Bubble entrainment by a sphere falling through a horizontal soap foam
aa r X i v : . [ c ond - m a t . s o f t ] J a n Bubble entrainment by a sphere falling through ahorizontal soap foam
S.J. Cox, I.T. DaviesDepartment of Mathematics, Aberystwyth University,Aberystwyth SY23 3BZ, UK.
Abstract
We simulate the quasi-static motion of a spherical particle through a stable, horizontalsoap film. The soap film subtends a fixed contact angle, in the range 10 − ◦ , where itmeets the particle. The tension and pressure forces acting on the particle are calculated in twocases: when the film is held within a vertical cylinder, trapping a bubble but otherwise freeto move vertically, and when the outer rim of the film is held in a fixed circular wire frame.Film deformation is greater in the second case, and the duration of the interaction thereforeincreases, increasing the contact time between particle and film. As the soap film returnstowards its equilibrium shape following the passage of the particle a small bubble is trappedfor contact angles below a threshold value of 90 ◦ . We show that this bubble is larger when theparticle is larger and when the contact angle is smaller. Aqueous foams interact with particles in a number of important situations [1, 2]; at high particledensity the particles can even replace surfactant and stabilise the foam [3]. At the other extreme,foam films can be used to separate individual particles based on their size [4]. In between, pro-cesses such as froth flotation and explosion suppression [2, 5] rely on the extent to which particlesare trapped by foam films. Once in the film, particles may rotate and, depending on parameterssuch as the contact angle, may cause rupture [6].Le Goff et al. [7] found that small millimetric-sized particles falling on to a soap film at speedsof about 1 m/s do not break the film. That is, after the particle has passed through the soap filmthe film “heals” itself [8]. This arrangement of a stable soap film held horizontally while a smallspherical particle falls onto it permits an investigation of the forces that the soap film exerts on theparticle and the consequent changes to the particle’s velocity. The soap film can be considered torepresent one repeating unit of a more extensive “bamboo” foam [9], in which successive impactsbetween the particle and different soap films could bring the particle to rest, representing a micro-scopic approach to the way in which a foam can be used in impact protection [5]. In the following,we choose the particle’s weight sufficiently large that it is never trapped by a single soap film. Then1he film is pulled into a catenoid-like shape as it is stretched by the particle, until, similar to theusual catenoid instability [10], the neck collapses and the soap film returns to its horizontal state.We will show that the forces exerted on the particle depend strongly on the contact angle alongthe triple line (Plateau border) where the liquid, gas and solid particle meet. In an experimentthis contact angle could be adjusted by coating the particle [11]. We allow the contact angle atwhich the soap film meets the spherical particle to vary: the equilibrium case is a contact angle of θ c = ◦ [12], in which the sphere is assumed to be coated with a wetting film that allows the soapfilm to move freely. However, experimental photographs [7, 13] show that the soap film wrapsaround the particle, with a contact angle far from 90 ◦ , before forming a catenoid-like neck. Thissuggests that the particle’s motion is faster than the mechanical relaxation of the foam. Here wenonetheless employ quasistatic simulations, and presume that the only effect of the dynamic natureof the experiments is to adjust the contact angle between particle and film. We consider severalvalues of θ c down to 10 ◦ .In experiments, the collapse of the catenoidal neck above the particle generates a small bub-ble [7], as for the impact of a liquid drop on a liquid surface [14, 15] and the collapse of an isolatedsoap-film catenoid [16]. This small bubble was not seen in previous simulations with a 90 ◦ contactangle [9, 12]. Our new simulations make clear why this is the case: only with a contact anglesmaller than 90 ◦ does the film curve around the particle sufficiently before detachment to enclosesuch a volume of gas.The particle in our simulations, described below, is a sphere of given radius R s and mass m grams, and hence with density ρ = m / ( / π R s ) . It falls, initially under its own weight, towards afilm with interfacial tension γ = / s (so a film tension of 2 γ ). We consider two cases:1. the soap film is held in a cylinder of radius R cyl = H = . H π R cyl = . π cm , i.e. that fills the lower half of thecylinder. In this case both the tension in the film and the pressure in the bubble exert a forceon the sphere once it touches the film.2. the soap film is held by a fixed ring of radius R cyl = Bo = ρ gR s / γ . In the simulations in §3 we ensure that theBond number is just greater than one, indicating that gravitational forces should exceed the re-tarding force due to surface tension. Making the density (and hence the Bond number) smallerwould lead to the sphere being trapped by the film, while increasing it would mean that the qua-sistatic approximation that we employ would be less appropriate. The maximum vertical tensionforce that the soap film could exert on the sphere to counteract its weight occurs when the filmmeets the sphere on its equator and pulls vertically upwards; then the film tension multiplied bythe sphere circumference is 4 πγ R s . So, roughly speaking, if the particle density is constant then topass through the film requires that the particle radius must be greater than p γ / ( ρ g ) whereas ifthe particle mass is constant the condition is R s < mg / ( πγ ) .2 (0,z ) s R θ (r , z ) sf sf V bub rz c (R ,z ) cfcyl Figure 1: The axisymmetric structure under consideration, shown in the ( r , z ) plane. In case 1 thereis a bubble of fixed volume V bub and the vertex at position ( R cyl , z c f ) is free to move, while in case2 there is no volume constraint and the vertex is fixed. We use the Surface Evolver [17] to compute the shape of the soap film. Since this software givesinformation about static situations, we assume that the motion is overdamped, and therefore thatthe sphere and soap film move through a sequence of equilibrium positions determined by theforces acting.By symmetry the sphere must remain in the centre of the film, so we perform an axisymmetriccalculation in the ( r , z ) plane. See figure 1. The film is represented by a curve whose endpointstouch, respectively, the sphere (or the axis of the cylinder before attachment and after detachment)and the outer cylinder / ring. We discretize the curve into short straight segments of length d l andwrite the energy of the system as E f ilm = γ ∑ segments π r d l . (1)We restrict segments to have lengths in the range 0.01 to 0.05 which balance the need for accuracywith short computational time.To include a contact angle θ c we add a further term to the energy representing a spherical capof film with tension 2 γ cos θ c that covers the lower part of the sphere. This is based on the height z s f of the film where it meets the sphere: E θ c = γ cos θ c . π R s (cid:0) z s f − ( z s − R s ) (cid:1) , (2)3here z s is the height of the centre of the sphere. This energy is set to zero before attachment andafter detachment.In case 1 we must also account for the volume V bub of the bubble trapped beneath the soap film.We calculate this volume based on the shape of the film and the positions of its endpoints. Thereare three terms required: V = ∑ segments π r d zV = z s f < z s − R s π R s ( z s f − z s ) − π ( z s f − z s ) + π R s z s − R s ≤ z s f ≤ z s + R s π R s z s f > z s + R s (3) V = π R cyl z c f , with V bub = V − V − V The first term ( V ) is the volume of revolution about the z axis of the filmbetween its endpoints, and the second term ( V ) is the volume of the spherical cap below the thepoint of contact between the film and the sphere. These are both subtracted from the third term( V ), which is the total cylindrical volume enclosed by the outer wall of the cylinder beneath thepoint of contact z c f between the film and the cylinder wall. We consider two forces in addition to the weight mg acting in the negative z direction. The tensionforce F γ is due to the pull of the soap film around its circular line of contact with the sphere andthe pressure force F p , which is only relevant in case 1, is due to the pressure in the trapped bubblewhich acts over the surface of the sphere below the contact line. We are interested only in thevertical component of these forces, since by symmetry the other components cancel.We define the angle θ that the film subtends with the centre of the sphere, tan θ = ( z s f − z s ) / r s f ,and then the z − components of the forces are F γ = γ . π r s f cos ( θ − θ c ) (4)and F p = π r s f p bub , (5)where p bub is the pressure in the bubble. We perform a quasi-static simulation in which the position of the sphere is held fixed while theequilibrium shape of the film is found, and then the sphere is moved a small distance in the directionof the resultant force. In case 1 the bubble pressure is found from the Lagrange multiplier of thevolume constraint, eq. (3).We start the simulation with the sphere above a horizontal film, and move the sphere down-wards in steps of ∆ z s = − ε mg , with the small parameter ε taken equal to 1 × − (which we find4a) (b)Figure 2: Film shapes in a frame of reference moving with the sphere, with contact angle θ c = ◦ ,shown every 100 iterations. (a) Case 1, where a wetting film on the outer cylinder wall allows thefilm to slip there and hence meet the wall at 90 ◦ . (b) Case 2, where the film is fixed at the outercylinder wall.(a) S phe r e P o s i t i on Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 (b) S phe r e P o s i t i on Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 Figure 3: The height of the centre of the sphere under the action of its weight and the forces thatthe foam exerts on it. The horizontal axis corresponds to time, in units of ε . (a) Case 1. (b) Case 2.is sufficiently small not to change the results), until contact is made. The inner end of the film thenjumps to a new position on the sphere and then the change in its vertical position obeys ∆ z s = ε (cid:0) F γ + F p − mg (cid:1) . (6)Detachment occurs when the film nears the top of the sphere and becomes unstable, at which pointit jumps back to being horizontal, and we then end the simulation. Note that ∆ z s is always negativein our simulations, since the weight of the sphere is large enough that it always exceeds the tensionforce. In this section we consider a sphere of radius R s = . m = . ρ ≈ g / cm and the Bond number is Bo ≈
2. An example of the shape of the5lm at different times is shown in figure 2.
The vertical position of the centre of the sphere is shown in figure 3. Before attachment the spherefollows the same path for all contact angles. Following attachment (at an iteration number close to150) we observe a shallower curve for smaller contact angles, indicating that the forces retard themotion of the sphere to a greater extent when the contact angle is small. When the contact angleis larger, for example with θ c greater than about 45 ◦ , the sphere motion is at first accelerated, asthe film pulls it downwards. In case 1, the bubble pressure is also negative at first (see figure 7below), adding to this effect. For the contact angle of θ c = ◦ this significantly reduces the timeof interaction before the film detaches from the top of the sphere.After detachment the slope of each curve returns to the same value as before attachment (datanot shown) for all contact angles. In case 2, without a volume constraint, the interaction time (whenthe film and sphere are in contact) is longer for each value of contact angle compared to case 1,and the sphere descends further before detachment. Hence the overall effect of constraining thevolume rather than the outer rim of the film is to retard the sphere.Detachment occurs before the inner end of the soap film reaches the top of the sphere. Instead,there is a sort of “pre-emptive” instability [18]: the curved soap film becomes unstable, the lineof contact jumps upwards, and a new configuration consisting of a flat film above the sphere isreached. This is seen, for example, in the abrupt jump in the surface area of the film, shown infigure 4, at the point of detachment.When the sphere first meets the film the film area is reduced because it contains a circular holethat is filled by the sphere. As the sphere descends further, the film is deformed, in order to obeythe volume constraint (in case 1) or the fixed rim at the cylinder wall (in case 2) and to satisfy thecontact angle where they meet. This causes the film area to increase, until the film approaches thepoint of detachment. For a contact angle of 135 ◦ (and presumably greater) the area of the filmnever exceeds its equilibrium value, A = π R cyl , indicating that it is not greatly deformed and thatdetachment occurs quickly. Comparing case 1 to case 2, for all other contact angles simulated, thefilm is slightly more deformed when its outer rim is fixed (case 2).Just as there is a sudden jump during detachment, there is also a jump in the vertical position ofthe circular line of contact when the film first meets the sphere (figure 5). The contact line rises toa new position to satisfy the contact angle (without, in case 1, violating the volume constraint), to adegree that increases with the contact angle. This end of the film is then pulled down by the sphere,more so for large contact angles, and the decrease is monotonic until detachment, whereupon thefilm is suddenly released. In case 1, the film returns to a higher position after the sphere has passed,because the volume enclosed beneath the film is augmented by the volume of the sphere.Fixing the outer rim of the film (case 2) leads to a greater deformation of the film (figure 4) andhence to the film becoming unstable when the line of contact is further from the top of the sphere(figure 5 insets).In case 1 the outer rim of the film, where it touches the cylinder wall, behaves slightly differ-ently (data not shown). It at first drops suddenly, i.e. in the opposite sense to the inner contact line,and then increases until the inner contact line approaches the top of the sphere. It then descends6a) F il m a r ea ( A / π R cy l ) Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 (b) F il m a r ea ( A / π R cy l ) Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 Figure 4: The area of the soap film as the sphere passes through it. (a) Case 1. (b) Case 2.(a) H e i gh t o f f il m - s phe r e c on t a c t Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c =135 -1-0.5 0 0.5 1 0 400 800 1200 R e l a t i v ehe i gh t (b) H e i gh t o f f il m - s phe r e c on t a c t Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c =135 -1-0.5 0 0.5 1 0 400 800 1200 R e l a t i v ehe i gh t Figure 5: The vertical position z s f of the line where the film touches the sphere. The inset showsthis position relative to the height of the centre of the sphere, ( z s f − z s ) / R s . (a) Case 1. (b) Case 2.7a) -1.5-1-0.5 0 0.5 1 1.5 0 200 400 600 800 1000 1200 T en s i on f o r c e / F il m t en s i on Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 (b) -1.5-1-0.5 0 0.5 1 1.5 0 200 400 600 800 1000 1200 T en s i on f o r c e / F il m t en s i on Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 Figure 6: Tension forces exerted on the sphere, determined by the direction in which the film pullsmultiplied by its tension. (a) Case 1. (b) Case 2. -0.06-0.04-0.02 0 0.02 0.04 0.06 0 200 400 600 800 1000 1200 P r e ss u r e f o r c e / F il m t en s i on Iteration number θ c = 10 θ c = 25 θ c = 45 θ c = 60 θ c = 75 θ c = 90 θ c = 135 Figure 7: Pressure forces exerted on the sphere in Case 1.again before suddenly returning to the same vertical position as the inner contact line when thefilm detaches and becomes flat.
We show the forces acting on the sphere in figures 6 and 7. For large contact angles the film pullsthe sphere downwards, accelerating its motion. The opposite occurs for small contact angles, andso the time over which the sphere contacts the sphere is extended. Just before the abrupt drop inforce at the point of detachment, there is a slight reduction in the tension force as the perimeter ofthe contact line becomes small, ameliorating the pull from the film.In case 1, the pressure in the bubble can be either positive or negative, depending on the cur-vature of the film. The pressure force on the sphere is determined by this pressure multipliedby the vertically-projected area of the sphere over which the bubble touches the sphere, eq. (5).The pressure force is much smaller in magnitude than the tension force. For the contact angle of135 ◦ the bubble pressure is large and negative for much of the passage of the sphere, because ofthe curvature induced by the contact angle, so in this case the pressure force “sucks” the spheredownwards and detachment occurs earlier than in case 2.8or smaller contact angles, for example θ c = ◦ , the pressure is always positive, opposing thedownward motion of the sphere. Yet it is still the case that detachment occurs sooner in case 1,even though for a given contact position the tension force is similar in both cases. Further, thefilm becomes unstable at a lower position in case 2. The resolution of this apparent paradox isthat when the contact line is at a certain position on the sphere, the sphere is at a different heightin the two cases, because of the need to satisfy the different constraints and for the film to meetthe sphere at the same contact angle. In particular, before the contact line passes the equator ofthe sphere ( z s f < z s ), it moves around the sphere more slowly in case 2, while above the equator itmoves more quickly (but over a shorter distance). Although our quasistatic simulations do not resolve the rapid film motion during detachment, wecan gain an idea of the size of the small bubble that is trapped [7] by examining the shape of thesoap film immediately before detachment, as shown in figure 8. We calculate the area of the regionin the ( r , z ) plane that is shaded in the figure, between the soap film and a radial line through thepoint of the soap film closest to the vertical axis, and rotate this region about the vertical axis toestimate the bubble volume. This is likely to be an underestimate, as the curvature of the filmaround the catenoidal neck is likely to increase during detachment.Figure 8 shows that for small contact angles the bubble size can reach almost 0 . . Thelimit in which the contact angle tends to zero appears to give a well-defined value for the maximumsize of this small satellite bubble. For contact angles of 90 ◦ and above there is no bubble becausethe point on the soap film nearest to the vertical axis is where the film touches the particle.There is a small effect of the choice of boundary conditions: in case 2, without a pressure force,the bubble is about 30% larger for θ = ◦ (although this difference decreases as the contact angleincreases). This is because, as noted above, in case 2 the instability that causes the film to detachoccurs earlier, when the line of contact is closer to the equator of the sphere.In case 1 with a fixed contact angle of 10 ◦ we varied the size of the spherical particle and againestimated the size of the trapped bubble. For a sphere of a given radius, we must choose betweena fixed particle mass (weight) or a fixed particle density. In the former case, the tension forceopposing the descent of the particle increases with particle radius, but since the sphere does notincrease in weight, it is brought to rest by the soap film once the particle exceeds a critical radius(in our case with R s ≈ . Bo ≈ R s greater than 0.1cm) does it pass through the soap film. Figure 9 showsthat the size of the bubble that is trapped is the same in both cases. So it is determined by the shapeof the soap film only, which in turn arises from the film meeting the sphere, of whatever radius, atthe given contact angle. Therefore the size of the trapped bubble increases with sphere size, sincethe film is more greatly deformed when the sphere is larger.There is also a small dependence of the size of the trapped bubble on the cylinder size. Asthe cylinder becomes larger, the sphere descends further before detachment, and so greater filmdeformation is possible. In addition, the pressure force is reduced in a larger cylinder, so the resultshould be closer to case 2. Thus, the trapped bubble is slightly larger if the cylinder radius is larger.9a) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) θ ’ sf (r , z ) sf φ (r , z ) (b) V o l u m e [ c m ] Contact angle [degrees] Case 1Case 2
Figure 8: (a) Close to the contact line between the soap film and the sphere, at the last iterationbefore detachment, we calculate the shaded area to estimate the volume of the small bubble that isleft behind. (b) The bubble volume depends strongly on the contact angle, depends only weaklyon whether we consider case 1 or case 2, and vanishes for contact angles greater than 90 ◦ .Figure 9 also shows the minimim and maximum sphere radius for which the sphere passesthrough the film, based on the predictions in §1. The lower bound (for constant particle density)is just below the numerical data while the upper bound (for constant particle mass) is about 20%above the upper limit of the data in that case. The bounds are predicated on the soap film pullingvertically upwards around the equator of the sphere, but despite this approximation appear to workwell.To validate our predictions, we compare with the image in Figure 1 of [7], which shows asphere of radius 0.16cm falling through a soap film trapping a bubble. (The cylinder radius andsphere mass are not recorded.) The bubble is trapped against the upper part of the sphere, butappears to be roughly hemispherical with radius 0.08cm, and hence a volume of 0.001cm . Thedata point, shown in figure 9, lies close to our prediction. We have explained the effect of contact angle on the forces that act on a spherical particle passingthrough a soap film. The duration of the interaction is determined by the contact angle and also theway in which the film is deformed; for example, with low contact angles the particle moves moreslowly, and stays in contact with the soap film for longer. Further, the interaction depends uponthe details of the experiment: greater deformation is induced by holding the film in a fixed circularwire frame than in a cylindrical tube, where it traps a bubble but where the outer circumference ofthe film is not fixed, such as in a soap-film meter [13]. In the latter case there is an additional forceon the particle due to the pressure in the bubble, but this is negligible in determining the dynamicsof the system.Analysing the shape of the soap film just before detachment allows us to predict the size ofthe small bubble that is formed when a particle passes through a film. The entrapment of this airand the formation of interface could play a role in determining the efficacy of using foams for the10 B ubb l e v o l u m e [ c m ] Sphere radius [cm]Constant densityConstant mass 0 0.005 0.01 0.015 0 1 2 3 4Cylinder radius [cm]
Figure 9: With contact angle θ c = ◦ in case 1, the volume of the bubble that is trapped by thefilm increases with the size of the particle and (inset) depends weakly on the size of the cylindercontaining the soap film. With fixed mass m = . g only spheres with radius up to R s = . cm pass through the film; with constant density ρ ≈ / cm only spheres with radius larger than R s = . cm pass through the film; the size of the trapped bubble is the same in both cases, indicatingthat it is determined by the geometry of the soap film. The vertical lines indicate the radius boundsestimated at the end of §1 and the solid circle is experimental data [7].suppression of explosions. We find that the bubble increases in size as the particle gets larger, andcan exceed 10mm .Extending our predictions to more general cases, such as oblique impact and non-sphericalparticles [6, 12], will require more computationally-intensive three-dimensional simulations. Acknowledgements
The late J.F. Davidson inspired us to work on this problem. We are also grateful to C. Raufaste foruseful discussions, and to K. Brakke for provision and support of the Surface Evolver software.SJC acknowledges financial support from the UK Engineering and Physical Sciences ResearchCouncil (EP/N002326/1).
References [1] D. Weaire and S. Hutzler.
The Physics of Foams . Clarendon Press, Oxford, 1999.[2] I. Cantat, S. Cohen-Addad, F. Elias, F. Graner, R. H¨ohler, O. Pitois, F. Rouyer, and A. Saint-Jalmes.
Foams - structure and dynamics . OUP, Oxford, 2013.[3] B.P. Binks and R. Murakami. Phase inversion of particle-stabilized materials from foams todry water.
Nature Mat. , :865–869, 2006.[4] B.B. Stogin, L. Gockowski, H. Feldstein, H. Claure, J. Wang, and T.-S. Wong. Free-standingliquid membranes as unusual particle separators. Science Advances , 4:eaat3276, 2018.115] M. Monloubou, M. A. Bruning, A. Saint-Jalmes, B. Dollet, and I. Cantat. Blast wave atten-uation in liquid foams: role of gas and evidence of an optimal bubble size.
Soft Matter , 12:8015–8024, 2016.[6] G. Morris, S.J. Neethling, and J.J. Cilliers. Modelling the self orientation of particles in afilm.
Minerals Engng. , :87–92, 2012.[7] A. Le Goff, L. Courbin, H.A. Stone, and D. Qu´er´e. Energy absorption in a bamboo foam. Europhys. Lett. , :36001, 2008.[8] L. Courbin and H. A. Stone. Impact, puncturing, and the self-healing of soap films. Physicsof Fluids , 18:091105, 2006.[9] I.T. Davies and S.J. Cox. Sphere motion in ordered three-dimensional foams.
J. Rheol. , :473–483, 2012.[10] S.A. Cryer and P.H. Steen. Collapse of the soap-film bridge: quasistatic description. J. Coll.Interf. Sci. , :276–288, 1992.[11] M. A. C. Teixeira, S. Arscott, S. J. Cox, and P. I. C. Teixeira. When is a surface foam-phobicor foam-philic? Soft Matter , 14(26):5369–5382, 2018.[12] I.T. Davies. Simulating the interaction between a descending super-quadric solid object anda soap film.
Proc. Roy. Soc. A , :20180533, 2018.[13] C.-H. Chen, A. Perera, P. Jackson, B. Hallmark, and J.F. Davidson. The distortion of ahorizontal soap film due to the impact of a falling sphere. Chem. Eng. Sci. , 206:305–314,2019.[14] H. N. Oguz and A. Prosperetti. Bubble entrainment by the impact of drops on liquid surfaces.
J. Fluid Mech. , 219:143179, 1990.[15] S. T. Thoroddsen, K. Takehara, T. G. Etoh, and Y. Hatsuki. Puncturing a drop using surfac-tants.
J. Fluid Mech. , 530:295304, 2005.[16] N.D. Robinson and P.H. Steen. Observations of singularity formation during the capillarycollapseand bubble pinch-off of a soap film bridge.
J. Coll. Interf. Sci. , :448–458, 2001.[17] K. Brakke. The Surface Evolver. Exp. Math. , :141–165, 1992.[18] S. Hutzler, D. Weaire, S.J. Cox, A. Van der Net, and E. Janiaud. Pre-empting Plateau: thenature of topological transitions in foam. Europhys. Lett. ,77