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

Abstract 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.


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;Deike et al. 2018;Gordillo & Rodríguez-Rodríguez 2018;Lai, Eggers & Deike 2018;Gordillo & Rodríguez-Rodríguez 2019;Berny et al. 2020) because it plays a key role in the production of the sea spray aerosol (MacIntyre 1972;Bigg & Leck 2008;Veron 2015;Wang et 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;Rein 1996;Ray, Biswas & Sharma 2015;Michon et al. 2017;Thoroddsen et al. 2018;Yang, Tian & Thoroddsen 2020) share a common 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 −1 in 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;Yang et al. 2020), whereas in the case of bubble bursting jets, Blanco-Rodríguez & Gordillo (2020) have reported maximum velocities of σ/μ, with σ and μ respectively 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  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 * ∝ Oh 1/2 (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 V jet and an initial radius R jet . The impact of a drop shown in (b), with Fr = 600, We = 90 and Mo = Mo w , reveals a similar jet ejection process to that depicted in (a), with the main difference being that the dimensionless radius of the cavity is r c 0.5Fr 1/4 (Prosperetti & Oguz 1993;Jain et 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 § 2 the 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 § 2 are compared with the theoretical predictions in § § 3 and 4 for the cases of bubble bursting jets and of drops impacting a deep liquid pool, respectively. The main results are summarised in § 5.

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(Popinet , 2009) assuming that the surrounding gaseous atmosphere is air with a density and a dynamic viscosity of 1.2 kg m −3 and 1.8 × 10 −5 Pa · 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, μ and σ will 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 R b = (3V b /(4π)) 1/3 , with V b indicating the bubble volume, will be presented in terms of dimensionless variables defined using R b , the capillary velocity √ σ/(ρR b ) and the capillary pressure σ/R b as 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 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 a radius R d falling over a pool of the same type of liquid with a velocity V will be expressed in terms of the following dimensionless variables, defined using R d , V and ρV 2 as characteristic values of length, velocity and pressure 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 = Mo w . 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 replicate those in Ray et 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 = Mo w given 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 = Mo w does not imply a constant value of Oh d . 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 r c . Whereas r c = 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;Jain et al. 2019) DP : r c 0.5Fr 1/4 . (2.4) In both the BB and DP cases, capillary waves with a characteristic wavelength λ * propagate with a dimensionless velocity v λ ; see figure 2. Figures 4 and 5 illustrate the way the values of v λ have been determined numerically: the shapes of the cavities are calculated at different instants of time separated at regular intervals Δt at which the increments in the angular positions of the maximum elevation of the capillary waves, Δθ, are also determined with the purpose of calculating v λ as v λ = r c Δθ Δt . (2.5) The results obtained from the analysis, depicted in figure 6, reveal that, for the case of BB jets, v λ 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 V λ = Vv λ , with v λ given in (2.5), divided by the capillary velocity based on the radius of the cavity, σ/(ρR d 0.5Fr 1/4 ). 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/(ρR d 0.5Fr 1/4 ); 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 * , 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 * ∝ Oh 1/2 . 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 to the case of BB jets (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 r c = 0.5Fr 1/4 . 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 ∼ r c are quite similar to the values of the 916 A37-6 On the jets produced by the capillary collapse of cavities 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 r c = 0.5Fr 1/4 . 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, z fs indicates the vertical position of the flat free interface.  (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 R b . This result is also valid for arbitrary values of Bo; see the Appendix A. 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 or R c = 0.5R d Fr 1/4 for DP jets. Since the Weber number based on the wave velocity is such that (V λ /V c ) 2 (V r /V c ) 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  the case of BB jets and arbitrary values of Oh and 10 for the case of DP jets; 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 4 will 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 z min (t) ≤ z ≤ s(t); and (iii) the drop, located at z = s(t). In the limit of low-viscosity liquids of interest here, Oh 1 and Oh d 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, with v(r, z, t) = ∇φ, satisfies the Laplace equation ∇ 2 φ = 0. The bulk region ends at z = z min (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/Dt indicating the material derivative. This momentum equation indicates that the flow within the jet is ballistic; namely, that fluid particles conserve the v (r, z, t) z min (t) r min (t) Figure 9. Sketch showing the meaning of the different variables used to characterise the jet ejection process as well as the different spatio-temporal regions in which the flow can be divided; namely, the bulk, jet and drop regions. These variables are defined in the main text, except for the jet radius r j and the velocity v j at the spatio-temporal boundary where the jet meets the drop: and v tip (t) and b(t) calculated using integral balances of mass and momentum, while u(z, t) and χ(z, t) are calculated using the method of characteristics described in Gekle & Gordillo (2010); see also Blanco-Rodríguez & Gordillo (2020), where the integral balances of mass and momentum at the drop are solved. The decomposition of the flow in three regions and the very good agreement between the predictions and the numerical results presented in Blanco-Rodríguez & Gordillo (2020) reveal that the jet is driven by the bulk velocity field v: once the value of the bulk velocity field is known at the base of the jet, both the spatio-temporal evolution of the jet and the drop radius and velocity can be predicted using the equations in Gekle & Gordillo (2010), Blanco-Rodríguez & Gordillo (2020). velocities they possess at the base of the jet. These velocities are prescribed by v since where v z indicates the vertical component of the bulk velocity field. As explained in Blanco-Rodríguez & Gordillo (2020), the time evolution of the radius of the drop, b(t), and of the jet tip velocity, v tip (t), can be determined by applying integral balances of mass and momentum at z = s(t) once the functions u(z, t) and χ(z, t), calculated using the method described in Gekle & Gordillo (2010), are particularised at z = s(t); see figure 9. Moreover, the criterion derived in Blanco-Rodríguez & Gordillo (2020) determines the instant t * at which the drop detaches from the jet and, hence, the drop radius and velocity can be predicted as v d = v tip (t * ), r d = b(t * ). Figure 9, together with the explanations given above, reveal that once v is known, the functions describing the jet radius and velocity χ(z, t) and u(z, t), the jet tip radius and velocity b(t) and v tip (t), and even the drop radius and velocity r d and v d , can be calculated as a function of the control parameters using the results described in Blanco-Rodríguez & Gordillo (2020). Since our main interest is in describing the origin of the jets, this section focuses on providing analytical expressions for the irrotational bulk velocity field v(r, z, t), and special attention is paid to the value of v z at r = 0 and z = z min (t ), with t the instant at which the minimum value of r min (t) is attained; see figures 9 and 10. With that purpose in mind, it proves convenient to now define z jet = z min (t ), r jet = r min (t ), and also v jet = v z (r = 0, z jet , t ), which represents the vertical component of the bulk velocity at the axis when the radius of the base of the collapsing cavity is the minimum. The relevance of v jet can be understood because this is the maximum velocity with which the liquid flows into the jet and, thus, at the usual limit at which the capillary force pulling back the tip of the jet is small compared with inertial terms in the momentum balance, v tip (t) v jet and v d v jet : this is due to the fact that material points conserve their velocities when flowing along the jet and because the jet tip barely decelerates through capillary forces at the usual limit when the dynamic pressure is much larger than the capillary pressure. As shown by the results reported in Blanco-Rodríguez & Gordillo (2020), the capillary forces pulling the tip of the jet downwards will make v d < v jet , but it will be shown here that these differences are small for low values of the Ohnesorge number. In summary, in this contribution the values of v jet and v d will be reported, but those of the velocity field u(z, t) will not -see figure 9 -which, in contrast, could be calculated using the theoretical framework presented in Blanco-Rodríguez & Gordillo (2020). Moreover, this contribution will focus on the description of bubble-bursting jets in the range Oh ≤ 0.02, for which a bubble is not entrapped before the jet is produced and for which v d v jet . The cases corresponding to the ejection of jets after a bubble is entrapped for 0.02 < Oh < 0.04, which has been studied carefully in Blanco-Rodríguez & Gordillo (2020), reveal that the capillary deceleration at the tip of the jet can no longer be neglected and, as a consequence of this, v d is noticeably smaller than v jet .
Recalling now that the dimensionless variables in this section are defined using R b , the capillary velocity √ σ/(ρR b ) and the capillary pressure σ/R b as the characteristic values of length, velocity and pressure, respectively, our analysis starts by noticing that Gordillo & Rodríguez-Rodríguez (2019) reported that the bulk velocity field v giving rise to BB jetssee figure 9 -can be calculated as the velocity field produced by a line of sinks of length and intensity q(z), with ∝ λ * ∝ Oh 1/2 , λ * the characteristic wavelength of the waves travelling along the cavity walls and q(z) the flow rate per unit length induced by the collapse of the void with a radius with β the opening semiangle of the truncated cone from which the jet is ejected (see figure 10a). The flow rate induced by the sinks located at a vertical position z 0 can thus be calculated as (see figure 10b) with v r calculated from the balance between the inertial and capillary terms in the Euler-Bernoulli equation (Zeff et al. 2000;Sierou & Lister 2004;Lai et al. 2018), The substitution of (3.1) and (3. with K the proportionality constant arising from the balance in (3.3). Therefore, the velocity field at (z, ) (see figure 10) generated by a line of sinks of length extending from z = z s = z min + αr min to z = z s + , with α an order unity constant, independent of time, and r min (t) and z min (t) = r min (t)/ tan β indicating the radial and vertical coordinates of the base of the cavity (see figure 10) can be expressed as with e r and e z indicating the unit vectors in the radial and axial directions, respectively. The substitution of (3.4) into (3.5) yields the following expressions for the vertical and radial components of the velocity: Equations (3.6) and (3.7) depend on and also on the constants α and K, which are to be determined in what follows. First, the result in Gordillo & Rodríguez-Rodríguez (2019) is used, where it was reported that the length of the line of sinks is proportional to the wavelength of the capillary waves that trigger the ejection of the jet and, therefore, ∝ λ * ∝ Oh 1/2 . Moreover, notice that the integration limits in (3.6) and (3.7) can be expressed as (see figure 10): with β = π/4 the opening semiangle of the truncated cone, ≈ Oh 1/2 and r min (t, Oh) indicating the radius of the base of the cavity from which the jet emerges, which is a time-varying quantity, and is not to be confused with r jet , which only depends on Oh. Indeed, let it be insisted that the maximum jet velocity, v jet (Oh), is reached at the axis of symmetry when the radius of the truncated cone is r jet (Oh), with r jet indicating the minimum value of r min (t, Oh). For Oh 0.03, Blanco-Rodríguez & Gordillo (2020) found that r jet varies linearly with the wavelength of the capillary wave as: with the values of the two free constants, 0.2215 and 0.0305, adjusted by fitting the theoretical prediction with the numerical results. For 0.03 Oh 0.04, a bubble is entrapped before the jet emerges and, in these cases, a viscous cut-off imposes that The role played by viscosity when a bubble is entrapped for Oh 0.03 will be clarified in § 4 and, as pointed out above, the focus here will mainly be on the analysis of the jets produced within the range of values of Oh for which no bubbles are entrapped and, thus, (3.9) provides the value of r jet (Oh).
In order to determine the values of the constants K and α, the results in Duchemin et al. (2002) and Lai et al. (2018) are used, where it is found that the time evolution of the jets can be described assuming the inertio-capillary balance, implying that lengths vary in time as τ 2/3 , with τ the instant to or from the ejection of the jet and, therefore, the velocities scale as τ −1/3 . Under this hypothesis, the value of the local Weber number, defined as the radial position of a point on the interface and v s (t) the corresponding liquid velocity, remains constant in time. From the results in figures 6 and 8 in § 2, where it is shown that capillary waves travel at a velocity which is five times the capillary velocity, it is deduced that the value of the local Weber number figure 11 reveals that the value of the local Weber number for La = Oh −2 = 4000 at the point on the interface located at r s = 1.25r min is roughly constant in time and equal to 25; a similar result is obtained for other values of r s provided that r s /r min < 1.5. Then, the values of the two free constants, K and α, are determined from the following two conditions: (i) We local = 25 at the point on the interface located at r s = 1.25r jet (Oh = La −1/2 = 4000 −1/2 ), with r jet given in (3.9); and (ii) v z (r = 0, z = r jet / tan β, Oh = La −1/2 = 4000 −1/2 ) = v jet (Oh = La −1/2 = 4000 −1/2 ), with v jet calculated numerically and v z calculated by means of (3.6). The solutions to these two equations yield the following values for the two free constants: K = 11.3 and α = 0.6. Figures 12 and 13 show that the velocity field predicted by (3.6) and (3.7) with = 1.24Oh 1/2 , K = 11.3 and α = 0.6 for arbitrary values of Oh are in good agreement with the numerical results at different instants of time after the jet is ejected. Here, the value of the free constant relating the length of the sinks with the wavelength of the capillary wave in ∝ λ * ∝ Oh 1/2 has been set to 1.24 (Gordillo & Rodríguez-Rodríguez 2019): it was checked that the modification of this multiplicative constant between 1 and 1.5 only has a noticeable effect on the velocity fields calculated for Oh 0.01.
It is noted that the velocity fields calculated using (3.6) and (3.7) depend parametrically on time through r min (t), which is given by (3.9) only at the instant the jet is ejected. Since figures 12 and 13 show that the velocity fields are accurately predicted by the theory once the jet is ejected, the time evolution of r min (t) could have been calculated using the initial shape of the interface obtained numerically as a starting point, with this shape being updated in time by means of the kinematic boundary condition using the predicted velocity fields. However, the results shown in figures 12 and 13 have been obtained using the values of r min (t) given by the numerical simulations. Notice also that the time evolution of the tip of the jet as well as the drop formation process could be predicted using the velocity fields depicted in figures 12 and 13. As pointed out above, the time evolution of the jets can be described using just the local flow structure near the base of the jet; see Blanco-Rodríguez & Gordillo (2020) for details.
Once the values of α, K and are known, the maximum jet velocity can be calculated as v jet = v z (r = 0, z jet = r jet / tan β) with r jet and v z respectively given in (3.9) and (3.6), which possesses the following analytical solution, already provided in Gordillo & Rodríguez-Rodríguez (2019): where x 2 1 = z s /z jet , z s = z jet + αr jet , z jet = r jet / tan β and x 2 2 = (z s + )/z jet , with = 1.24Oh 1/2 , α = 0.6, K = 11.3, β = π/4 and r jet given by (3.9).   the velocities of the drops emitted, v d . 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 v jet and v tip 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), r jet decreases with Oh 1/2 and, 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, v jet also increases with Oh because the length of the line of sinks increases as ∝ Oh 1/2 . Therefore, the increase in v jet with Oh in the region where no bubbles are entrapped, Oh ≤ 0.02, is caused by the combination of two  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, r 0 and 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;Day et al. 1998;Burton et 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, r 0 (t) and r −1 1 (t), which characterise the time-evolving local shape of the interface h(z, t) = r 0 (t) + z 2 r 1 (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) > r 0 (t) with b r 0 / √ r 0 r 1 (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/a and the capillary velocity √ σ/(ρa) read (Gordillo & Pérez-Saborid 2006;Gordillo 2008): The solutions for (4.1), the latter of which is deduced from dr 0 /dt =ṙ 0 = r 0ṙ0 /r 0 , with γ an order unity constant and s = − ln r 0 , depend on: 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 ∝ r 0ṙ0 placed at the axis of symmetry along a distance b ∝ r 0 / √ r 0 r 1 ; see figure 16. In fact, the first logarithmic term in (4.1) results from ln(r 0 / b ) = 1/2 ln(r 0 r 1 ) for b = r 0 / √ r 0 r 1 > r 0 . The system solution (4.1) in the inviscid limit Oh 1 predicts that r 0 ∝ τ α c with τ the 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., r 0 ∝ τ 2/3 (Day et al. 1998;Duchemin et al. 2002;Lai et 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 r 1 ∝ r −1 0 and the capillary terms were not subdominant, b = r 0 / √ r 0 r 1 ∝ r 0 and the first logarithmic term in (4.1), ln(r 0 / 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 r 0 ∝ τ 2/3 would 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 b that, in general, will be different from the value b ∝ r 0 required 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 § 3 in 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 r 0 ∝ τ 2/3 can be understood in simple terms because the length of the sinks b (t) ∝ r 0 / √ r 0 r 1 is larger than r 0 (t) and, also, the ratio b (t)/r 0 (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 r 0 is, 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 α c in r 0 ∝ τ α c are associated with the larger values of the time-dependent ratio b (t)/r 0 (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, r 0 ∝ τ 2/3 . 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 r 0 (0),ṙ 0 (0) and r 1 (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): which, when expressed in terms of the time to pinch-off τ , yield to solutions of the type r 0 ∝ τ α c with 1/2 < α c < 2/3. Indeed, and, therefore: which, in the limit d ln q −1 /ds 1 reads: Then, the substitution of the expression of q in (4.2a,b) into (4.5) yields: which indicates that the exponent α c decreases 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, ln(r 0 r 1 (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 α c decreases when the slenderness of the initial shape of the cavity increases; i.e., when [ln(r 0 r 1 (0))] 2 increases 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 r 0ṙ0 decreases in time as r 0 → 0, a fact meaning that the local value of the Reynolds number, given by Oh −1 r 0ṙ0 30r 0ṙ0 , 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 r 0 (τ ) ∝ τ α c , with 1/2 < α c < 2/3, will be found, with smaller values of α c ; i.e., with larger values of the velocities for the larger values of the ratio b /r 0 = 1/ √ r 0 r 1 , see figure 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 § 3 and sketched in figure 10. Indeed, two different length scales, r min (t) and b , with r min and b playing the roles of r 0 and r 0 / √ r 0 r 1 in (4.1), respectively, are also required to describe the collapse of these cavities. Therefore, since b is imposed by the capillary waves reaching the base of the cavity and, thus, is constant in time, the ratio b /r min (t) will increase in time and the collapse will be faster than the self-similar collapse r min ∝ τ 2/3 , which would only take place if b (t)/r min (t) remained constant in time, as explained above. Moreover, the first logarithmic term in (4.1) corresponds to ln(r min / b ) (Gordillo 2008); therefore, the equation describing the time evolution of the minimum radius of the cavity r min (t), which plays the same role as r 0 (t) in the equations above, can be written as: Defining s = − ln(r min / b ) and the intensity of the sinks as q = r minṙmin , (4.7) yields: and, therefore, the result found in Gordillo et al. (2005) is recovered: where the initial conditions at s(0) ≈ 1 have been imposed, and it has also been taken into account that, as illustrated in figure 8, r min (0)/q 2 (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 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 r min (t) as in § 3 but 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(z 0 ) = −2πq dz 0 , (4.10) the substitution of (4.10) into (3.5) yields: and with q the 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 q in (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 r c , q(0) v λ r c 5 (see figure 6a). Therefore, assuming that the unknown length b in (4.9) is the initial radius of the cavity, b = r c = 1, (4.9) reads: (4.13) Notice that the approximation for the flow rate per unit length in (4.13) is asymptotically valid in the limit r min 1 regardless 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 r min | | ln b | and, therefore, ln(r min / b ) = ln r min − ln b ln r min . 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), z s = r min / tan β + αr min and z = z s + , 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, = COh 1/2 , with C a multiplicative constant. The value C = 2.8 used here is determined using the condition that the vertical velocity predicted by (4.11) at the axis of symmetry, v z (r = 0, z = r jet / tan β), with r jet 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, = 2.8Oh 1/2 and q given by (4.13) for all values of Oh and t. As explained at the beginning of § 3, the maximum jet velocity is attained when r min (t) = r jet with r jet given in (3.9); therefore, v jet can be predicted solving (4.11)  Figure 18. Comparison of the numerical and analytical velocity fields calculated using (4.11) and (4.12) for α = 0.6, = 2.8Oh 1/2 and q given by  Figure 19. Comparison of the numerical and analytical velocity fields calculated from (4.11) and (4.12) for α = 0.6, = 2.8Oh 1/2 and q given by (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 = 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 green line at the axis indicates where the line of sinks is located. particularised at z = r jet / tan β, = 0, which yields: (4.14) The substitution of α = 0.6, = 2.8Oh 1/2 and the value of q in (4.13) into (4.14) yields: (4.15) with r jet given in (3.9). Figure 20 compares the values of v jet calculated numerically with those predicted by (4.15). The excellent agreement between the predictions, experiments and the numerical results depicted in figure 20 indicate that (4.15) can be used to calculate the initial jet velocity for all ranges of Oh considered here. Notice that (4.15) predicts the jet velocities slightly more accurately than the analogous (3.10) for the smaller values of Oh. However, the differences between the two approaches regarding the smaller range of values of Oh is extremely small, which it can be appreciated in figure 21. In fact, the larger differences, albeit also small, between the two approaches are observed for values of Oh for which a bubble is entrapped; see the case of La = 1000 in figures 13 and 19, where it can be noted that the analysis of § 3 predicts the numerical velocity field for this particular value of La more accurately.
The results shown in this section reveal that the agreement between the predictions, numerical results and experiments is as good as the agreement depicted in § 3. Therefore, the velocity field produced by the bursting of bubbles for arbitrary values of Oh and for times close to the jet ejection instant can be predicted using the theoretical framework put forward in Lai et al. (2018), Gordillo & Rodríguez-Rodríguez (2019). It should also be noted that the time evolution of the radius and of the jet tip velocity could be quantified in terms of the time-varying local velocity fields depicted in figures 12 and 13 or 18 and 19 during the instants close to the jet ejection instant; specifically, when r min (t) = r jet using the theoretical framework provided in Blanco-Rodríguez & Gordillo (2020).
However, the results in figures 18-20 clearly indicate that the ejection of bubble bursting jets can also be described when the inertio-capillary balance is relaxed. Then, because the expressions deduced in this section are simpler than the analogous ones in § 3, because the equations deduced in this section do not depend on the semiangle β illustrated in figure 10 and also because the ejection of the faster DP jets can only be described when the cavity collapse is driven by inertia as will be shown below, from now on the velocity fields of BB and DP jets will be quantified making use of (4.11), (4.12), (4.14) and (4.15). The Appendix A demonstrates that the equations deduced in this section can also be used to describe the ejection of BB jets for arbitrary values of Bo and Oh ≤ 0.02. The Appendix A also provides equations that predict the radii of the ejected droplets and also the droplet velocities v d with a special emphasis on the case of water.
Thanks to the analogy that exists between BB and DP jets, which has already been pointed out in § 2 and illustrated in figures 4 and 5, the results in this section are used to predict the velocity fields and the initial velocities of the jets produced when a drop falls onto a deep liquid pool. Indeed, the flow rate per unit length q(0) in (4.9) can also be calculated in the case of DP as the product of the velocity of the capillary waves multiplied by the radius of the cavity, r c , as: where it has been taken into account that r c = 0.5Fr 1/4 (Prosperetti & Oguz 1993;Jain et al. 2019). The results in figure 6(b) indicate that the velocity of the capillary waves can be approximated as: v λ 3.5(0.5Fr 1/4 We) −1/2 , (4.17) and, therefore, the substitution of the result of (4.17) into (4.16) yields: q(0) 3.5 0.5Fr 1/4 We . (4.18) Consequently, the flow rate per unit length of the sinks that describe the ejection of DP jets can be expressed, using the results in (4.9) and (4.18) as:  Figure 21. Comparison between the numerical velocity fields (in red) and the analytical velocity fields calculated using either (4.11) and (4.12) for α = 0.6, = 2.8Oh 1/2 and q given by (4.13) (in green) or (3.6) and where b is also taken here as the radius of the cavity, b = r c = 0.5Fr 1/4 . The same comment made in the case of BB jets applies here: (4.19) is expected to provide an approximate value for the flow rate per unit length q whenever r min b r c . Another difference with the case of BB jets is that, in the case of DP jets, λ * ≈ 1; see figure 5, and, therefore, (4.11) and (4.12) are solved using = 1 and α = 0.6, with q calculated through (4.19). The results, depicted in figures 22-24 for Mo = Mo w , for the ranges of Fr, We indicated in figure 3 and several instants of time after the jet is issued indicate that, BB and DP jets can indeed be described in terms of the same physical ideas using the theoretical framework presented in Gordillo & Rodríguez-Rodríguez (2019).
The initial velocity of DP jets can also be predicted by making use of (4.14), which, using the expression of q given in (4.19), reads: with α = 0.6 and where it has been taken into account that λ * = 1 and also that, in this case, variables are made dimensionless using R d , V and ρV 2 as characteristic values of length, velocity and pressure. The predictions of (4.20) are expected to explain the measured experimental values of the velocities of the very fast jets ejected after the impact of a drop on a liquid pool (Michon et al. 2017;Thoroddsen et al. 2018). Indeed, (4.20) reveals that V jet = Vv jet diverges when r jet → 0 as v jet ∝ (r jet − ln r jet ) −1 and, hence, in comparison with the case of bubble bursting jets, extremely fast jets can be ejected if the capillary waves travelling along the cavity walls deform the base of the void in such a way that r jet 1. However, in contrast to the case of bubble bursting jets, this type of event happens under very specific values of the control parameters and it is barely repeatable (Michon et al. 2017) because the collapsing cavity caused by an impacting drop easily losses axisymmetry, resulting in a reduction in the jet velocity (Gekle & Gordillo 2010). In spite of the fact that these are very rare events, Thoroddsen et al. (2018) and Yang et al. (2020) report experiments on DP jets with velocities up to 48 m s −1 , which can be depicted in figure 25, where the experimental measurements are compared with the jet velocities calculated as V jet = Vv jet , with v jet given by (4.20) and r jet measured from the experimental images included in the figure. It can be seen in figure 25 that the predicted velocity of the jets are, in some cases, in very close agreement with the measured values. It is particularly interesting to note that the fastest velocities reported by Thoroddsen et al. (2018) are very well predicted by (4.20). For the other cases depicted in figure 25, the predicted value is close but below the measured value. The reason why the agreement is not equally good for all the cases shown is because the predicted jet velocity depends on r jet , which is taken from the images inserted into figure 25, which admit a relative error in the measurements from the pictures provided in Thoroddsen et al. (2018) and Yang et al. (2020) of ±20 %. Therefore, the predicted jet velocity depends highly on the instant at which those images were taken. Thus, if the image corresponds to an instant shortly before the jet is issued, r jet in (4.20) will be larger and the calculated value of the velocity v jet will be smaller. This fact explains the different degrees of agreement between the measurements and predictions in figure 25. It should be pointed out that the images in figure 25 that correspond to the experiments in Thoroddsen et al. (2018) reflect the instant at which the jet is issued, as explained in the original paper. Notice the close agreement between the predicted and the experimental values for the jet      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 α c 0.53 ± 0.02 in r min ∝ τ α c when the collapse of the cavity gives rise to the emergence of jets with velocities of ∼ 50 m s −1 and a bubble is not entrapped. Under these conditions, r min /r c = 2r min Fr −1/4 4 × 10 −3 → s = − ln(r min /r c ) 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) is such that q −1 ∝ √ s and, therefore, (4.5) predicts that: 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 case of jets being ejected after a bubble has been entrapped. As pointed out above, the exponent α c depends highly on the slenderness of the entrapped bubble as (4.6) clearly shows: the more elongated the entrapped bubble is, the smaller α c is. 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 b is constant in time for the case of the jets ejected when a bubble is not entrapped, the ratio b /r min 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 α c = 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 § 3 were applied to describe the jet ejection process in the DP case, the jet velocities predicted would be far smaller than those observed experimentally, a conclusion which can also be deduced by comparing (3.10) and (4.20): while in the former case v jet ∝ r −1/2 jet , the values of v jet predicted by (4.20), which are in good agreement with the experimental measurements in figure 25, diverge as v jet ∝ (r jet − ln r jet ) −1 r −1/2 jet when r jet → 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 r jet > 0 and, hence, v jet 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 ≥ Mo w , We/Fr 1/4 and values of the Foude number 200 Fr 2000. The radii r d of the droplets formed under these conditions, which are a good proxy of r jet 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) for Mo = Mo w . The jet velocities calculated by means of (4.20) when r jet is approximated linearly to the data for r d in Michon et al. (2017) are compared with the experimental results in figure 26(b), where it can be seen that, in spite of approximating r jet by r d , 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), V jet ∝ √ σ/(ρR d ). The reason for the deviation from the experimental data is that v jet also depends on r jet , which can also be inferred from the analysis of the encircled regions in figure 26. The larger r d is, the smaller the dimensionless jet velocity is. In contrast, the dependence of v jet with r jet is well captured by (4.20), even when r jet is approximated by the experimental value of r d . The jet velocities predicted by (4.20) for DP jets resort to the value of r jet , 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  Michon et al. (2017) for the radii r d of the droplets ejected, which is expected to be a good approximation to r jet and, therefore, r jet r d . Right-hand panel: A comparison 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 We 1/2 with r jet 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 V jet / √ σ/(ρR d ) = 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 r jet and, therefore, that the jet velocity is proportional to the capillary velocity but with corrections associated with the precise value of r jet , as expressed by (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 r jet as a function of Oh; see (3.9). The determination of the function r jet (Mo, Fr, We) for the case of DP jets, 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 r jet (Fr, We), even in the case of water, Mo = Mo w . This fact justifies why the determination of r jet (Mo, Fr, We) is far from being trivial and why this extra effort should be left for future study.

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 −1 of 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 ∼ 1 m s −1 of the thicker and much slower jets commonly seen in experiments. The main physical idea in this contribution can be summarised in simple 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.  Princen (1963).
are compared with the results obtained numerically. Notice that, in agreement with the numerical results, the values of v jet predicted through (4.15) decrease with Bo as a consequence of the fact that r jet in (A1) increases with Bo. The calculated values of v jet are, however, larger than v d because of the capillary forces that decelerate the tip of the jet; see Blanco-Rodríguez & Gordillo (2020) or the beginning of § 3 for details. This is clearly shown on the right of figure 30, where the experiments for the case of water in Ghabache et al. (2014) and Spiel (1995) are compared with the numerical results and also with the predicted value of v jet for Bo = 0. In spite of v jet − v d increases with Bo, figure 31 shows that (4.15) for v jet calculated using the expression of r jet in (A1) provides an estimation of v d for the case of water with acceptable relative errors, even for Bo 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 r d given in Blanco-Rodríguez & Gordillo (2020) for Bo 1: 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 La d = La r d is represented as a function of La. The discontinuous dependence of r d with 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 r d with Oh. Our simplified expression for r d in (A2), however, predicts that r d cannot be zero and v d cannot tend to infinity when a bubble is entrapped. Indeed, the equations for r jet , v jet , v d and r d deduced in Gordillo & Rodríguez-Rodríguez (2019), Blanco-Rodríguez & Gordillo (2020), as well as those deduced here, are not singular, which is in contrast with the analogous expressions deduced in Gañán Calvo 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' with the flow generated by a line of sinks. However, it has been shown here that: (i) the emergence of BB and DP jets can indeed be described by means of the velocity field induced by a line of sinks; and (ii) neither velocities reach infinite values, nor is the diameter of the jets produced zero for any combination of the control parameters. This is because the singularity is prevented by the viscous cut-off length μ 2 /(ρσ ) (Gordillo & Rodríguez-Rodríguez 2019), and this is the reason why the minimum size of the droplets produced is 10μ 2 /(ρσ ) and why the maximum drop velocities are ∼ σ/μ (Blanco-Rodríguez & Gordillo 2020). Notice also another reason that prevents infinite jet velocities being reached when a bubble is entrapped is the overpressure required to evacuate the gas from the axisymmetric collapsing cavity which, as pointed out in Gordillo (2008), is responsible for the generation of tiny satellite bubbles. In addition, in the cases in which a bubble is not entrapped, the velocity at which the base of the truncated cone or the cylindrical surface from which the jet is issued shrinks in the axial direction more quickly 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. Finally, it should be pointed out that, in a real system, the convergence of interfacial asymmetries towards the axis reduces the jet velocity with respect to the case in which axisymmetry is preserved all along the cavity collapse process, as discussed at length in Gekle & Gordillo (2010).