Bubble Bursting: Universal Cavity and Jet Profiles

Ching-Yao Lai, Jens Eggers, and Luc Deike

Phys. Rev. Lett. 121, 144501 – Published 2 October 2018

Abstract

After a bubble bursts at a liquid surface, the collapse of the cavity generates capillary waves, which focus on the axis of symmetry to produce a jet. The cavity and jet dynamics are primarily controlled by a nondimensional number that compares capillary inertia and viscous forces, i.e., the Laplace number $La = ρ γ R_{0} / μ^{2}$ , where $ρ$ , $μ$ , $γ$ , and $R_{0}$ are the liquid density, viscosity, interfacial tension, and the initial bubble radius, respectively. In this Letter, we show that the time-dependent profiles of cavity collapse ( $t < t_{0}$ ) and jet formation ( $t > t_{0}$ ) both obey a $| t - t_{0} |^{2 / 3}$ inviscid scaling, which results from a balance between surface tension and inertia forces. Moreover, we present a scaling law, valid above a critical Laplace number, which reconciles the time-dependent scaling with the recent scaling theory that links the Laplace number to the final jet velocity and ejected droplet size. This leads to a self-similar formula which describes the history of the jetting process, from cavity collapse to droplet formation.

Received 20 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.144501

Physics Subject Headings (PhySH)

Interfacial flows Surface tension effects

Aerosols Microbubbles Microfluidics

Navier-Stokes equation

Fluid DynamicsNonlinear Dynamics

Authors & Affiliations

Ching-Yao Lai¹, Jens Eggers², and Luc Deike^1,3,*

¹Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
²School of Mathematics, University of Bristol, University Walk, Bristol BS8 1 TW, UK
³Princeton Environmental Institute, Princeton University, Princeton, New Jersey 08544, USA

^*ldeike@princeton.edu

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Vol. 121, Iss. 14 — 5 October 2018

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Images

Figure 1
The time evolution of the liquid-gas interface of a bursting bubble of initial radius $R_{0}$ at $La = 2000$ and $Bo = 10^{- 3}$ . The color bar indicates the time corresponding to different profiles. (a) The formation of a jet after the collapse of capillary waves. The time difference between the curves is $δ t / τ = 0.038$ , where $τ = \sqrt{ρ R_{0}^{3} / γ}$ is the inertio-capillary timescale. The time when a jet ejects a drop is $t_{d}$ . (b) The interface profiles right before a bubble is entrained. The time difference between the curves is $δ t / τ = 0.0003$ . As the front of the capillary wave approaches $r = 0$ the interface steepens, snaps, entrains a bubble, and forms a jet at $t = t_{0}$ . The bottom location of the bubble at the entrapment is denoted with $h_{0}$ .
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Figure 2
Comparison between the numerical result (dots) and the scaling arguments (solid line) in Eqs. (4) and (5). When the jet drop detaches from the jet at $t = t_{d}$ , the length of the jet and the radius of the jet drop are denoted $L_{d}$ and $R_{d}$ , respectively. (a) The numerical results show that the jet length $L_{d} \propto φ$ obeys Eq. (4) with a fitted prefactor $k_{ℓ} \approx 50$ (solid line). (b) The timescale $t_{d} - t_{0}$ , during which the jet forms and produces a jet drop, agrees well with the solid line $(t_{d} - t_{0}) \propto φ^{7 / 4}$ [Eq. (5) with fitted prefactor $k_{t} \approx 2$ ].
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Figure 3
(a) The time evolution of the liquid-gas interface during jet formation ( $t > t_{0}$ ) for $La = 1000$ , 1500,2000,5000,20 000,50 000 at times $t - t_{0} = t_{j}, 3 / 2 t_{j}, 2 t_{j}, 5 / 2 t_{j}$ , where $t_{j} \equiv φ^{7 / 4} ℓ_{μ} / V_{μ}$ is the characteristic time of jet formation. (b) The dimensionless jet profiles rescaled using Eq. (7). The profiles are shifted in the axial direction with respect to the bottom of the jet $h_{b}$ . The dimensionless profiles for different times and $La = 5000 - 50 000$ collapse, except for the region of the rounded tip. When $La \leq 2000$ (i.e., $φ \leq 45$ ) the timescale $t_{j}$ used here deviates slightly from the jet lifetime $t_{d} - t_{0}$ [see Fig. 2], and therefore affects the collapse of the jet onto the universal profile.
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Figure 4
(a) The time evolution of the interface during cavity collapse ( $t < t_{0}$ ) for $La = 1000$ , 1500, 2000, 5000, 20 000 at times $t_{0} - t = 6 t_{c}, 8 t_{c}, 10 t_{c}$ , where $t_{c} \equiv φ^{3 / 2} ℓ_{μ} / V_{μ}$ is the characteristic time of the horizontal capillary wave. (b) The rescaled cavity profiles nondimensionalized according to Eq. (8). The bottom of the bubble when bubble entrapment occurs ( $t = t_{0}$ ) is denoted $h_{0}$ . The dimensionless profiles collapse onto a universal curve for $La = 1000 - 5000$ during the time window $t_{0} - t = 6 t_{c} - 10 t_{c}$ . Near the bubble entrapment time $t_{0} - t < 6 t_{c}$ the profile steepens and snaps, and thus deviates from the universal shape. When $La \geq 5000$ , multiple capillary waves are observed [see (a)], and Eq. (8) does not collapse the profiles for $La > 5000$ .
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