On the jets produced by drops impacting a deep liquid pool and by bursting bubbles

J. Fluid Mech. (2021), vol.916, A37, doi:10.1017/jfm.2021.207 On the jets produced by drops impacting a deep liquid pool and by bursting bubbles Francisco J. Blanco–Rodríguez1and J. M. Gordillo1,† 1Área de Mecánica de Fluidos, Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n 41092, Sevilla, Spain (Received 12 September 2020; revised 20 January 2021; accepted 3 March 2021) Here we provide a unified theoretical description of two different physical situations in which liquid jets are expelled out of the bulk of a liquid as a consequence of the capillary collapse of a void. We demonstrate that the velocity field giving rise to the emergence of these jets can be calculated as the flow generated by a line of sinks with a length and an intensity that can be expressed in terms of the initial cavity radius and the wavelength and velocity of the capillary waves propagating along the cavity walls. The predicted jet speeds, which are expressed through algebraic equations, are in good quantitative agreement with those obtained from experiments and from the simulations of bubbles bursting on a free surface or after the implosion of the crater formed when a drop impacts a liquid pool. Key words: bubble dynamics, drops 1. Introduction The iconic and quite familiar image in figure 1 of a drop falling on a liquid pool and producing a vertical jet from which a droplet is emitted upwards synthesises in a visual and straightforward way the beauty and complexity of liquid flows, a fact which could explain its widespread use in artistic photography or advertising campaigns to evoke freshness, stimulating flavours, and so on (Michon, Josserand & Séon 2017). It will become clear in what follows that the liquid jets produced in this way originate in a similar manner to those emitted after the bursting of a bubble, a process that has received much attention in the recent literature (Duchemin et al. 2002; Ghabache et al. 2014; Gañán Calvo 2017; Brasz et al. 2018;Deikeet al. 2018; Gordillo & Rodríguez-Rodríguez 2018; Lai, Eggers & Deike 2018; Gordillo & Rodríguez-Rodríguez 2019;Bernyet al. 2020) because it plays a †Email address for correspondence: jgor[email protected] © The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. 916 A37-1 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo (a)(b)(c)(d)(e) Figure 1. Sequence of images showing the impact of a water drop of radius Rdfalling on a deep liquid pool with a velocity Vsuch that We =ρV2Rd/σ ≃90, Fr =V2/(gRd)=565, with g,ρand σbeing the gravitational acceleration, the density and the interfacial tension coefficient, respectively, at different instants of time T,withT=0 the instant the drop touches the surface: (a)t=TV/Rd≃−2.3(b)TV/Rd≃41, (c)TV/Rd≃46, (d)TV/Rd≃57, (e)TV/Rd≃68.7. The value of We has been calculated using the material properties of water. In this experiment, the thin jet initially observed in (c)breaks into tiny droplets and widens with time. key role in the production of the sea spray aerosol (MacIntyre 1972;Bigg&Leck2008; Veron 2015;Wanget al. 2017; Blanco-Rodríguez & Gordillo 2020) and in the dispersion of contaminants and bacteria (Walls, Bird & Bourouiba 2014). Recently, it has been also pointed out that the drops emitted from the tip of the jets ejected by the collapse of bubbles might be used in technological applications related with the design of novel printing devices (Castrejón-Pita, Castrejón-Pita & Martin 2012; Basaran, Gao & Bhat 2013; Ismail et al. 2018). Indeed, the high-speed jets formed after the bursting of bubbles (MacIntyre 1972; Duchemin et al. 2002; Ghabache et al. 2014) or after a drop impacts a free surface (Prosperetti, Crum & Pumphrey 1989; Prosperetti & Oguz 1993;Rein1996; Ray, Biswas & Sharma 2015; Michon et al. 2017; Thoroddsen et al. 2018; Yang, Tian & Thoroddsen 2020) share acommon feature since they both emerge as a consequence of the axial convergence of the capillary waves that propagate along the collapsing cavity walls. Moreover, the largest jet velocities measured in each of these physical situations are quite similar:∼50 m s−1in the case of the collapse of Faraday waves (Zeff et al. 2000) or when a drop impacts a deep liquid pool (Thoroddsen et al. 2018;Yanget al. 2020), whereas in the case of bubble bursting jets, Blanco-Rodríguez & Gordillo (2020) have reported maximum velocities of ≃σ/μ,withσand μrespectively indicating the interfacial tension coefficient and the liquid viscosity,which, in the case of water properties, imply maximum jet speeds of ∼70 m s−1. It is the main purpose of this contribution to provide conclusive evidence showing that the velocities of the high-speed, thin jets produced following the bursting of a bubble or after the impact of a drop on a deep liquid pool can be quantified using a common theoretical framework, which has already been put forward in Gordillo & Rodríguez-Rodríguez (2019), where the flow field is represented as the one produced by a line of sinks. Moreover, the flow rate per unit length and also the length of the line of sinks will be expressed as a function of the initial radius of the crater from which the jet emerges 916 A37-2 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities g g z z r rc r rc≡1 2rd≡2 t=0 t= 0.15 h= 0.04 t= 0.30 t= 0.40 t=0 t=2 t=10 t=20 t≈tjet 2rjet 2rjet vjet v≡1 vjet t≈tjet ρ, μ ρ, μ ρ, μ λ∗λ∗λ∗ λ∗ (a) (b) Figure 2. (a) Sequence of events following the bursting of a bubble at a free interface for Oh =0.012. The retraction of the rim causes capillary waves of wavelength λ∗∝Oh1/2(Gordillo & Rodríguez-Rodríguez 2019) that propagate along the cavity walls and which, when reaching the base of the void, trigger the formation of a fast jet of an initial velocity Vjet and an initial radius Rjet. The impact of a drop shown in (b), with Fr =600, We =90 and Mo =Mow, reveals a similar jet ejection process to that depicted in (a), with the main difference being that the dimensionless radius of the cavity is rc≃0.5Fr1/4(Prosperetti & Oguz 1993;Jainet al. 2019); see (2.4). Figure (b) also shows that, in contrast with the case of bubble bursting jets, λ∗≈1. and of the wavelength and velocity of the capillary waves travelling along the cavity walls. The integrals expressing the vertical velocity field can be solved analytically, providing algebraic expressions for the jet velocities as a function of the control parameters of each of the two physical situations at hand. The theoretical velocity fields, as well as the initial jet velocities, will be shown to be in quantitative agreement with both the numerical results and experimental measurements. The paper is structured as follows: in § 2the ejections of the jets produced after the bursting of a bubble or after the impact of a drop on a liquid pool are simulated numerically. The velocity fields computed in §2are compared with the theoretical predictions in §§ 3and 4for the cases of bubble bursting jets and of drops impacting a deep liquid pool, respectively. The main results are summarised in §5. 2. Numerical simulations The numerical results in figure 2, which illustrate the generation and propagation of the capillary waves that give rise to the emergence of the jets produced by the bursting of a bubble or by the impact of a drop falling on a deep pool, have been obtained, as well as the rest of numerical results shown in this contribution, using the open-source package GERRIS (Popinet 2003,2009) assuming that the surrounding gaseous atmosphere is air with a density and a dynamic viscosity of 1.2kgm −3and 1.8×10−5Pa ·s, respectively. From now on, dimensionless variables will be written using lower-case letters to differentiate them from their dimensional counterparts, written in capitals, and ρ,μand σwill denote the liquid density, viscosity and interfacial tension coefficient, respectively. Moreover, the acronyms BB and DP will be used in what follows to indicate variables or results that correspond either to the bursting of a bubble or to the impact of a drop on a liquid pool. Furthermore, the numerical results that correspond to the bursting of a bubble with a radius Rb=(3Vb/(4π))1/3,withVbindicating the bubble volume, will be presented in 916 A37-3 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo terms of dimensionless variables defined using Rb, the capillary velocity √σ/(ρRb)and the capillary pressure σ/Rbas the characteristic values of length, velocity and pressure, respectively – see figure 2(a). This physical situation is characterised by two dimensionless parameters; namely, the Bond (Bo) and Ohnesorge (Oh) numbers – or, equivalently, the Bond and Laplace (La) numbers, which are defined as Bo =ρgR2 b σ,Oh =μ √ρRbσwith La =ρRbσ μ2=Oh−2.(2.1a,b) The Ohnesorge number varies within a range of values 0.006 ≤Oh ≤0.032 and, except in the Appendix A, the value of the Bond number is kept constant and equal to Bo =0.01. Figure 2(a) shows that the unperturbed air-liquid interface located far from the bubble is flat, and this horizontal boundary has been used to divide the computational domain into two identical cylinders with a height and radius of 5 dimensionless units. The present numerical simulations reproduce those which have already been published in Brasz et al. (2018), Blanco-Rodríguez & Gordillo (2020) and have been carried out by imposing symmetry conditions at the axis and zero flux at the rest of the boundaries. The dynamically adaptive numerical grid has been refined up to a level of 14, which means that each grid cell can be subdivided up to 14 times in order to appropriately describe the flow at those regions with high gradients or large values of the interfacial curvature. From now on, the results of the simulations that correspond to the impact of a drop with aradiusRdfalling over a pool of the same type of liquid with a velocity Vwill be expressed in terms of the following dimensionless variables, defined using Rd,Vand ρV2 as characteristic values of length, velocity and pressure Fr =V2 gRd ,We =ρV2Rd σ,Mo =gμ4 ρσ3,Mow=25.74 ×10−12,(2.2a–d) with gdenoting the acceleration of gravity, Fr the Foude number, We the Weber number, Mowthe value of the Morton number corresponding to the physical properties of water and Mo the Morton number, which is related with the Ohnesorge number based on Rd, Ohd=μ/√ρσRd,as: Ohd=(MoFr/We)1/4.(2.3) The leftmost panel in figure 2(b) shows that the drop is initially placed at a distance of 0.04 dimensionless units above the gas-liquid interface. This flat boundary is used to divide the numerical domain into two identical cylinders with a height and a radius of N dimensionless units, with N=27 for Mo =Mow. The numerical calculations have been performed by imposing a free outflow boundary condition at the top part of the numerical region and free-slip and impermeable boundary conditions at the axis of symmetry and at the lateral and the bottom surfaces of the domain. The maximum grid refinement level varies between 12 and 14,and it was verified that the results obtained for Fr =600 replicatethoseinRayet al. (2015). Unless otherwise specified, the value of the Morton number in the simulations presented here correspond to the value of the Morton number Mo =Mowgiven in (2.2a–d). The values of Fr and We explored here are indicated in figure 3, where the two solid lines with equations provided in Prosperetti & Oguz (1993) delimit the bubble entrapment region described in Pumphrey, Crum & Bjorno (1989). In view of (2.3), the fact that Mo =Mowdoes not imply a constant value of Ohd. Figure 2 shows that both the BB and DP jets emerge after the capillary waves excited by the rim retraction process reach the base of the cavity with an initial radius rc. 916 A37-4 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 400 100 20 102103 50 100 150200 500 100 600 700 We Fr Figure 3. The squares and circles in the figure indicate the values of the Froude and Weber numbers used in the simulations for the case of DP jets when Mo =Mowsee (2.2a–d). The area limited by the curves We =18.24Fr0.179 and We =20.35Fr0.247 given in Prosperetti & Oguz (1993) indicates the bubble entrapment region. Whereas rc=1 for BB jets, the impact of a drop on a liquid pool produces a crater with a dimensionless radius given by (Prosperetti & Oguz 1993;Jainet al. 2019) DP : rc≃0.5Fr1/4.(2.4) In both the BB and DP cases, capillary waves with a characteristic wavelength λ∗ propagate with a dimensionless velocity vλ;seefigure 2.Figures 4 and 5illustrate the way the values of vλhave been determined numerically: the shapes of the cavities are calculated at different instants of time separated at regular intervals Δtat which the increments in the angular positions of the maximum elevation of the capillary waves, Δθ, are also determined with the purpose of calculating vλas vλ=rc Δθ Δt.(2.5) The results obtained from the analysis, depicted in figure 6, reveal that, for the case of BB jets, vλ≃5, a value which is independent of Oh and which coincides with the one reported in Gordillo & Rodríguez-Rodríguez (2019). For the case of DP jets, figure 6 represents the velocity of the waves Vλ=Vvλ,withvλgiven in (2.5), divided by the capillary velocity based on the radius of the cavity, σ/(ρRd0.5Fr1/4). The result in this figure reveals, also in this case, that the waves propagate with a velocity which is proportional to the capillary velocity based on the radius of the cavity. The proportionality constant, however, varies slightly with Fr and We, and it is somewhat smaller than in the case of the BB jets. In the remainder of this contribution, the variations with Fr and We depicted in figure 6 will be neglected, and the speed of the capillary waves will be approximated as 3.5σ/(ρRd0.5Fr1/4); namely, 3.5 times the capillary velocity based on the radius of the deformed cavity. Moreover, it can be seen from figure 4 that the maximum radii of curvature of the capillary waves excited for the case of BB jets, which are a proxy to the wavelength λ∗, 916 A37-5 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 0.5 –0.5 –0.5 –1.0 –1.5 –2.0 –1.0 –0.5 0.5 1.00 0.5 0 –0.5 –1.0 –1.5 –2.0 –1.0 –0.5 0.5 1.00 0.5 0 –1.0 –1.5 –2.0 –1.0 –0.5 0 0.5 1.0 0 0.5 –0.5 –1.0 –1.5 –2.0 –1.0 –0.5 0 0.5 1.0 0 z–zfs z–zfs rr (a)(b) (c)(d) Figure 4. Time evolution of the capillary waves excited after the bursting of a bubble for Bo =0.01 and Oh =0.006 (a), Oh =0.012 (b), Oh =0.020 (c)andOh =0.032 (d). The shape of the bubble at t=0.100 is represented in all figures in black. Notice that the radii of curvature of the travelling capillary waves, which is a proxy for the wavelength, λ∗, increase with Oh. Here, zfs indicates the vertical position of the flat free interface. increase with Oh. This result was already reported in Gordillo & Rodríguez-Rodríguez (2019), where, in addition, it was predicted and later confirmed from an analysis of the numerical results that λ∗∝Oh1/2. In contrast, in the case of DP jets, the wavelength of the capillary waves is somewhat similar to the initial radius of the drop, which can be noted in figure 5. Therefore, from now on, λ∗≈1 for the case of DP jets. Notice that the reason for the different scaling for λ∗is due to the fact that, in the case of DP jets, the initial thickness of the retracting rim that induces the generation of the capillary waves that travel along the cavity walls is not negligible, which is in contrast tothecaseofBBjets (Gordillo & Rodríguez-Rodríguez 2019). Instead, this rim possesses an initial thickness which is similar to the wavelength of the wave that, after the impact, displaces the initially flat interface upwards. This fundamental difference from the case of BB jets is the reason why, for the case of DP jets, λ∗does not scale with the Ohnesorge number based on rc=0.5Fr1/4. This result is further confirmed in figure 7, where it is shown that λ∗and the wave propagation velocity are insensitive to changes of Mo. Figure 8 shows a key result for our subsequent purposes: the radial velocities at the base of the deformed cylindrical cavity at the scale ∼rcare quite similar to the values of the 916 A37-6 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 2 (a)(b) (c)(d) (e)(f) rd rc 1 0 –1 –2 z – zfs –3 –4 240 rr –2–4 240–2–4 240–2–4 240–2–4 240 –2–4 240 –2–4 –5 2 1 0 –1 –2 –3 –4 –5 2 1 0 –1 –2 z – zfs –3 –4 –5 2 1 0 –1 –2 –3 –4 –5 2 1 0 –1 –2 z – zfs –3 –4 –5 2 1 0 –1 –2 –3 –4 –5 Figure 5. Time evolution of the capillary waves excited after a drop impacts a deep liquid pool for Mo =Mow. Top row: Fr =300 and We =60 (a), We =75 (b). Middle row: Fr =600 and We =75 (c), We =90 (d). Bottom row: Fr =3000 and We =90 (e), We =120 ( f). The cavity shape at t=10 is represented in all figures in black. The radii of the circles in dashed lines are rc=0.5Fr1/4. Notice that the radii of curvature of the travelling capillary waves do not noticeably change with Fr or We and are similar to those of the impacting drop; namely, λ∗≈1. Here, zfs indicates the vertical position of the flat free interface. 916 A37-7 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 0.01 0.05 Oh 3.0 3.5 4.0 4.5 5.0 vλ 50 100 150 2.5 3.0 3.5 4.0 4.5 5.0 vλ√0.5 Fr1/4 We Fr = 300 Fr = 600 Fr = 3000 We (a)(b) Figure 6. Propagation velocity of the capillary waves for the cases of (a)BBjetsand(b)DPjets.Thevalues represented have been calculated using (2.5), with Δθobtained from the analysis of figures 4 and 5.Theresult in figure (a)is the same as the result already reported in Gordillo & Rodríguez-Rodríguez (2019), where it was found that the velocity of capillary waves is independent of Oh and is equal to five times the capillary velocity based on Rb. This result is also valid for arbitrary values of Bo;seetheAppendix A. 2 1 0 –1 –2 –3 –4 –5 z – zfs –4 –2 024 r Figure 7. Comparison between the time evolutions of the capillary waves excited after a drop impacts a deep liquid pool at the same values of tfor Fr =600, We =114 and three different values of Mo:Mo1=1.75Mow (dashed line), Mo2=16Mo1(dashed-dotted line) and Mo3=81Mo1(solid line), which implies a threefold variation of Ohd=(MoFr/We)1/4. In contrast to the BB case depicted in figure 4, the wavelength of the wave travelling along the cavity walls does not depend on Mo and, thus, it does not depend on Ohd;see(2.3); the velocity of the capillary waves does not depend on Ohdeither. Here, Mowindicates the value of the Morton number corresponding to the physical properties of water given in (2.2a–d). capillary wave velocities depicted in figure 6, and this result applies to the arbitrary values of Oh for the BB case and of Fr,We and Mo for DP jets. The main consequence of the fact that the capillary waves propagate at velocities which are clearly larger than the capillary velocity based on the crater radius for both BB and DP jets is that the values of the Weber number based on the wave velocity and on the unperturbed radius of the cavity are ≈25 for 916 A37-8 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities t = 0.424 (a)(b) t = 33.5 −1.5 −3.0 0 −1 −2 −3 −4−5 Vr/Vc ≈ 5 Vr/Vc ≈ 3.7 Figure 8. The radial velocity Vrat the base of the deformed cavity is very similar to the values of the wave velocities depicted in figure 6; namely, Vr≃Vλ.Inthisfigure, Vc=√σ/(ρRc),withRc=Rbfor BB jets or Rc=0.5RdFr1/4for DP jets. Since the Weber number based on the wave velocity is such that (Vλ/Vc)2≃ (Vr/Vc)21, the radial collapse of the void that gives rise to the ejection of the jet is driven by inertia, and not by capillarity. Figure (a) shows a BB case with Oh =0.032 and (b)showsaDPcasewithFr =600, We =90 and Mo =Mow. the case of BB jets and arbitrary values of Oh and 10forthecaseofDPjets;see figure 8. Therefore, in both cases, the dynamic pressure associated with the velocities induced by the propagation of the capillary waves is an order of magnitude larger than the capillary pressure, a fact that suggests that the implosion of the base of the cavity and the subsequent ejection of the jet could be described as a purely inertial process in which capillarity plays a subdominant role. Section 4will be devoted to exploring this possibility; however, before this is done, the computed velocity fields for the case of BB jets will be compared with the ones predicted by the theory in Gordillo & Rodríguez-Rodríguez (2019). 3. Comparison of the numerical results corresponding to the case of bubble bursting jets with theoretical predictions assuming an inertio-capillary balance Figure 9 illustrates that the velocity field can be divided into three well-defined spatio-temporal regions: (i) the bulk, with a velocity field v(r,z,t); (ii) the jet region, which extends along the spatio-temporal region zmin(t)≤z≤s(t); and (iii) the drop, located at z=s(t). In the limit of low-viscosity liquids of interest here, Oh 1and Ohd1, see (2.1a,b)and(2.3), vorticity is confined within thin regions located very close to the interface and, therefore, the bulk velocity field is irrotational.The velocity potential φ,withv(r,z,t)=∇φ, satisfies the Laplace equation ∇2φ=0. The bulk region ends at z=zmin(t); namely, just at the beginning of the jet region, where the equations governing the flow can be notably simplified, as reported recently in Blanco-Rodríguez & Gordillo (2020). Indeed, due to the fact that the jet geometry is slender and the dynamic pressure is much larger than the capillary pressure since, otherwise, a jet would not be formed (Gordillo & Rodríguez-Rodríguez 2019), pressure gradients within the jet can be neglected and, therefore, the vertical momentum equation for u(z,t)reduces to Du/Dt=0, with D/Dtindicating the material derivative. This momentum equation indicates that the flow within the jet is ballistic;namely, that fluid particles conserve the 916 A37-9 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 2.5 1.0 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.9 0.8 0.7 z 0.6 0.5 2.0 1.5 1.0 0.5 0 –0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin 2.5 2.0 1.5 1.0 0.5 0 –0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin 2.5 2.0 1.5 1.0 0.5 0 –0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin –1 0 r/rmin 1–10 r/rmin 1–10 r/rmin 1 –1 0 r/rmin 1–10 r/rmin 1–10 r/rmin 1 –1 0 r/rmin 1–10 r/rmin 1–10 r/rmin 1 |v| = 35 |v| = 45 |v| = 120 |v| = 30 |v| = 40 |v| = 90 |v| = 25 |v| = 40 |v| = 60 –0.2 0.2–0.1 0.10 r–0.15 0.15–0.05 0.050 r–0.15 0.15–0.05 0.050 r (a) (d) (g) (j) (b) (e) (h) (c) (f) (i) (k)(l) Figure 13. Comparison between the numerical and the analytical velocity fields calculated from (3.6)and (3.7)forα=0.6, =1.24Oh1/2,β=π/4andK=11.3 for the case of BB jets at the different instants of time and different values of Oh corresponding to the bubble shapes depicted in the top row. The red arrows indicate the numerical results, whereas the blue ones correspond to those computed using (3.6)and(3.7). Time advances from top to bottom, the results are grouped in columns and, from left to right, correspond to the following values of the Ohnesorge number: Oh =La−1/2=4444−1/2,Oh =La−1/2=2500−1/2=0.02 and Oh =La−1/2=1000−1/2. The last column corresponds to conditions for which a bubble is entrapped and the fastest jets with the smaller drops are produced (Brasz et al. 2018; Blanco-Rodríguez & Gordillo 2020). The blue line at the axis indicates where the line of sinks is located. 916 A37-16 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 0.001 0.01 0.02 Oh 100 101 vjet, vtip, vd 0.001 0.01 0.02 Oh 10–2 10–1 rjet Numerical results Figure 14. This figure compares the droplet velocities vdfor Bo ≤0.05 reported in Ghabache et al. (2014) (green open squares), in Krishnan, Hopfinger & Puthenveettil (2017) (purple open triangles), the droplet velocities calculated in Blanco-Rodríguez & Gordillo (2020)forBo =0.01 (solid blue dots) and Bo =0.05 (solid red dots), the numerical results in figure 6 in Deike et al. (2018)forvtip,withvtip defined in figure 9 (blue and cyan open symbols, corresponding respectively to Bo =10−3and 10−2) and also the values for the velocities vjet calculated numerically at r=0, z=zjet =rjet/tanβwhen rmin(t)=rjet (solid black squares) with the values of vjet (blue continuous line) predicted by (3.10)with=1.24Oh1/2,α=0.6, K=11.3, β=π/4andrjet given by (3.9), represented in the inset. Notice that the values of vdare slightly smaller than vjet as a consequence of the capillary forces pulling the tip of the jet back (Blanco-Rodríguez & Gordillo 2020). the velocities of the drops emitted, vd.As explained above, this is a consequence of the capillary deceleration experienced by the tip of the jet, an effect already quantified in Blanco-Rodríguez & Gordillo (2020), where a detailed description of the drop formation process is also provided. However, as explained at the beginning of this section, the values of vjet and vtip are very similar to each other within this range of values of Oh because material points move ballistically along the jet and also because of the relatively small deceleration caused by the capillary forces acting on the jet tip. Equation (3.10) reveals that the origin of the large velocities of the jets produced by bursting bubbles is caused by the combination of two effects: as expressed by (3.9), rjet decreases with Oh1/2and, therefore, the term r−1/2 jet increases with Oh. This means that one of the reasons for the large velocities of the jets produced after the bursting of bubbles is that the capillary waves that propagate along the walls of the crater transform the initially rounded cavity into a truncated cone with a radius at its base that is substantially smaller than that of the original bubble. In addition, vjet also increases with Oh because the length of the line of sinks increases as ∝Oh1/2. Therefore, the increase in vjet with Oh in the region where no bubbles are entrapped, Oh ≤0.02, is caused by the combination of two 916 A37-17 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 0.9 1.25rmin Welocal 45 50 40 35 30 25 20 15 10 5 0.8 0.7 0.6 0.5 0.4 0.3 –0.2 –0.1 –0.03 –0.02 –0.01 0 0.01 0.02 0.030 1.25rmin Welocal 45 50 40 35 30 25 20 15 10 5 –4 –2 0 2 4 t – tjet z 0.1 0.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 –0.2 –0.1 0 r z 0.1 0.2 (×10–3) (b)(a) (c)(d) Figure 15. The local Weber number is only approximately constant in time for the case of BB jets. Top and bottom figures correspond, respectively, to Oh =0.01 and Oh =0.02. effects: rjet decreases with Oh1/2, and also because the wavelength of the capillary waves increases with Oh as λ∗∝Oh1/2. The findings in this section support the results in Gordillo & Rodríguez-Rodríguez (2019), where the flow field is calculated by being induced by a line of sinks with a length prescribed by the wavelength of the capillary waves, and their flow rate per unit length is fixed by the same inertio-capillary balance as the one which leads to the self-similar solution in Lai et al. (2018), which is proven to be an excellent approximation. Indeed, the self-similar results in Lai et al. (2018) are approximate but not exact because if they were, the value of the local Weber number would not vary in time, whereas figures 11 and 15 show that the value of the local Weber number does experience slight variations in time. This fact is the motivation for the results presented in next section, where whether or not the ejection of BB and DP jets can be described using the theoretical framework in Gordillo & Rodríguez-Rodríguez (2019) without resorting to the inertio-capillary balance implicitly assumed in (3.3)is explored. 4. Comparison of the numerical results with predictions relaxing the assumption of an inertio-capillary balance While the agreement between the predictions and the numerical results in § 3is fairly good when the intensities of the sinks are calculated using (3.3), this section is devoted to 916 A37-18 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities h(z, t) = r0(t) + z2r1(t) 2r0(t) q(t) = r0r 0 z b ∼ (r0/r0r1) Figure 16. Sketch showing that the local geometry of a collapsing bubble can be described using a parabola of the type h(z,t)=r0(t)+r1(t)z2. The blue line at the axis indicates the location of the line of sinks. exploring how the predictions of the velocity field are modified when the assumption of an inertio-capillary balance is relaxed. This discussion begins by noticing that the implosion of a void that is located near a free interface of a low-viscosity liquid such as water shares many similarities with the pinch-off of bubbles which, unlike the case of the pinch-off of drops in air (Eggers 1993;Day, Hinch & Lister 1998; Burton, Rutledge & Taborek 2004; Castrejón-Pita et al. 2015), is not universal. Indeed, it is well known that the final instants of the collapse of an axisymmetric bubble are not universal because they are strongly dependent on the liquid viscosity (Doshi et al. 2003; Suryo, Doshi & Basaran 2004; Burton, Waldrep & Taborek 2005; Thoroddsen, Etoh & Takehara 2007), on the gas flow rate through the neck (Gordillo et al. 2005) and, most importantly, also on the initial geometry of the cavity, which is characterised, as it is shown in figure 16, by two different length scales, r0and r−1 1(Bergmann et al. 2006; Gordillo & Pérez-Saborid 2006). The latter and main difference from the analogous case of the pinch-off of drops in air, together with the slow logarithmic convergence of the system of equations that describe the collapse of a cavity (Gordillo & Pérez-Saborid 2006), prevent the purely inertial asymptotic limit being reached. This asymptotic regime is described in Eggers et al. (2007), Gordillo & Fontelos (2007) but it is never observed in practice (Bolaños Jiménez et al. 2008; Gordillo 2008). Therefore, the dynamics governing the final instants of bubble pinch-off do not converge in a real experiment to a universal solution, as it is the case for drops (Eggers 1993;Dayet al. 1998;Burtonet al. 2004; Castrejón-Pita et al. 2015),instead needing to be described as solving the system of two ordinary differential equations deduced in Gordillo & Pérez-Saborid (2006), Gordillo (2008) for the two lengths, r0(t)and r−1 1(t), which characterise the time-evolving local shape of the interface h(z,t)=r0(t)+z2r1(t);see figure 16. These two equations are deduced by making use of the Euler-Bernoulli equation for a time-dependent velocity field generated by a line of sinks of length b(t)>r0(t)with b≃r0/√r0r1(Bergmann et al. 2006) which, when written in dimensionless variables using, as characteristic values of length, pressure and velocity, an outer length scale a, the capillary pressure σ/aand the 916 A37-19 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo capillary velocity √σ/(ρa)read (Gordillo & Pérez-Saborid 2006; Gordillo 2008): ln(r0r1)dln(r0˙r0) ds−1+γΛ ˆp r0r1+4Oh r0˙r0+2r0(1−2r0r1) (r0˙r0)2=0 (ln(r0r1)+2Oh r0˙r0)dln(r0r1) ds−1+Λ 2 p r0r1+4Oh r0˙r0+r0 (r0˙r0)2=0 dt ds=−e−2s r0˙r0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .(4.1) The solutions for (4.1), the latter of which is deduced from dr0/dt=˙r0=r0˙r0/r0,with γan order unity constant and s=−ln r0, depend on: (i) the initial conditions; (ii) on the Ohnesorge number Oh =μ/√ρaσ; (iii) the gas to liquid density ratio Λ; and (iv) on the Reynolds number based on the gas material properties through the functions pand ˆpthat express gas overpressure. The system of equations (4.1) accurately describes the experimental observations, using liquids with different viscosities, bubbles of different diameters and different initial conditions; see Bolaños Jiménez et al. (2008) and Bolaños Jiménez et al. (2009). It was also shown in Gordillo (2008)that(4.1)reproduced the experimental observations reported in Thoroddsen et al. (2007) regarding the time evolution of r0(t)and regarding the formation of tiny satellite bubbles,and also recovered the Stokes limit deduced in Doshi et al. (2003), Suryo et al. (2004), as demonstrated by the experiments in Bolaños Jiménez et al. (2009). The ability of (4.1) to reproduce experimental observations is a clear consequence of the correctness of the different approaches made in the modelling, which start by neglecting the vorticity produced at the gas-liquid interface and by expressing the irrotational velocity field as a line of sinks of dimensionless flow rate per unit length ∝r0˙r0placed at the axis of symmetry along a distance b∝r0/√r0r1;see figure 16. In fact, the first logarithmic term in (4.1) results from ln(r0/b)=1/2ln(r0r1)for b=r0/√r0r1>r0. The system solution (4.1) in the inviscid limit Oh 1predicts that r0∝ταcwith τthe time to pinch-off and 1/2<α c<2/3, a fact indicating that the collapse of bubbles is faster than in the analogous case of the pinch-off of drops, for which the inertio-capillary balance holds; i.e.,r0∝τ2/3(Day et al. 1998; Duchemin et al. 2002;Laiet al. 2018). In order to understand the reasons why the faster dynamics govern the pinch-off of bubbles,it should be noted that if the initial conditions governing the collapse were such that r1∝r−1 0and the capillary terms were not subdominant, b=r0/√r0r1∝r0and the first logarithmic term in (4.1), ln(r0/b), would be a constant. In this particular case, the axial and radial lengths characterising the local shape of the bubble around its minimum (see figure 16) would scale in the same way, and only the former of the equations in (4.1) would be required to describe the bubble collapse process. Moreover, a self-similar solution of the type r0∝τ2/3would be obtained, and the value of the local Weber number would remain constant in time. However, as pointed out above, the (4.1)system converges very slowly (logarithmically) to the asymptotic solution (Gordillo & Pérez-Saborid 2006); therefore, it retains a memory of the initial conditions, which, in the case at hand, are represented by the capillary waves propagating towards the base of the bubble. These waves will impose an initial value for bthat, in general, will be different from the value b∝r0required for the system to evolve in a self-similar manner and, moreover, capillary terms are subdominant as depicted in figure 8. These facts explain the findings in § 3in 916 A37-20 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities which the numerical solutions are not, strictly speaking, self-similar; indeed, as depicted in figures 11 and also in 15, the local Weber number is not constant in time. The reason why the bubble collapse proceeds faster than r0∝τ2/3can be understood in simple terms because the length of the sinks b(t)∝r0/√r0r1is larger than r0(t)and, also, the ratio b(t)/r0(t)increases in time, a fact meaning that the cavity shape becomes more slender as the cavity collapses (Gordillo et al. 2005). Clearly, the longer the line of sinks for a given r0is, the larger the velocities are and the faster the collapse process will be. From the point of view of the equations governing the collapse of cavities, the smaller values of the exponent αcin r0∝ταcare associated with the larger values of the time-dependent ratio b(t)/r0(t)>1, which grows in time for the case of bubbles, but remains constant in time for the self-similar case in which the inertio-capillary balance holds, r0∝τ2/3. The left column in figure 17 shows a comparison between the predicted and the numerical time evolution of the local shapes of the interface for the values Fr =600, We =95 within the bubble entrapment region depicted in figure 3. The system (4.1)is integrated in the limit Λ=0 using, as initial conditions, the values of r0(0),˙r0(0)and r1(0)taken from the simulations, with x(0)indicating the initial value of the generic variable x.It should be highlighted that the solution for the system (4.1) largely depends on the initial geometry of the collapsing bubble. Indeed, the solutions for (4.1) in the limits of negligible gas, viscous and capillary effects read (Gordillo 2008): q=r0˙r0=r0˙r0(0) r0r1(0)×e−√2s−2s(0)+[ln(r0r1(0))]2,r0r1=e−√2s−2s(0)+[ln(r0r1(0))]2, (4.2a,b) which, when expressed in terms of the time to pinch-off τ, yield to solutions of the type r0∝ταcwith 1/2<α c<2/3. Indeed, r0∝ταc→q∝r0˙r0∝τ2αc−1∝r2−1/αc 0→−ln q∝(2−1/αc)s→(2−1 αc )=dlnq−1 ds, (4.3) and, therefore: αc=1 21−1 2 dlnq−1 ds−1 ,(4.4) which, in the limit d ln q−1/ds1 reads: αc≃1 21+1 2 dlnq−1 ds.(4.5) Then, the substitution of the expression of qin (4.2a,b)into(4.5) yields: αc=1 2+1 42[s−s(0)]+[ln(r0r1(0))]2,(4.6) which indicates that the exponent αcdecreases with time and converges very slowly (logarithmically) to 1/2. Interestingly, (4.6) also reveals that this convergence depends on the initial conditions through the term representing the initial shape of the cavity, 916 A37-21 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 3.2 0.46 0.44 0.42 0.40 0.38 0.36 3.4 3.6 3.8 zfs – z r0 qq r0 4.0 4.2 4.4 0.35 0.020 0.016 0.012 0.008 0.004 0.30 0.25 0.20 0.15 0.10 0.14 0.50 0.45 0.40 0.35 0.30 0.25 0.13 0.12 0.11 0.10 0.09 0.08 0 0.1 0.2 0.3 0.4 0 0.10.20.30.4 012345 –0.6 –0.4 –0.2 –0.04 –0.02 0 0.02 0.040 0.2 0.4 0.6 rr z t – t0 012345 t – t0 (×10–4) (×10–4) (e) (b)(a) (c)(d) (f) Figure 17. Left-hand column: The top panel shows the comparison between the numerical local shapes of the bubble entrapped in the DP case corresponding to Fr =600, We =95, Mo =Mow(continuous lines) and the calculated shapes resulting from the integration of the system (4.1) in the limit Λ=0 (dashed black lines). Below, a comparison between the numerical values of r0(t)and of q(t)(squares) with the values calculated by integrating the equations in (4.1). Right-hand column: The top panel shows a comparison between the numerical local shapes of the bubble entrapped in the BB case corresponding to Oh =0.032 (colour lines) and the shapes predicted by (4.2a,b) (dashed black lines). Below, a comparison between the numerical values of r0(t)and q(t)(squares) with the values calculated using the equations in (4.2a,b). 916 A37-22 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities ln(r0r1(0)), this being one of the reasons why the different exponents appear in the literature to describe the pinch-off of bubbles (Gordillo 2008). Therefore, (4.6)predicts that αcdecreases when the slenderness of the initial shape of the cavity increases;i.e., when [ln(r0r1(0))]2increases or the entrapped bubble is more elongated. The collapse of the bubble entrapped in the case of BB jets for Oh =1000−1/2≃0.032 is depicted in figure 17, where the (4.2a,b)solution, which is valid for low-viscosity liquids when the value of the local Weber number is much larger than the unity, is superimposed into the numerical solution, finding an almost perfect match. Figure 17 also shows that, in agreement with (4.2a,b), the sink strength r0˙r0decreases in time as r0→0, a fact meaning that the local value of the Reynolds number, given by Oh−1r0˙r0≃30r0˙r0, also decreases in time. Therefore, viscous effects become relevant when the bubble splits in two and the jet is ejected;this is the reason why, as mentioned above, for 0.04 Oh 0.03 there exists a viscous cut-off and the minimum radius from which the jet emerges is different from zero.Thus, the existence of the viscous cut-off length avoids singular values of the jet velocity (Gordillo & Rodríguez-Rodríguez 2019). In summary, in general, two different length scales are required to describe the time evolution of the shape of a bubble that splits in two. Moreover, if the ratio of these two length scales is not kept constant along the collapse process, self-similarity is lost and solutions of the type r0(τ) ∝ταc,with 1/2<α c<2/3, will be found, with smaller values of αc;i.e., with larger values of the velocities for the larger values of the ratio b/r0=1/√r0r1,seefigure 16. Similar ideas to those expressed above are still applicable for the cases in which a cavity collapses without breaking into two parts, as is the case of the jets emerging from the bases of truncated cones considered in §3and sketched in figure 10. Indeed, two different length scales, rmin(t)and b,withrmin and bplaying the roles of r0and r0/√r0r1in (4.1), respectively, are also required to describe the collapse of these cavities. Therefore, since bis imposed by the capillary waves reaching the base of the cavity and, thus, is constant in time, the ratio b/rmin(t)will increase in time and the collapse will be faster than the self-similar collapse rmin ∝τ2/3, which would only take place if b(t)/rmin(t) remained constant in time, as explained above. Moreover, the first logarithmic term in (4.1) corresponds to ln(rmin/b)(Gordillo 2008); therefore, the equation describing the time evolution of the minimum radius of the cavity rmin(t), which plays the same role as r0(t)in the equations above, can be written as: ln rmin bdrmin˙rmin dt+1 2˙r2 min −1 rmin =0.(4.7) Defining s=−ln(rmin/b)and the intensity of the sinks as q=rmin˙rmin,(4.7)yields: −2sdlnq ds−1+2be−s q2=0⇒d dssq2 2+be−s=0 (4.8) and, therefore, the result found in Gordillo et al. (2005) is recovered: q=q(0) √−ln(rmin/b)s(0)+2rmin(0)−rmin q2(0)1/2 ≈q(0) √−ln(rmin/b),(4.9) where the initial conditions at s(0)≈1have been imposed, and it has also been taken into account that, as illustrated in figure 8,rmin(0)/q2(0)1. The expression for the flow rate in (4.9) has been calculated under the assumption that the collapsing cavity is a cylinder instead of a truncated cone and that the Euler-Bernoulli 916 A37-23 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo equation is particularised at the plane of symmetry of the cylinder. Therefore, the result in (4.9) is just an approximation whose validity will be checked a posteriori by comparing the predictions with the numerical results. The physical meaning of (4.9) is that the strength of the sinks driving the jet ejection process is not fixed by an inertio-capillary balance at the scale of rmin(t)as in § 3but by the flow rate q(0)imposed by the capillary waves travelling towards the pole of the bubble at the scale of the initial radius of the cavity;see figure 8. Then, since dQ(z0)=−2πqdz0,(4.10) the substitution of (4.10)into(3.5) yields: vz(z,)=−q 2zs+ zs (z−z0) [(z−z0)2+2]3/2dz0,(4.11) and vr(z,)=−q 2zs+ zs  [(z−z0)2+2]3/2dz0,(4.12) with qthe flow rate per unit length in (4.9), which is constant along the vertical direction. In order to express the flow rate per unit length qin (4.11)and(4.12) as a function of the control parameters, it must be taken into account that since capillary waves propagate in the case of BB jets at five times the capillary velocity at the scale of the radius of the cavity rc,q(0)≃vλrc≃5(see figure 6a). Therefore, assuming that the unknown length bin (4.9) is the initial radius of the cavity, b=rc=1, (4.9) reads: q=q(0) √−ln(rmin/b)=5 √−ln rmin .(4.13) Notice that the approximation for the flow rate per unit length in (4.13) is asymptotically valid in the limit rmin 1regardless of the exact value of the outer length b. This is because, for an arbitrary value of b∼O(1), which is a fixed length independent of time, |ln rmin||ln b|and, therefore, ln(rmin/b)=ln rmin −ln b≃ln rmin. It should be pointed out that (4.13) expresses that the flow rate of the sinks controlling the jet ejection process is fixed at the scale of the radius of the unperturbed bubble and also that it can be calculated as the product of the velocity of the capillary waves travelling along the cavity walls towards the base of the bubble times the radius of the bubble, with small logarithmic corrections. The integration limits in (4.11)and(4.12), zs=rmin/tanβ+αrmin and z=zs+, depend on α, which is fixed here to the value found in § 3,α=0.6, and also depend on the length of the sinks, =COh1/2,withCa multiplicative constant. The value C=2.8usedhereisdeterminedusing the condition that the vertical velocity predicted by (4.11) at the axis of symmetry, vz(r=0,z= rjet/tanβ),withrjet given in (3.9), coincides with the numerical value obtained for the smallest value of Oh explored here, Oh =6×10−3, at the instant of time the vertical velocity is maximum. The results depicted in figures 18 and 19 show remarkable agreement between the numerical results and the velocity fields calculated through (4.11)–(4.13) using α=0.6, =2.8Oh1/2and qgiven by (4.13) for all values of Oh and t. As explained at the beginning of §3, the maximum jet velocity is attained when rmin(t)=rjet with rjet givenin(3.9); therefore, vjet can be predicted solving (4.11) 916 A37-24 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 0.9 0.8 0.7 z rr 0.6 0.5 0.4 2.5 2.0 1.5 1.0 0.5 0 –0.5 0.9 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.8 0.7 0.6 0.5 0.4 –0.2 –0.30 –0.15 0 0.05 0.30–0.1 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 0 0.1 0.2 –0.2 –0.1 0 0.1 0.2 |v| = 15 |v| = 25 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 30 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 15 |v| = 20 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 25 (z – zmin)/rmin 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 15 |v| = 15 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 20 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin r/rmin r/rmin r/rmin r (a) (d) (g) (j) (b) (e) (h) (c) (f) (i) (k)(l) Figure 18. Comparison of the numerical and analytical velocity fields calculated using (4.11)and(4.12)for α=0.6, =2.8Oh1/2and qgivenby(4.13) for the case of BB jets at the different instants of time and different values of Oh corresponding to the bubble shapes depicted in the top row. The red arrows indicate the numerical results, whereas the green ones correspond to those computed using (4.11)and(4.12). Time advances from top to bottom, the results are grouped in columns and, from left to right, the results correspond to the following values of the Ohnesorge number: Oh =La−1/2=27777−1/2,Oh =La−1/2=10000−1/2=0.01 and Oh =La−1/2=7200−1/2. The green line at the axis indicates where the line of sinks is located. 916 A37-25 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 2.4 2.6 2.8 rr 3.2 3.0 3.4 3.6 2.5 2.0 1.5 1.0 0.5 0 –0.5 1.8 2.0 1.8 2.0 2.2 2.4 2.8 2.6 2.2 2.4 2.6 2.8 3.0 –0.6 –0.4 –0.6–0.4–0.2 0 0.2 0.4 0.6–0.2 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 0 0.2 0.4 0.6 –0.6 –0.4–0.2 0 0.2 0.4 0.6 |v| = 2 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 (z – zmin)/rmin 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin r/rmin r/rmin r/rmin r (a) (d) (g) (j) (b) (e) (h) (c) (f) (i) (k)(l) zfs – z Figure 23. Comparison between the numerical and analytical velocity fields calculated from (4.11)and(4.12) for Fr =300, Mo =Mow,α=0.6, =1andqgivenby(4.19) for the case of DB jets at the different instants of time and different values of We that correspond to the bubble shapes depicted in the top row. The red arrows indicate the numerical results, whereas the green ones correspond to those computed using (4.11)and(4.12). Time advances from top to bottom, the results are grouped in columns and, from left to right, the results correspond to the following values of the Weber number: We =55, We =70, We =85. The green line at the axis indicates where the line of sinks is located. 916 A37-32 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 2.8 3.0 3.2 rr 3.6 3.4 3.8 4.0 2.5 2.0 1.5 1.0 0.5 0 –0.5 2.4 2.6 2.8 3.0 3.2 3.4 3.8 4.0 3.6 2.8 3.0 3.2 3.4 3.6 –0.6 –0.4 –0.6–0.4–0.2 0 0.2 0.4 0.6–0.2 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 0 0.2 0.4 0.6 –0.6 –0.4–0.2 0 0.2 0.4 0.6 |v| = 2 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 4 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 4 (z – zmin)/rmin 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 2 |v| = 2 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 4 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin r/rmin r/rmin r/rmin r (a) (d) (g) (j) (b) (e) (h) (c) (f) (i) (k)(l) zfs – z Figure 24. Comparison between the numerical and the analytical velocity fields calculated from (4.11)and (4.12)forFr =600, Mo =Mow,α=0.6, =1andqgiven by (4.19) for the case of DB jets at the different instants of time and different values of We corresponding to the bubble shapes depicted in the top row. The red arrows indicates numerical results, whereas the green ones correspond to those computed using (4.11)and (4.12). Time advances from top to bottom, the results are grouped in columns and, from left to right, the results correspond to the following values of the Weber number: We =60, We =75, We =99. The green line at the axis indicates where the line of sinks is located. 916 A37-33 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 102 00 2 Rjet 101 100 100 200 2 We 300 400 Vjet (m s–1) 500 600 700 800 –2 2 Figure 25. Comparison between the experimental measurements in Thoroddsen et al. (2018) (in red) and Yang et al. (2020) (in blue) with the jet velocity calculated as Vjet =Vvjet,withvjet given by (4.20)andrjet measured from the experimental images in the figure. The experimental values of rjet are indicated in red in each of the insets and the black scale bar indicates 5 ×10−4m for the experiments in Thoroddsen et al. (2018)and 2×10−4m for the experiments in Yang et al. (2020). In the case of the experiments in Yang et al. (2020), the jet velocity is calculated using the material properties of water. The velocities of the jets produced when a bubble is entrapped in Thoroddsen et al. (2018) are taken as that of the downward jets in their figure 4 because the velocities of the upward jets, once they are seen above the free interface, are much smaller as a consequence of the jet tip deceleration caused by the capillary retraction and by the air drag force (Blanco-Rodríguez & Gordillo 2020). velocity, which is also pretty close to the measured vertical velocities before the jet emerges in figure 6 in Thoroddsen et al. (2018). Moreover, in their figures 5 and 6, Thoroddsen et al. (2018)provide the value of the exponent αc≃0.53 ±0.02 in rmin ∝ταcwhen the collapse of the cavity gives rise to the emergence of jets with velocities of ∼50 m s−1and a bubble is not entrapped. Under these conditions, rmin/rc=2rminFr−1/4≃4×10−3→ s=−ln(rmin/rc)≃5.5 during the final instants of time of the radial collapse before the jet emerges. In the present theoretical description, the flow rate per unit length of the sinks that induce the collapse of the cavity and the emergence of the jet given by (4.9)issuch that q−1∝√sand, therefore, (4.5) predicts that: αc=1 2+1 8s≃0.522,(4.21) a value which is quite close to the one measured experimentally. Notice that the values of the exponents predicted by (4.21) are smaller than those in (4.6)that correspond to the caseofjetsbeing ejected after a bubble has been entrapped. As pointed out above, the 916 A37-34 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities exponent αcdepends highly on the slenderness of the entrapped bubble as (4.6) clearly shows: the more elongated the entrapped bubble is, the smaller αcis. These conclusions are in agreement with the results shown in figures 5 and 6 in Thoroddsen et al. (2018). Clearly, since the length of the line of sinks bis constant in time for the case of the jets ejected when a bubble is not entrapped, the ratio b/rmin is larger than in the case where a bubble is entrapped, which explains why the value of the exponent (4.21) calculated using (4.5) is smaller than the value provided in (4.6) for the case where a bubble is entrapped. It should be pointed out that, in contrast with the case of BB jets, which could be equally well described using either the approximation described in this section or the one in §3, the velocities of the jets produced in the DP case can only be accurately predicted using the results in this section. Indeed, if the inertio-capillary balance were to be preserved during the cavity collapse process, the value of the exponent would be αc=2/3 and not slightly larger than 0.5 as seen in (4.21), a conclusion which had already been emphasised by Thoroddsen et al. (2018) and Yang et al. (2020). Therefore, if the results in § 3were applied to describe the jet ejection process in the DP case, the jet velocities predicted would be far smaller than those observed experimentally, aconclusion which can also be deduced by comparing (3.10)and(4.20): while in the former case vjet ∝r−1/2 jet ,the values of vjet predicted by (4.20), which are in good agreement with the experimental measurements in figure 25, diverge as vjet ∝(rjet−ln rjet)−1r−1/2 jet when rjet →0. Nonetheless, as discussed in the Appendix A, where the effect of the Bond number on the bursting of bubbles is also analysed, jet velocities do not diverge to infinity because rjet >0 and, hence, vjet is finite. In this regard, it can be noted that the base of the truncated cone from which the jet is issued shrinks in the axial direction faster than in the radial direction by a factor of 1/α =1/0.6>1;see(4.14)and(4.20). Therefore, the length of a cylinder with an aspect ratio ∼1 will shrink to zero before its radius is zero, thus preventing the appearance of infinite velocities.This is a purely irrotational mechanism that also avoids the existence of infinite jet velocities. The case of the much slower and much thicker jets appearing outside the bubble entrapment region depicted in figure 3 is analysed by making use of the experimental data in Michon et al. (2017). Indeed, Michon et al. (2017) report experiments on the velocities and the diameters of the drops emitted from the tip of of DP jets formed with liquids of different viscosities, Mo ≥Mow,We/Fr ≃1/4 and values of the Foude number 200 Fr 2000. The radii rdof the droplets formed under these conditions, which are a good proxy of rjet because, for this range of values of Fr and We, the jets are mostly cylindrical, are provided in figure 4(b) of Michon et al. (2017)forMo =Mow. The jet velocities calculated by means of (4.20) when rjet is approximated linearly to the data for rdin Michon et al. (2017)are compared with the experimental results in figure 26(b), where it can be seen that, in spite of approximating rjet by rd, the predicted jet velocity is in fair agreement with experiments. Figure 26 also shows that the experimental results deviate from the scaling proposed in Michon et al. (2017), Vjet ∝√σ/(ρRd). The reason for the deviation from the experimental data is that vjet also depends on rjet,which can also be inferred from the analysis of the encircled regions in figure 26. The larger rdis, the smaller the dimensionless jet velocity is. In contrast, the dependence of vjet with rjet is well captured by (4.20), even when rjet is approximated by the experimental value of rd. The jet velocities predicted by (4.20) for DP jets resort to the value of rjet, which, unlike the case of BB jets, could not explicitly be expressed in this case as a function of the control parameters We,Fr and Mo.Indeed,(3.10)and(4.15), which are the analogous 916 A37-35 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 2.5 10 8 6 4 2 0 500 600 700 800 900 1000 1100 1200 1300 2.0 1.5 1.0 0.5 0 800400 1200 Fr Fr rjet 1600 2000 2400 Vjet /σ/(ρRd) (b)(a) Figure 26. Left-hand panel: In this figure, rjet is approximated by a linear fit to the data in figure 4(b)of Michon et al. (2017)fortheradiirdof the droplets ejected, which is expected to be a good approximation to rjet and, therefore, rjet ≃rd.Right-hand panel: Acomparison between the jet velocities given in figure 7(a)of Michon et al. (2017)in the case of water and the jet velocities calculated using (4.20) multiplied by We1/2with rjet given by the linear fit to the data depicted in the left-hand panel and α=0.6 (red line). In the right-hand panel, the blue dashed line indicates the scaling Vjet/√σ/(ρRd)=constant proposed by Michon et al. (2017). Notice also that the encircled region of points in the left-hand panel, which represents the drop radii above the trend of the rest of experimental data, produces slower jets. These two facts indicate that the jet velocity depends on rjet and, therefore, that the jet velocity is proportional to the capillary velocity but with corrections associated with the precise value of rjet,as expressedby(4.20). expressions for (4.20) for the case of BB jets, depend explicitly on Oh because, in this case, Gordillo & Rodríguez-Rodríguez (2019) succeeded in expressing rjet as a function of Oh;see(3.9). The determination of the function rjet(Mo,Fr,We)forthecaseofDPjets,a task which is to be left for future study, could also help to improve our understanding of the precise conditions under which bubbles are entrapped as a consequence of the crater collapse, a process with implications for, among others, the origin of the noise of rain (Prosperetti & Oguz 1993). Let it be emphasised here that the papers published on the noise of rain since the original work of Pumphrey et al. (1989), including the seminal contributions by Prosperetti et al. (1989), Prosperetti & Oguz (1993), were not able to provide an equation for rjet(Fr,We),even in the case of water, Mo =Mow.This fact justifies why the determination of rjet(Mo,Fr,We)is far from being trivial and why this extra effort should be left for future study. 5. Conclusions In this contribution, we present a quantitative and predictive model that unifies the description of the jets produced by bursting bubbles and by the collapse of the crater formed when a drop impacts a liquid pool. Our results make use of the theory in Gordillo & Rodríguez-Rodríguez (2019), where the velocity field is calculated as the one produced by a line of sinks. In the case of bursting bubbles, the jet velocities are expressed in a closed form using two different approximations, their predictions being in remarkable agreement with experimental and numerical results. It is also shown that the same type of theoretical description applies in the case of jets formed after a drop impacts a liquid pool. In this case, an algebraic equation has also been deduced for the velocities of the jets,which effectively predicts the velocities of ∼50 m s−1of the very fast jets produced after the impact of a drop on a liquid pool that takes place under very specific and well-controlled conditions, as well as the velocities ∼1ms −1of the thicker and much slower jets commonly seen in experiments. The main physical idea in this contribution can be summarised in simple 916 A37-36 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities terms as follows: jets are produced as a consequence of the flow rate imposed by the radial velocity field induced by the capillary waves travelling along void walls and these waves also deform the bottom of the cavity from which the jet is issued. Since the flow rate is imposed by the converging capillary waves, the smaller the radius of the bottom of the cavity from which the jet is issued is, the larger the jet velocity will be. This rather simple physical idea is quantified using the theoretical framework presented in Gordillo & Rodríguez-Rodríguez (2019), which provides algebraic equations that are in good agreement with experimental results and numerical simulations. In the case of bubble bursting jets, the dimensionless jet velocities can be calculated by making use of equations that depend explicitly on the Ohnesorge number. In the case of jets produced after a drop impacts a liquid pool, the dimensionless jet velocity is expressed as a function of Fr,We and of the minimum radius of the cavity, which we aim to express as a function of the Weber, Froude and Morton numbers in a future contribution. Supplementary materials. Supplementary materials are available at https://doi.org/10.1017/jfm.2021.207. Acknowledgements. The authors wish to thank Professor S.T. Thoroddsen for kindly providing them with valuable experimental information and also Francisco del Campo-Cortés for the experimental images shown in figure 1. Funding. This work has been supported by the Spanish MINECO under Project DPI2017-88201-C3-1-R, partly financed through European funds. F.J.B.-R. acknowledges the funding received from the Spanish Government program Juan de la Cierva-Incoporación through grant IJCI 2016-30126. Declaration of interest. The authors report no conflict of interest. Author ORCIDs. Francisco J. Blanco–Rodríguez https://orcid.org/0000-0001-5762-4613; J. M. Gordillo https://orcid.org/0000-0003-1431-3780. Appendix A. Bond number effect for the case of bursting bubbles in water This appendix is devoted to analysing the way the results for the bubble bursting case in the limit Bo 1 studied in sections §§3and 4are modified for larger values of the Bond number. With that purpose in mind, the static shapes of the bubbles are first computed, solving the Laplace-Young equation following the method described in Lhuissier & Villermaux (2012) which, as figure 27 shows, significantly deviate from a sphere for increasing values of Bo. These initial shapes are taken as the initial condition for the numerical simulations in GERRIS (Popinet 2003,2009). In the supplementary materials, which are available at https://doi.org/10.1017/jfm.2021.207,it has been verified that vλ=5 irrespective of the value of Bo and, thus, capillary waves travel along the cavity walls at five times the capillary velocity based on Rbfor arbitrary values of Bo and Oh. Hence, it is expected that (4.11)and(4.12)withα=0.6, =2.8Oh1/2and qgiven by (4.13) reproduce the numerical velocity fields.This is confirmed in figures 28 and 29, a fact indicating that our results can indeed be extended to describe the ejection of BB jets for finite values of Bo. This conclusion is further supported by the result depicted on the left in figure 30, where the values of vjet calculated using (4.15) and the equation for rjet including the effect of Bo rjet =0.2215(1+0.4Bo)1−Oh 0.0305(1+0.3Bo),(A1) 916 A37-37 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 1.5 Bo = 0.672 Bo = 0.392 Bo = 0.218 Bo = 0.094 Bo = 0.01 1.0 0.5 0 1.5 1.0 0.5 R/Lc Z/Lc 0 Figure 27. Initial shapes of the bubbles calculated by solving the Young-Laplace equation for different values of Bo =ρgR2 b/σ,withRb=(3Vb/(4π))1/3and Vbthe bubble volume. Here, Lc=√σ/(ρg)denotes the capillary length. The results shown in this figure have been validated by comparing them with the ones provided in Princen (1963). are compared with the results obtained numerically. Notice that, in agreement with the numerical results, the values of vjet predicted through (4.15) decrease with Bo as a consequence of the fact that rjet in (A1) increases with Bo. The calculated values of vjet are, however, larger than vdbecause of the capillary forces that decelerate the tip of the jet; see Blanco-Rodríguez & Gordillo (2020) or the beginning of § 3for details. This is clearly shown on the right of figure 30, where the experiments for the case of water in Ghabache et al. (2014)andSpiel(1995) are compared with the numerical results and also with the predicted value of vjet for Bo =0. In spite of vjet −vdincreases with Bo,figure 31 shows that (4.15)forvjet calculated using the expression of rjet in (A1)provides an estimation of vdfor the case of water with acceptable relative errors,evenforBo ≃1. Notice that the reason why the focus here has been on the case of water is because the effect of the Bond number on the drop ejection process has already been considered in detail by Berny et al. (2020). It should be pointed out, however, that the simplified equation for rdgiven in Blanco-Rodríguez & Gordillo (2020)forBo 1: rd=0.22(1−(Oh/0.031)1/2)if Oh <0.03, rd=18.45Oh2if 0.03 <Oh <0.04,(A2) with Oh =La−1/2, is in very good agreement with the numerical and experimental results reported in Berny et al. (2020) for arbitrary values of Bo and also with our own numerical results, as shown in figure 32, where the value of Lad=La rdis represented as a function of La. The discontinuous dependence of rdwith Oh exhibited in (A2) is a consequence of the simplifications of the full theoretical model presented in Blanco-Rodríguez & Gordillo (2020), which predicts a continuous dependence of rdwith Oh. Our simplified expression for rdin (A2), however, predicts that rdcannot be zero and vdcannot tend to infinity when a bubble is entrapped. Indeed, the equations for rjet,vjet,vdand rddeduced in Gordillo & Rodríguez-Rodríguez (2019), Blanco-Rodríguez & Gordillo (2020), as well as those deduced here,arenot singular, which is in contrast with the analogous expressions deduced in Gañán Calvo 916 A37-38 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 2.2 2.1 rr 1.9 2.0 1.8 1.7 2.5 2.0 1.5 1.0 0.5 0 –0.5 1.9 1.8 1.6 1.5 1.4 1.3 1.1 1.0 1.2 1.7 1.6 1.5 1.4 1.3 –0.3 –0.2 –0.3–0.2–0.1 0 0.1 0.2 0.3–0.1 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 0 0.1 0.2 0.3 –0.3 –0.2–0.1 0 0.1 0.2 0.3 |v| = 10 |v| = 10 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 10 –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 10 |v| = 10 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 10 (z – zmin)/rmin 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin –1 0 1 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 10 |v| = 10 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1 0 1 |v| = 10 2.5 2.0 1.5 1.0 0.5 0 –0.5 (z – zmin)/rmin r/rmin r/rmin r/rmin r (a) (d) (g) (j) (b) (e) (h) (c) (f) (i) (k)(l) z Figure 28. Comparison between the numerical and analytical velocity fields calculated using (4.11)and(4.12) for α=0.6, =2.8Oh1/2and qgiven by (4.13) for the case of BB jets at the different instants of time and different values of Bo corresponding to the bubble shapes depicted in the top row. The red arrows indicate the numerical results, whereas the green ones correspond to those computed using (4.11)and(4.12). Time advances from top to bottom, the results are grouped in columns and, from left to right, the results correspond to the following values of Bo and La:Bo =0.6718 and La =Oh−2=131140, Bo =0.3924 and La =Oh−2= 141750 and Bo =0.2177 and La =Oh−2=149290. 916 A37-39 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 F. J. Blanco–Rodríguez and J. M. Gordillo 0 0.1–0.1–0.2–0.3 1.7 1.8 1.9 2.0 2.1 2.2 0.2 0.3 z r0 0.1–0.1–0.2–0.3 1.7 1.8 1.9 2.0 2.1 2.2 0.2 0.3 r0 0.1–0.1 1.70 1.80 1.90 1.95 1.85 1.75 2.00 r 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin (z–zmin)/rmin |v|=15 |v|=25 |v|=40 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin (z–zmin)/rmin |v|=15 |v|=25 |v|=40 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin 01 –1 –0.5 0.5 1.5 2.0 1.0 0 2.5 r/rmin (z–zmin)/rmin |v|=15 |v|=25 |v|=40 (a) (d) (g) (j) (b) (e) (h) (c) (f) (i) (k)(l) Figure 29. Comparison between the numerical and analytical velocity fields calculated using (4.11)and(4.12) for α=0.6, =2.8Oh1/2and qgivenby(4.13) for the case of BB jets with Bo =0.6718 at the different instants of time and different values of La =Oh−2corresponding to the bubble shapes depicted in the top row. The red arrows indicate the numerical results, whereas the green ones correspond to those computed using (4.11)and(4.12). Time advances from top to bottom, the results are grouped in columns and, from left to right, the results correspond to the following values of La:La =Oh−2=32786, La =Oh−2=8196 and La =Oh−2=2049, for different instants of time corresponding to the shapes depicted at the top of each column. 916 A37-40 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207 On the jets produced by the capillary collapse of cavities 0.001 0.01 0.02 Oh 1 10 vjet 10 –2 10 –1 10 0 0.001 0.01 0.02 Oh 1 10 vjet,vd Bo (a)(b) Figure 30. (a) Comparison between the numerical values of vjet and the predicted jet velocities calculated using (4.15)and(A1). (b)Acomparison between the numerical values of vjet (squares) and vd(circles), the experiments for the case of water in Ghabache et al. (2014)(∗)andSpiel(1995)(+) and the predicted value of vjet using the equation for rjet givenin(A1)forBo =0 (blue line). The black line represents the equation for vdgiven in (A3). The results are made dimensionless using the capillary velocity. The colour code,which serves to indicate the value of Bo, is the same as that used by Deike et al. (2018), Berny et al. (2020). 10–2 10–1 100 Bo 0 0.1 0.2 0.3 0.4 0.5 e Figure 31. Relative error between the drop velocities measured by Ghabache et al. (2014)(∗)andSpiel(1995) (+) and the jet velocity vjet calculated using (4.15)whenrjet is given by (A1). The relative error is defined here as e=|vd−vjet|/vjet. The results shown correspond to the case of water, with physical properties given by ρ=103kg m−3,σ=7×10−2Nm −1. The vertical line corresponds to Rb=1.2×10−3m. (2017), given in Berny et al. (2020)as: vd=19Oh1/2(1/√550 −Oh)−3/4,Lad=La rd=0.6[√La(La/540 −1)]5/4, (A3a,b) Equations (A3a,b) are represented using black lines in figures 30 and 32. Let us point out here that the theory in Gañán Calvo (2017) predicts as equations (A3) express that the jet velocity could be infinite and the radii of the droplets could be zero within certain ranges of the parameter space – and, moreover, Gañán Calvo & Lopez-Herrera (2019) also assert that their own numerical results are ‘starkly inconsistent’ 916 A37-41 Downloaded from https://www.cambridge.org/core. Universidad de Sevilla, on 13 May 2021 at 14:58:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2021.207