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Research Article

Restoring universality to the pinch-off of a bubble

Amir A. Pahlavan, Howard A. Stone, View ORCID ProfileGareth H. McKinley, and View ORCID ProfileRuben Juanes
PNAS July 9, 2019 116 (28) 13780-13784; first published June 17, 2019; https://doi.org/10.1073/pnas.1819744116
Amir A. Pahlavan
aDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139;
bDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540;
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  • For correspondence: pahlavan@princeton.edu juanes@mit.edu
Howard A. Stone
bDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540;
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Gareth H. McKinley
aDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139;
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  • ORCID record for Gareth H. McKinley
Ruben Juanes
cDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139;
dDepartment of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139
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  • ORCID record for Ruben Juanes
  • For correspondence: pahlavan@princeton.edu juanes@mit.edu
  1. Edited by Osman A. Basaran, Purdue University, West Lafayette, IN, and accepted by Editorial Board Member John D. Weeks May 13, 2019 (received for review November 19, 2018)

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  • Fig. 1.
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    Fig. 1.

    Displacement of a partially wetting liquid from a microcapillary tube. As the glycerol (white) is withdrawn from the right end of the tube with a constant flow rate Q, air (black) invades the tube from the left end at atmospheric pressure and entrains a thin film of the glycerol on the tube walls (the white stripe in the middle of the tube is due to light refraction; SI Appendix, section 1). The entrained liquid film then starts receding along the tube axis with a velocity Ucl, forming a growing dewetting rim ahead of the contact line, where the liquid, solid, and air meet at a nonzero apparent contact angle θap. As the liquid rim grows, the bubble neck diameter shrinks and ultimately leads to pinch-off and the formation of a bubble.

  • Fig. 2.
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    Fig. 2.

    Evolution of the neck diameter versus time τ=(t0−t) to pinch-off at t0. (A) Data from 12 different experiments are shown: light blue symbols correspond to d=750 μm and μ=1.4 Pa.s; the cyan symbols correspond to d=280 μm and μ=1.4 Pa.s; the dark blue symbols correspond to d=750 μm and μ=0.2 Pa.s. Each color represents data corresponding to four different flow rates with Ca=μU/γ∈[0.008,0.02], where U=4Q/(πd2) and Q is the liquid flow rate. While changing the flow rate does not influence the evolution of the bubble neck diameter (2r0), changing μ or d shifts the curves. (B) When time and length scales are nondimensionalized with the visco-capillary time scale t*=μd/γ and the tube diameter d, respectively, the data corresponding to all 12 experiments collapse onto a single curve (τ̃=τ/t*=γτ/(μd), and r̃0=r0/d). Here, two self-similar regimes can be observed: an early-time regime, which follows a 1/5 power-law scaling in time, and a late-time regime very close to the point of pinch-off, which follows a linear scaling in time.

  • Fig. 3.
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    Fig. 3.

    Self-similarity of the neck profile. (A) The evolution of the bubble neck profile in time (data corresponding to d=750 μm, μ=1.4 Pa.s, and Ca=0.008); blue and green symbols represent the data corresponding to the early and late-time self-similar regimes, and red symbols represent the transition between the two. (B) Scaling the neck profile with the minimum neck diameter collapses the data corresponding to the early-time self-similar regime, where R̃(ξ)=r̃(z̃,τ̃)/τ̃1/5 and ξ=(z̃−z̃0)/τ̃1/5. The dashed line, overlaying the blue symbols corresponding to the early-time self-similar regime, represents the self-similar solution of the long-wave model (SI Appendix, section 2). The data corresponding to the late-time self-similar regime, however, deviate from the predictions of the long-wave model. (C) The definition of parameters used to characterize the bubble neck profile. (D) The evolution of the axial length scale defined as ζ̃=r̃0r̃c versus time to pinch-off. In the early-time regime, ζ̃=r̃cr̃0∼τ̃1/5, consistent with the predictions of the long-wave model. In the late-time regime, however, r̃0∼τ̃ and ζ̃∼τ̃1/2, which indicates that the axial radius of curvature becomes constant, i.e., the neck profile becomes a parabola that simply translates in time (13, 42). (E) Scaling the axial length scale with the expressions obtained in D leads to the collapse of all bubble neck profiles during the entire pinch-off process (shown in A) onto a single parabolic curve: r/r0=1+[(z−z0)/ζ]2 (dashed line).

  • Fig. 4.
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    Fig. 4.

    Axial radius of curvature versus time. The evolution of the axial radius of curvature rc versus time to the pinch-off shows that as the point of pinch-off is approached, the curvature asymptotes to a constant value (symbols are the same as in Fig. 2). (A) The time evolution of axial radius of curvature is independent of the flow rate but changes when the liquid viscosity or the tube diameter is varied. (B) The data corresponding to all 12 experiments collapse on a single curve when the nondimensional axial radius of curvature is plotted against the nondimensional minimum neck diameter. The nondimensional late-time axial radius of curvature asymptotes to a universal constant limτ̃→0r̃c(τ̃)=r̃cf≈0.07.

Data supplements

  • Supporting Information

    • Download Appendix (PDF)
    • Download Movie_S01 (AVI) - Evolution of the dewetting rim and the ultimate breakup of the bubble in a capillary tube with the diameter d = 280 μm, and Ca = 0:016. The imaging is done at 20 kfps.
    • Download Movie_S02 (AVI) - Motion of a microbubble in the vicinity of the bubble neck. The microbubble here acts as a tracer, showing the flow direction, which is axially-dominant at early times and crosses over to a radially-dominant flow at late times.
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Restoring universality to the pinch-off of a bubble
Amir A. Pahlavan, Howard A. Stone, Gareth H. McKinley, Ruben Juanes
Proceedings of the National Academy of Sciences Jul 2019, 116 (28) 13780-13784; DOI: 10.1073/pnas.1819744116

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Restoring universality to the pinch-off of a bubble
Amir A. Pahlavan, Howard A. Stone, Gareth H. McKinley, Ruben Juanes
Proceedings of the National Academy of Sciences Jul 2019, 116 (28) 13780-13784; DOI: 10.1073/pnas.1819744116
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