0.009, We=0.026 and a static angle of 13 degrees.

Download Ding et al. supplementary movie(Video)
Video 3.1 MB

Ding et al. supplementary movie

Movie 1. Typical sequence of a second-stage pinchoff for a drop of water and glycerin mixture. Oh=0.009, We=0.026 and a static angle of 13 degrees.

Download Ding et al. supplementary movie(Video)
Video 426.7 KB

Ding et al. supplementary movie

Movie 2. Six-stage coalescence cascade after 1st-stage pinchoff for Oh=0.006, We=0.033 and a static angle of 12+/- 2 degrees.

Download Ding et al. supplementary movie(Video)
Video 45.5 MB

Ding et al. supplementary movie

Movie 2. Six-stage coalescence cascade after 1st-stage pinchoff for Oh=0.006, We=0.033 and a static angle of 12+/- 2 degrees.

Download Ding et al. supplementary movie(Video)
Video 1.1 MB

Ding et al. supplementary movie

Movie 3. Propagation of the capillary wave from the contact line (right end), in terms of the radial coordinate at the drop surface as a function of the polar angle, as defined in figure 4. Oh=0.008, We=0.016, contact angle is 30 degrees.

Download Ding et al. supplementary movie(Video)
Video 17.2 MB

Ding et al. supplementary movie

Movie 3. Propagation of the capillary wave from the contact line (right end), in terms of the radial coordinate at the drop surface as a function of the polar angle, as defined in figure 4. Oh=0.008, We=0.016, contact angle is 30 degrees.

Download Ding et al. supplementary movie(Video)
Video 1.5 MB