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New Concepts at the Interface: Novel Viewpoints and Interpretations, Theory and Computations
The ejection of droplets from a bursting bubble on a liquid surface—A dimensionless criterion for ‘jet’ droplets Ryan Mead-Hunter, Monica M. Gumulya, Andrew (A.) King, and Benjamin J Mullins Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00664 • Publication Date (Web): 08 May 2018 Downloaded from http://pubs.acs.org on May 9, 2018
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The ejection of droplets from a bursting bubble on a
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free liquid surface—A dimensionless criterion for
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‘jet’ droplets
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Ryan Mead-Hunter*,†,‡, Monica M. Gumulya⁑, Andrew J.C. King‡,⁑ and Benjamin J. Mullins†,‡
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†
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6845, Australia
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‡
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⁑
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University, Perth WA, 6845, Australia
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Occupation, Environment & Safety, School of Public Health, Curtin University, Perth WA,
Fluid Dynamics Research Group, Curtin University, Perth WA, 6845, Australia Western Australian School of Mines: Minerals, Energy and Chemical Engineering, Curtin
⁑
School of Civil and Mechanical Engineering, Curtin University, Perth WA, 6845, Australia
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KEYWORDS
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Jet Droplets, Bubble Bursting, Computational Fluid Dynamics
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ABSTRACT
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This work examines the ejection of droplets from a bursting gas bubble on a free liquid surface,
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both experimentally and numerically. We explore the physical processes which govern the
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bursting of bubbles and the subsequent formation of ‘jet’ droplets. We present new relationships
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regarding the dependence of jet drop formation on bubble diameter. Furthermore, we propose a
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new dimensionless parameter to describe the region of properties where ‘jet’ drops will occur.
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This parameter, termed the droplet number (Dn), complements existing parameters defining jet
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drop formation—namely a maximum Ohnesorge number and a maximum Bond number.
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INTRODUCTION
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The generation of droplets from bursting bubbles is a commonly known and well-studied
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phenomenon1-3. This process gives rise to a significant portion of atmospheric aerosols4-7 and has
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been described as an important process in the enjoyment of a glass of sparkling wine1. Despite
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being well studied, there are still a number of questions surrounding the role of liquid properties
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and their contribution to the mechanisms of bubble bursting.
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Two distinct mechanisms of droplet formation via bubble bursting exist2,3. The first is when a
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bubble collapses at the liquid surface, giving rise to a liquid column, which breaks into droplets
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which are ejected vertically with significant momentum—these are termed ‘jet’ droplets (and are
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shown in Figure 1). The other case is where a liquid film surrounding the bubble, breaks apart.
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Giving rise to ‘film’ droplets.
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The formation of ‘jet’ droplets only occurs over a specific range of droplet sizes for a given
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liquid. Bubbles bursting in water for example will only produce ‘jet’ droplets at diameter of 4.2
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mm or less8. The relevant properties of the liquid are its viscosity and surface tension, which
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dictate the nature of the bubble collapse. The range of properties for which ‘jet’ droplets are
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produced have been described in terms of the dimensionless Ohnesorge number (Oh) and Bond
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number (Bo). Systems with Oh ≲ 0.037 and Bo ≲ 3, have typically been said to produce ‘jet’
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droplets8.
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In recent work, Walls et al.8 identified an additional zone, where systems that would be
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expected to produce ‘jet’ droplets, based on values of Oh and Bo, do not. This was explored
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through the use of computational fluid dynamics (CFD) and ascribed to the combined effects of
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gravity and viscosity.
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The influence of viscosity was also highlighted by Ghabache et al.1 who conducted experiments using 150 to 1000 µm radius droplets, of varying concentrations of glycerol in
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water. In their work they considered the role of the bubble bursting process in the control of the
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production of droplets, and noted two regimes. A surface tension dominated regime and a
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viscosity dominated regime, with the transition between the two occurring at a Morton number
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(Mo) (Eq. 3) of approximately 3 x 10-8. The process of bubble bursting and the production of
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‘jet’ droplets may therefore be seen as a function of surface tension, viscosity, gravity and bubble
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size.
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In addition to Oh, Bo, and Mo, the Weber number (We) has also been used to describe the
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dynamics of bubble bursting. Ghabache et al.1 use this value to determine a dimensionless jet
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velocity term (given as WeBo1/2), which clearly demonstrates the surface tension and viscosity
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driven regimes, as evidenced in Figure 4. The value of the dimensionless terms are defined as
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follows,
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We =
ρVtip2 r γ
(1)
Bo =
ρgr 2 γ
(2)
Mo =
gµ 4
ργ
3
(3)
Oh =
µ (4) ργr
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where, ρ is the liquid density, r is the bubble radius, γ the liquid surface tension, g acceleration
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due to gravity, µ the liquid viscosity and Vtip is the tip velocity. The value of tip velocity, as used
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by Ghabache et al.1 and in this work, is the velocity of the liquid column at the point where it
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reaches the surface of the bulk liquid.
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Figure 1. Images of the bubble collapse and ejection of droplets in salt water. The 4 frames on
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the left show the beginning of the collapse of the bubble after it has reached the surface. Moving
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right we can see the cavity beginning to collapse as it is drawn in from the base as the jet begins
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to form. The final two frames, are from a camera orientation an and show the ejection of a
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droplet from the rising liquid column.
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While the motivations for studying bubble-bursting are many and varied, one application is to
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the (re-)entrainment of liquid aerosol from coalescing filters11. In previous work9 we have
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observed this (re-)entrainment and have proposed two possible mechanisms. These have been
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further explored by Hotz et al.10 who measured the force required to detach a droplet from a
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fibre, and Wurster et al.11 who identified that the bursting of bubbles was (in most cases) the
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detachment mechanism for filter re-entrainment.
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When utilising coalescing media to treat liquid aerosols, the captured liquid has the ability to
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coalesce and flow within the media. As the filter becomes saturated the liquid moves through the
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filter to the rear face, where it can drain from the filter. This process is accompanied by the
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formation of a liquid film on the rear face, though which air must pass. In cases where the
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saturation level of the filter is high, the air may need to break through the rear liquid film, as
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process which results in the formation of bubbles, which burst releasing droplets.
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In this work we simulate the bubble bursting process and the subsequent formation of ‘jet’
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droplets using experimental data generated as part of this work, and data from Ghabache et al.1
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for validation. If the key physical processes can be reliably captured then the simulation methods
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used can be more broadly applied to our coalescing filter case—an application where
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experiments are considerably more difficult. Through our simulations we will also consider
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parameters for ‘jet’ droplet formation and attempt to define a parameter to identify the region
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identified by Walls et al.8 where ‘jet’ droplets do not occur, despite meeting Oh and Bo criteria.
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The volume-of-fluid (VoF) method has previously been used to simulate bubble motion12,13
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and to accurately resolve liquid interfaces on small scales14 and will be applied here.
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METHODOLOGY
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Experimental
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Experiments were conducted in a clear, liquid filled cylinder with bubbles introduced by
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passing compressed air through a fine nozzle (of varying size) in the base of the cylinder. The
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bubble rate was controlled using a multi-stage pressure regulator and fine control achieved using
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a manually adjustable needle valve. Bubbles were allowed to rise to the liquid surface and the
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bursting process was observed with the aid of a high speed camera (Photron FASTCAM SA 3
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Model 120K-M2) fitted with a macro lens (Nikon AF-S VR Micro-Nikkor 105 mm f/2.8G IF-
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ED). Only bubbles that clearly produced, or clearly did not produce jet droplets were chosen for
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further analysis. Bubbles that did not produce jet drops, were measured only for size, bubbles
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that did produce jet drops were measured for size and tip velocity.
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Experiments were conducted in (distilled) water, salt water (30 g NaCl/l in distilled H2O),
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ethanol (99.5%), and 40 and 50% propylene glycol-water mixtures.
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Results from Ghabache et al.1 were extracted using Plot Digitizer
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(plotdigitizer.sourceforge.net). These data were produced in a manner similar to that described
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above, however using glycerol-water mixtures.
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Model
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Simulations were conducted using the OpenFOAM computational fluid dynamics (CFD)
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software package (OpenFOAM Version 2.1.x, ESI Group). We utilised the volume-of-fluid
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(VoF) method to resolve the liquid-air interface, coupled with dynamic mesh refinement applied
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to all cells at the liquid-air interface (defined as cells with a liquid volume fraction above 0.999
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or below 0.001).
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For the purposes of simulation, we prepared a rectangular domain, partially filled with liquid
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and introduced a bubble of air below the liquid surface. The bubbles were allowed to rise due to
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buoyancy forces, i.e. the bubble had an initial velocity of 0. The simulations were run for
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sufficient time to allow the bubble to rise to the surface, collapse and eject a column of liquid,
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potentially producing ‘jet’ droplets. Figure 3 illustrates the latter part of this process where
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several droplets have been ejected from the collapsing liquid column. As this process is governed
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by the fluid properties, simulations with different liquids were run for different times.
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Simulations were considered complete, when the rise of the ejected ‘jet’ droplets stopped (i.e.
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they began to fall back towards the liquid surface).
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In order to ensure that domain size was not a factor in the rise of the bubble or its behaviour at
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the interface, simulations were conducted to assess domain size and level of mesh refinement
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required. This was critical to ensure that the break-up of the rising liquid column that forms after
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bubble collapse was resolved.
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A domain width of 10r was found to be sufficient, with lower values resulting in unrealistic
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velocity profiles at the interface, possibly due to wall effects or numerical artefacts. An initial
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mesh comprised of 20 cells per bubble diameter, was found to be sufficient, prior to initial
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refinement. The mesh is dynamically refined at the liquid interface, with a minimum refinement
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level of 26 times the number of cells in the refinement region. A fixed limit on the total number
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of cells in the domain was set at 12 million—a limit which was not reached in any of the results
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presented. This limit may need to be increased should larger droplets be simulated, however was
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sufficient for 250, 500 and 1000 µm radius droplets considered in this work. Use of lower levels
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of refinement, or lower limits on the total number of cells resulted in poor resolution of the liquid
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column.
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Given the rapid nature of the bubble bursting process, a sufficiently small time step, and write
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interval, had to be used to ensure that the behaviour at the interface was properly captured. This
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was important to allow the tip velocity to be extracted.
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To allow comparison with the experiments conducted by Ghabache et al.1, a number of
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different water-glycerol mixtures were simulated (water, ethanol and water with varying
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concentrations of glycerol), which correspond to the range of Morton numbers found to produce
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‘jet’ droplets as a result of bubble bursting. The properties of the liquids used are given in Table
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1.
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Table 1. Liquid properties used in simulations Liquid
Density (ρ) kg/m3
Surface (γ) Tension Viscosity (µ) Pa.s N/m
Water
998
0.072
0.00089
20% Glycerol
1043.5
0.071
0.00135
30% Glycerol
1068.6
0.0695
0.00187
40% Glycerol
1094.8
0.069
0.00272
50% Glycerol
1120.1
0.068
0.00421
55% Glycerol
1134.2
0.0675
0.00568
60% Glycerol
1148.3
0.067
0.00719
65% Glycerol
1162
0.0665
0.01064
70% Glycerol
1175.6
0.066
0.01410
Ethanol
789
0.0224
0.00104
7 8 9 10 11
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RESULTS & DISCUSSION
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To examine the resolution of the liquid-air interface, the results of simulations were compared
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visually to the images taken by Ghabache et al.1. For this qualitative validation the simulations of
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500 µm radius bubbles in water and 50% glycerol solutions were used. The images shown in
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Figure 2, show the shape of the interface below the bulk liquid surface just before and just after
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the formation of the liquid column resulting from the collapse of the bubble. These compare
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favourably to the images shown in Figure 4 of Ghabache et al.1
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The other important aspect in terms of resolving the interface is the ability to simulate the
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droplets ejected as part of the bubble collapse process. This is illustrated in Figure 3, for the
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bubble rising in water case. The expected behaviour (again with reference to the results of
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Ghabache et al.1 can be seen, in terms of the ejection of the ‘jet’ drops and the ripples on the
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liquid surface.
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Tip Velocity
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The ability to resolve the interface is an important aspect of these simulations, however before
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such simulation can be applied to liquid (re-)entrainment in coalescing filters, we need to
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conduct a more quantitative analysis. This has been achieved by considering the tip velocity of
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the liquid column that results from the collapse of the bubble. By extracting this value from the
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simulations, we were able to evaluate the values of We, Bo and Mo, as used by Ghabache et al.1,
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and were able to plot the results of our simulations against their experimental results. These
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values are shown in Figure 4.
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2 3
(a)
(b)
(c)
(d)
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Figure 2. Shape of the interface below the liquid surface, (a) shows the profile after the bubble
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has reached the surface and 130 µs before the formation of the liquid column. (b) shows the
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profile 30 µs before formation of the liquid column, and (c) at time 0, where the bubble has
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collapsed and the liquid column/jet is about to form. (d) shows the profile 30 µs after the bubble
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collapse, where the liquid column has formed, though is still below the surface of the bulk liquid.
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Figure 3. The bursting of a 1000 µm radius bubble in a glycerol-water mixture, the column/jet of
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liquid has formed and ejected a droplet.
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Figure 4. Experimental results from Ghabache et al.1 compared to our simulations. There is reasonable agreement between the simulated values and the experimental values in
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Figure 4. It is important to note that where we have used bubbles of either 250, 500, 1000 or
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2000 µm radius in our simulations, Ghabache et al.1 have used bubbles ranging from 150 to 2100
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µm. It is also evident from Figure 4, that our simulations have correctly captured the two
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different relationships with Morton number (as illustrated by considering the results either side
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of Mo = 3 x 10-8).
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The differences observed between the 250 µm, 500 µm and 1000 µm bubble diameter cases,
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also appear to be consistent, with the results for 1000 µm bubbles consistently appearing above
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the results for the 500 µm bubbles, which themselves appear above the results for 250 µm
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bubbles. A trend easily explained by considering that both We and Bo are a function of bubble
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size. This also explains the vertical orientation of the results of Ghabache et al.1, for liquids with
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the same value of Mo.
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There is some question over the ability to capture Vtip accurately, whether in experiment or
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simulation. In our simulations, we increase the temporal resolution to ensure that the point where
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the rising liquid column/jet meets the plane of the liquid surface is captured. This is necessary as
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the velocity of this liquid jet rapidly changes, as the column/jet develops. We must also consider
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difficulties in determining such values experimentally, as the movement of the liquid column
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between individual frames, would produce a similar issue.
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Bubble size and jet dynamics
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Earlier modelling work, conducted by Boulton-Stone et al.15 observed that larger bubbles
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produce proportionally wider jets, a trend we have observed in our simulations also. They also
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observed that smaller bubbles produce faster jets. If we consider the width of the rising jet in our
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simulations, we observe both wider jets forming with increasing bubble size and well as with
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increasing viscosity and decreasing surface tension. The full widths at half height of the fully
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developed jet were extracted from the simulations and are shown in Table 2.
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Table 2. Full width at half height of the fully developed liquid jet (µm)
17 % Glycerol
30
40
50
55
60
250
40
40
62
88
114
500
54
77
87
92
101
r (µm)
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If we consider the values in Table 2, we can see that larger bubbles produce wider jets and that
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an increase in viscosity (increase in glycerol concentration) also produces wider jets. This
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widening of the jet with increasing viscosity would appear to be evidence of the role of viscosity
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in inhibiting jet droplet formation. This can also be seen if we consider the number of droplets
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produced. Smaller bubbles, in water (high surface tension, low viscosity liquids) produce
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multiple jet droplets, something confirmed by the observations of Hayami and Toba17, where
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bubbles of less than 1000 µm radius produced multiple droplets and bubbles less than 2000 µm
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in radius consistently produced droplets. We see a similar pattern in our simulations, where
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multiple droplets result from the water and lower glycerol concentrations and only single
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droplets are given off form higher viscosity liquids, before droplet formation ceases at higher
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viscosities. We also note that the glycerol concentration at which droplet formation decreases
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reduces with increasing bubble size, confirming the role of gravity in the process.
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While the above indicates that qualitatively we are observing the correct behaviour in our
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simulations, we can also take a more quantitative approach, by considering the scaling
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relationships proposed by Gañán-Calvo19. In their work, scaling parameters were introduced and
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applied to the experimental data of a number of authors, showing that they could all be described
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by a power law. Here we consider two of the terms described by Gañán-Calvo19 a reduced radius
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and a reduced velocity, as shown in Eq. 5 and Eq. 6, respectively. 5
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Where, Rd is the radius of the first jet droplet ejected and lµ and Vµ are expressed as follows; = 7
= 8
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We can plot our simulated data in the same manner, to illustrate the power law relation. This is
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shown in Figure 5. Here we see that our simulations conform to the -5/3 power law relation
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shown by Gañán-Calvo19, indicating that our simulations do correctly reflect the behaviour of the
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physical system.
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Figure 5. Reduced radius and reduced velocities for a subset of our simulated data. The line shows the -5/3 power law described by Gañán-Calvo19.
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Parameters to define jet drop formation
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The range of Mo value covered by Ghabache et al.1 appears to cover the range of liquid
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properties where ‘jet’ droplets are produced. Additional simulations on 80% glycerol and a
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multicomponent oil (Mo > 1 x 10-6) did not produce ‘jet’ drops, which would appear to confirm
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this. Based on our simulations and the results of Ghabache et al.1, we propose that there is a
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range of Mo values where ‘jet’ drops may arise as a result of bubble bursting.
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This range starts with the low value of Morton number of the order of 10-11 (that corresponding
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to water) and increase to a transition point at approximately 3 x 10-8, where the viscosity
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dominant regime takes over. ‘Jet’ droplets still form under this regime, however the velocity with
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which they are ejected appears to decrease, to a point where the collapse of the liquid cavity,
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does not give rise to droplets. Based on our simulations, and existing literature this region occurs
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at Morton numbers between 1 x 10-6 and 1 x 10-5.
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Given that the Morton number is heavily dependent on the surface tension and viscosity of the
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liquid, it may also be meaningful to consider these two parameters alone. If we choose the
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viscosity to surface tension ratio µ/γ then our region of transition between the surface tension
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dominant regime (left of Mo = 3 x 10-8 in Figure 4) and the viscosity dominant regime (right of
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Mo = 3 x 10-8 in Figure 4) occurs at approximately 0.1, and the region where ‘jet’ droplets are no
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longer formed corresponds to approximately µ/γ = 0.2. Broader application of such limits would
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require further simulations of additional liquids, with properties around these limits. It may be
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the case, however, that a more complete (and non-dimensioned) limit can be found in the form of
21
a dimensionless number.
22
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The droplet number Given that Walls et al.8 observed that bubbles in liquids with Oh ≲ 0.037 and Bo ≲ 3 do not
3
always produce jet droplets, we chose to explore this behaviour with our simulations. As our
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initial work considered a smaller range of diameters than that of Walls et al.8, we chose to run
5
selected simulations on liquids containing rising bubbles up to radii of 4000 µm. We were also
6
able to also identify this additional region and sought to define it in terms of a dimensionless
7
term, initially considering Mo.
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While the Morton number can be used to provide some indication of when ‘jet’ drops may form,
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it does not include a term for bubble diameter, a parameter which is also known to influence ‘jet’
10
droplet formation. The use of a viscosity to surface tension term, described above, also presents
11
the same limitation. As we discussed in the introduction, the formation of jet droplets is related
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to the surface tension, viscosity, gravity and bubble size, so it makes sense to consider a
13
parameter that contains all these variables. Hence we define such a parameter, the droplet
14
number (Dn), where;
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Dn = f ( g , r , µ , γ )
(9)
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Given that a certain value of a ratio of viscosity to surface tension appears to correspond to a
17
regime where ‘jet’ droplets form, it is reasonable to assume that a terms governing the formation
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of ‘jet’ droplets would include such a term. The Ohnesorge number includes such a term, though
19
however, with γ1/2, and does not itself account for gravity. If we also require the parameters of g
20
and r to be included, it makes physical sense to include them together, as the influence of gravity
21
will ultimately be influenced by the size of the bubble.
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Based on these considerations we propose the following expression for Dn,
2
Dn =
3
rgµ 2
γ2
(10)
Upon evaluating this term for our simulations on 0.5, 1 and 2 mm diameter bubbles, we note
4
that there appears a value of Dn, above which ‘jet’ droplets no longer occur. This threshold
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occurs when Dn ≈ 0.000126, with values above this not appearing to produce ‘jet’ droplets. The
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implication here is that jet droplets only occur at very low values of Dn. This appears to reflect
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the role of viscosity—in Eq. 10 and in the physical system. If we consider the viscosities of the
8
water glycerol mixtures (see Table 1) the values are low for pure water and increase with
9
increasing glycerol concentration, effectively covering 2 orders of magnitude. The pattern we
10
have observed, and which is reflected in Figure 6, is that higher viscosities (for bubbles of the
11
same size, as indicated by Mo) tend not to produce jet droplets, which is certainly the case for the
12
70% glycerol solution. Given that for a fixed droplet radius the value of viscosity shows the
13
greatest range, it makes sense that this parameter has such a strong influence on the value of Dn.
14
With increasing viscosity there is an increase in the value of the numerator in Eq. 10, which
15
results in an increase in the value of Dn. Moving us towards the threshold value. If we consider
16
different size bubbles in a liquid of the same viscosity we will see a similar pattern with the value
17
of the numerator increasing as a function of r. This in agreement with the work of Walls et al.8,
18
who noted the role of viscosity and gravity in limiting the formation of jet droplets.
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Page 18 of 28
1 2
Figure 6. The droplet number for our experiments and simulations, as well as selected data from
3
Walls et al.8. The dashed vertical lines indicate two potential ‘threshold’ value of Dn, which will
4
be discussed further in text. The three points to the left of the first vertical line, correspond to 0.5
5
mm bubbles, for which Oh < 0.037, and would not be expected to produce jet droplets.
6 7
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Langmuir
Closer inspection of our parameter reveals that it may also be expressed in terms of a function
2
of Bo and Oh, where;
3
= !ℎ
(8)
4
In order to further explore the use of this parameter we consider the results of Walls et al.8 and
5
additional simulations conducted for larger bubbles. Our results and selected results from Wall et
6
al.8 (only results where ‘jet’ droplets occurred) are shown in Figure 6. The figure also includes
7
lines for Dn = 0.000126 and Dn = 0.000398, which based on our results present potential
8
threshold values for droplet formation. It should be noted that the 3 data points to the left of the
9
Dn = 0.000126 line, correspond to Oh values of less than 0.037, so are not expected to produce
10
jet droplets. Our intention therefore, is not to provide a single value which dictates jet droplet
11
formation, rather an additional term that works with the established values of Oh and Bo. What
12
we propose is that bubbles in liquids with Oh values above 0.037 and Bo values below 3 would
13
then also require a Dn number below a certain value. The intent of this term is to account for the
14
region where jet droplets would be expected to occur, based on Oh and Bo values, however do
15
not.
16
Given that the majority of our works considers smaller droplets, we chose to further explore
17
the suitability of our new dimensionless number, though consideration of larger droplets. This
18
also presented the opportunity to test different liquids which may influence the value of Dn for
19
which jet droplets occur. As Dn is a function of viscosity and surface tension, this is a possibility
20
that should be considered. Using two different concentrations of water-propylene glycol
21
solution, we tested bubbles larger than 1500 µm in radius, and observed cases where jet droplets
22
did occur and did not occur; and evaluated Dn for these. We found that bubbles with Dn >
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Page 20 of 28
1
0.000398 did not produce jet drops. Indicating that a limiting value of Dn exists for these cases
2
though the value differs slightly. Given that the viscosity to surface tension ratio of water-
3
propylene glycol mixtures is approximately twice that of water-glycerol mixtures of the same
4
concentration, this difference in terms of limiting Dn value may make sense, though it may also
5
be the case, that further experimental measurements and/or simulations are required to find a
6
single value. A second line for Dn = 0.000398 is given in Figure 6.
7
Our simulations and experiments can be seen to agree relatively well with the region defined
8
by the Dn lines in Figure 6. With our bubble simulations falling left of the Dn = 0.000126
9
vertical line, and our experiments with water-propylene glycol solutions sitting left of the Dn =
10
0.000398 line, for cases where jet drops were observed (black diamond and black circles in
11
Figure 5). There is however some variation in terms of the results of Walls et al.8 and where they
12
sit in relation to the line for Dn = 0.000126. The values which are close could simply be
13
explained by the value of 0.000126 being approximate, however some values are further
14
removed from the Dn = 0.000126 line. For these values there are two possible explanations.
15
Firstly, the manner in which the radius is determined differs between our work and the work of
16
Walls et al.8. In our work we us the radius of the imposed bubble in the simulations, which is
17
essentially the bubble equivalent radius used by Georgescu et al.16 and in a number of previous
18
experimental works2,15,16,17. In contrast Walls et al.8 have used the cap radius, i.e. the radius when
19
the bubble reaches the surface. At this point the bubble takes on a shape closer to an ellipsoid
20
and therefore has a larger radius than a sphere of equivalent volume. This means that the values
21
from Walls et al.8 shown in Figure 6 have effectively been shifted to the right. Had an equivalent
22
radius been determined based on the dimensions of a sphere with the same volume been
23
provided the results would be shifted to the left. Secondly, the methodology employed by Walls
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Langmuir
1
et al.8 is not fully described. The values for the properties given glycerol mixtures given by Walls
2
et al.8 vary from the values we have used in our simulations, in particular the value of surface
3
tension. This would influence the results of the simulations. It may be that these were evaluated
4
at different temperatures, however this information is not provided.
5
CONCLUSION
6
We have simulated the ejection of jet droplets from bubble bursting in liquids, with a range of
7
parameters. Using the VoF method and dynamic mesh refinement, we were able to capture the
8
key elements of the physical processes of bubble bursting and ‘jet’ droplet formation. The results
9
compare favourably with existing literature. Our simulations have provided refinement to
10
existing experimental data, to show a bubble size influence on jet drop formation. We have also
11
proposed a dimensionless term governing the formation of ‘jet’ droplets, which complements
12
existing limits, Oh ≲ 0.037 and Bo ≲ 3. Our term, Dn, provides the final boundary condition to
13
the system at a values of approximately 0.000126 and 0.00398. It is hoped that this parameter
14
will prove useful to those studying bubble bursting processes, or applications which involve the
15
bursting of bubbles to produce droplets.
16 17
AUTHOR INFORMATION
18
Corresponding Author
19
*Email:
[email protected]
20
Author Contributions
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Page 22 of 28
1
The manuscript was written through contributions of all authors. All authors have given approval
2
to the final version of the manuscript.
3
ACKNOWLEDGMENT
4
This work was supported by resources provided by the Pawsey Supercomputing Centre with
5
funding from the Australian Government and the Government of Western Australia, and funded
6
by an Australian Research Council (ARC) Linkage Grant (LP140100919). We would also like
7
to acknowledge Marion Arzt for assistance with some experiments and Stefan Wurster for useful
8
discussions on the topic.
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REFERENCES 1) Ghabache, E., Antkowiak, A., Josserand, C. & Seon T. On the physics of fizziness: How bubble bursting controls droplets ejection. Phys. Fluids 2014, 26, 121701. 2) Knelman, F., Dombrowski, N. & Newitt, D.M. Mechanism of the bursting of bubbles. Nature 1954, 173, 261. 3) Woodcock, A.H., Kientzlen, C.F., Arons, A.B. & Blanchard, D.C. Giant condensation nuclei from bursting bubbles. Nature 1953, 172, 1144-1145. 4) Andreas, E.L., Edson, J.B., Monahan, A.P., Rouault, M.P. & Smith, S.D. The spray
19
contribution to net evaporation form the sea – A review of recent progress. Bound.- Layer
20
Meteor. 1995, 72, 3-52.
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5) Resch, F.J. Darrozes, J.S. & Afeti, G.M. Marine liquid aerosol production from bursting of air bubbles. J. Geophys. Res.: Oceans 1986, 91, 1019-1029. 6) Spiel, D.E. More on the births of jet drops from bubbles bursting on seawater surfaces. J. Geophys. Res.: Oceans 1997, 102, 5818-5821. 7) Spiel, D.E. On the births of jet drops from bubbles bursting on seawater surfaces. J. Geophys. Res.: Oceans 1995, 100, 4995-5006. 8) Walls, P.L.L, Henaux, L. & Bird, J.C. Jet drops from bursting bubbles: How gravity and
8
viscosity couple to inhibit droplet production. Phys. Rev. E 2015, 92, 021002.
9
9) Mullins, B.J., Mead-Hunter, R., Pitta, R., Kasper, G. & Heikamp, W. Comparative
10 11
performance of philic and phobic oil-mist filters. AIChE J. 2014, 60, 2976-2984. 10) Hotz, C., Mead-Hunter, R., Becker, T., King, A.J.C., Wurster, S., Kasper, G. & Mullins,
12
B.J. Detachment of droplets from cylinders in flow using an experimental analogue. J.
13
Fluid Mech. 2015, 771, 327-340.
14
11) Wurster, S., Meyer, J., Kolb, H.E. & Kasper, G. Bubbling vs. blow-off – On the relevant
15
mechanism(s) of drop entrainment from oil mist filter media. Sep. Purif. Technol. 2015,
16
152, 70-79.
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12) Tomiyama, A., Zun, I., Sou, A. & Sakaguchi, T. Numerical analysis of bubble motion with the VOF method. Nucl. Eng. Des. 1993, 141, 69-82. 13) Xu, Y.G., Ersson, M. & Jonsson, P. Numerical simulation of single argon bubble rising in molten metal under a laminar flow. Steel Res. Int. 2015, 86, 1289-1297.
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14) Mead-Hunter, R., King, A.J.C. & Mullins, B.J. Plateau-Rayleigh instability simulation. Langmuir 2012, 28, 6731-6735. 15) Boulton-Stone, J.M. & Blake, J.R. Gas bubbles bursting at a free surface. J. Fluid Mech. 1993, 245, 437-466. 16) Georgescu, S.C., Achard, J.L. & Canot, E. Jet drops ejection in bursting gas bubble process. Euro. J. Mech. B: Fluids 2002, 21, 265-280. 17) Hayami, S. & Toba, Y. Drop production by bursting of ari bubbles on the sea surface (I). J. Oceanogr. Soc. Jpn. 1958, 14, 145-150. 18) Toba, Y. Drop production by bursting of air bubble on the sea surface (II). J. Oceanogr. Soc. Jpn. 1959, 15, 121-130. 19) Gañán-Calvo, A.M. Revision of bubble bursting: universal scaling laws of top jet drop size and speed. Phys. Rev. Lett. 2017, 119, 204502.
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1
Supplementary Information
2
If we consider the variables that govern the system, we have;
3
ρ
4
g
m
5
r
m
6
µ
kg
7
γ
kg
8
Bucking ham Pi yields 5 parameters -3 units = 2 variables in a valid non-dimensional system.
9
Conventional set
10
kg
m-3
m-1
s-2
s-1 s-2
Selecting σ, ρ, r as repeating variables; Π = $ % & ' (
11
Linear system (using primary variables) )(: + + - = 0 .: − 3- + 1 + 2 = 0 3: − 2+ − 22 = 0
12
Solution gives Π =
& = -
Π = $ % & ' 4 13
Linear system (using primary variables) )(: + + - + 5 = 0 .: −3- + 1 − 5 = 0 3: −2+ − 5 = 0
14
Solution gives
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Π =
6&
Page 26 of 28
= !ℎ
1
Our chosen set
2
If we choose different repeating variables, we get an alternate system. Π = $ (% & '
3
Linear system (using primary variables) )(: + + 2 = 0 .: - + 1 − 2 = 0 3: − 2+ − 2- − 2 = 0
4
Solution gives Π = (/ &/ =
5
From our manuscript = Π =
6
6&(
&(
For the second system we have Π = $ (% & ' 4
7 8
Linear system (using primary variables) )(: + + 5 = 0 .: - + 1 − 35 = 0 3: − 2+ − 2- = 0
9
Solution gives (& Π = =
10
This would suggest that the system can be described by Bo, Oh and Dn.
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Langmuir
1
Further iterations
2
Changing the repeating variables again, yields two further solutions 8 ( 9 =
3
1 :
And 1 = (&
4
For completeness
5 6
There are only 5 possible non-dimensional number form this set. The four considered so far are Bo, Oh, Mo and Dn. The final is (& 8
7 8 9
Which appears to be a dimensionless radius. We have not considered the tip radius, which would likely bring the Weber number into play. It would appear that Bo, Oh and Dn are sufficient to describe the system.
10 11
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