On the impulsive generation of drops at the interface of two inviscid fluids

K.K Tjan, W.R.C Phillips

Abstract

The numerical simulation of the deformation of an inviscid fluid–fluid interface subjected to an axisymmetric impulse in pressure is considered. Using a boundary integral formulation, the interface is evolved for a range of upper-fluid and lower-fluid density ratios under the influence of inertial, interfacial and gravitational forces. The interface is seen to evolve into axisymmetric waves or droplets depending upon the density ratio, level of surface tension and gravity. Moreover, the droplets may be spherical, tear shaped or elongated. These conclusions are expressed in a phase diagram of inverse Weber number We−1 versus Atwood number At at zero gravity, i.e. with the Froude number Fr−1→0, and complement the earlier findings of Tjan & Phillips, who present a phase diagram of We−1 versus Fr−1 for the case in which the upper fluid has zero density. They too report tear-shaped droplets; however, while, in their paper, they form as a result of gravity, those reported here form as a result of surface tension. It is also found that the pinch-off process which effects drops remains of the power-law type with exponent 2/3 irrespective of the presence of gravity and an upper fluid. However, the constant Embedded Image that relates the necking radius to the time from pinch off, which is universal in the absence of gravity and an upper fluid, is affected by the presence gravity, an upper fluid and the class of drops which form.

Keywords:

1. Introduction

In a recent study, Tjan & Phillips (2007; henceforth TP) considered the deformation of a free surface subjected to an axisymmetric impulse in pressure. Depending upon the level of gravity and surface tension, expressed through the inverse Froude, Fr−1, and Weber, We−1, numbers (later defined), the surface was seen to evolve into axisymmetric jets, waves or drops. The liquid under consideration was inviscid and the density of fluid above it was assumed to be zero, or at least sufficiently small relative to that of the liquid to play no role in the dynamics of the evolving surface. Our purpose here is to relax that restriction by introducing an adjacent layer whose density may differ from zero. We then characterize the density ratio of the adjacent fluid layers with an Atwood number, At, and, given Fr−1 and We−1, determine what role At has on the evolving interface.

The pressure impulse is assumed Gaussian, this being an approximation for the pressure field induced by focused ultrasound (Kino 1987). Moreover, the impulse is assumed to arise from a burst of ultrasound, the burst being long with respect to that of the period of the ultrasound but small with respect to the time scale of the evolving interface. The impulse is applied normal to the initial interface.

Liquid films excited by ultrasound have long been known to emit droplets, with studies dating from Wood & Loomis (1927) and Lang (1962; in the frequency range of 10–800 kHz) who report the formation of fog on the surface. On the other hand, focused ultrasound can be used to eject isolated droplets from the surface, and, to achieve this, Elrod et al. (1989) found bursts of ultrasound (in the range of 5–300 MHz) preferable to continuous forcing. In fact, free surfaces excited by focused ultrasound are observed to exhibit a variety of phenomena, including standing axisymmetric waves, ejected individual drops, multiple drops and a fog of tiny drops. All are evident in the (water to air) photographs of Sakai et al. (2003).

Considerable attention has been paid to the formation of drops and most particularly those induced by gravitational or periodic accelerative forcing. These include viscous dripping (Brenner et al. 1997; Wikes et al. 1999), liquid jet breakup (Lin & Reitz 1998) and parametric excitation (Faraday 1831; James et al. 2003), albeit at forcing frequencies well below those of ultrasound. On the other hand, little attention has been given to impulsively generated drops and, since this construct is significantly different, we expect the dynamics which affect droplet formation to likewise differ. We do not expect details of drop pinch off to differ, however, as these are probably ubiquitous in all studies.

Elrod et al. (1989) appear to be the first to have studied impulsively generated isolated drops and approach the topic from the viewpoint of ink-jet printing. Their work comprised both experiments and numerical modelling (to predict the time evolution of the surface). But they consider only water into air; there appear to be no studies of ultrasonically excited drops into fluids other than air.

Pinch off, on the other hand, has been studied on its own (e.g. by Eggers 1993; Brenner et al. 1996; Monika & Steen 2004), in the context of ink-jet printing (e.g. by Day et al. 1998; Basaran 2002; Leppinen & Lister 2003) and in the context of impulsively excited drops (TP).

In forming their model, TP include surface tension and gravity but ignore viscosity, arguing that because viscous terms (in the equation of motion) are small relative to their surface tension counterparts, they play little role until pinch off. They further note that subsequent to the pressure impulse, which gives rise to a vortex sheet on the interface, there is no mechanism to create further vorticity or sufficient time for vorticity in the sheet to diffuse, so that the interior fluid remains irrotational. Here, we follow suit and investigate the evolution of the interface of two unbounded inviscid irrotational fluids of different densities following an axisymmetric impulse in pressure, subject to the interplay of inertial, interfacial and gravitational forces.

Our problem involves solving the Laplace equation (2.1) subject to the Bernoulli equation (2.2) on the interface. For computational purposes, equation (2.1) is best recast into a boundary integral form and we do so in a manner akin to that of Baker et al. (1984). This technique has been recently employed by others (e.g. Hou et al. 1994, 1997, 2001; Nie & Baker 1998; Nie 2001), albeit on either a finite or a periodic domain. Here, however, the domain is semi-infinite and details of how we handle it are given in TP and outlined in §3. Results are given in §4, followed by a discussion of the finite-time singularity which occurs at pinch-off in §5 and remarks in §6.

2. Formulation

Our model is formulated as an initial-value problem in an unbounded axisymmetric domain. The governing equations and boundary conditions are introduced in §2a with non-dimensionalization discussed in §2b. Conversion of the governing equation to a Fredholm integral equation of the second kind is outlined in §2c.

(a) Governing equations

Consider an axisymmetric Embedded Image domain Embedded Image, where Embedded Image is radial and Embedded Image vertical, composed of two subdomains Embedded Image1 and Embedded Image2, which contain inviscid incompressible fluids of densities ρ1 and ρ2, respectively. Embedded Image1 and Embedded Image2 initially occupy Embedded Image and Embedded Image and are separated for all time Embedded Image by a sharp interface Γ, along which the coefficient of surface tension is σ.

Assuming irrotationality, velocity potentials φi may be defined in each Embedded Imagei, with the velocity vector given by Embedded Image, for i=1, 2, while incompressibility Embedded Image demands that φi satisfy the Laplace equationEmbedded Image(2.1)in each fluid. Under conservative body forces, the unsteady Bernoulli equationEmbedded Image(2.2)then precisely enforces conservation of momentum.

When a pressure field is applied on Embedded Image=0 over a short time scale with respect to the time scale Embedded Image of the evolving surface, the pressure and, from (2.2), the velocity potential φ1 (since Embedded Image) register as impulses relative to Embedded Image and thereby provide an initial condition to the problem.

The initial-value problem is completed by boundary conditions which require for all Embedded Image that Embedded Imagei vanish at infinity, while the usual kinematic and dynamic boundary conditions apply on the interface Γ. The former requires that the velocity normal to the fluid surface be identical to that of the fluid particle at the surface; the latter is the Laplace–Young condition (Young 1805), which requires that the pressure jump across Γ be proportional to the surface curvature Embedded Image.

In order to effect the Laplace–Young boundary condition on Γ, we introduce the scaled velocity potential φ and the dipole strength μ, which are defined on Γ asEmbedded Image(2.3)Accordingly, we note thatEmbedded Imagewhere Embedded Image is the direction normal to Γ pointing from fluid 1 to fluid 2, with Embedded Image the tangential direction along Γ. This demands that the velocities of both fluids normal to Γ be continuous while permitting a possible jump in tangential velocities, a necessary condition to satisfy the kinematic boundary condition on Γ. We then see on Γ that Embedded Image, Embedded Image and Embedded Image, where At is the Atwood ratio of the densities of the two fluids, viz.Embedded Image(2.4)

(b) Non-dimensionalization

We introduce velocity potential and length scales Φ and Embedded Image as per TP, in which Φ is a measure of the peak velocity potential and Embedded Image is a measure of the radial width of the impulse. Consequently, time and velocity scales follow as Embedded Image and ΦEmbedded Image−1, respectively, allowing us to write Embedded Image, Embedded Image, Embedded Image, Embedded Image, Embedded Image and Embedded Image, where r, z, κ, ϕ, μ, t and un are dimensionless quantities.

In terms of the scaled velocity potential and dipole strength, and on substituting the Laplace–Young boundary condition Embedded Image, the unsteady Bernoulli equation (2.2) valid on Γ then becomesEmbedded Image(2.5)Two additional dimensionless parameters are evident in (2.5): the Weber number, Embedded Image, which measures the relative importance of inertial to interfacial forces and the Froude number, Embedded Image, which measures the relative importance of inertial to gravitational forces. Note that μ→2ϕ in the limit At→0, leavingEmbedded Image

(c) Boundary integral equation

The Laplace equation is recast into a boundary integral equation yielding a Fredholm integral equation of the second kind as (see TP)Embedded Image(2.6)where the directional Green's function to (2.1) isEmbedded ImageEmbedded Imageand E(k) and K(k) are, respectively, complete elliptic integrals of the first and second kind.

Accordingly, the azimuthal component of the vector potential is given byEmbedded Image(2.7)whereEmbedded ImageFinally, from Bθ, we define a pseudo-streamfunction, ψ, asEmbedded Image(2.8)

Then, knowing the value of the triple (r, z, ϕ) at any instant, we can solve the Fredholm integral (2.6) for the dipole strength μ and subsequently use (2.7) to evaluate Bθ(s). Then, with the knowledge of the velocities from (2.8), we use the Bernoulli equation (2.5) to evolve ϕ forward in time, while enforcing the kinematic boundary condition to evolve the surface forward in time. The process can then be repeated with the updated ϕ and new surface profile as input.

Finally, to set the process in motion, we require an initial condition for ϕ on z=0; this may be any continuous C2 function that vanishes as r→∞, but for comparison with TP we set Embedded Image.

3. Numerics

Beyond the introduction of a further parameter, the formulation for handling two fluids is not greatly different from that with one fluid and presented no new numerical challenges. The numerical procedure thus follows exactly that in TP and hence only an outline is presented here. To proceed, we first introduce a mapping from the semi-infinite physical domain s[0,∞) to a finite computational domain η[−1,+1]. The dependent variables {r, z, ϕ, ψ, μ} are then expressed in terms of basis functions of order N over η and coefficients for {r, z, ϕ, ψ} are determined using collocation, while those for μ are found via a Galerkin method.

In order to follow the evolution of the (initially flat z=0) surface, we place on it N Lagrangian markers, each located by radial r(s) and vertical z(s) coordinates, and monitor them. The markers, together with the velocity potential ϕ(s), form the triple {r, z, ϕ} that defines the primary dependent variables of the system and these are supplemented by the further dependent variables ψ and μ. The numerical solution involves marching these variables forward in time.

For example, given the normal and tangential velocities u=(un, ut) at the surface, the location of the markers r and z are evolved kinematically asEmbedded Image(3.1)We note that while un is unambiguously given by (2.8), the choice for ut remains arbitrary as the Lagrangian markers may be advected with any choice of tangential velocity without changing the surface shape. In other words, while the Lagrangian markers necessarily have the same normal velocities as the fluid on either side of the interface, their tangential velocities will in general be different. The following choice for the tangential velocity is used, viz.Embedded Imageas it has the desirable property (see TP) that the distance between markers is preserved. Accordingly, such a choice for ut is especially useful for a Chebyshev collocation implementation, since now s for each collocation point is fixed in time. Various values for the basis function number, and thus the number of mesh points, N, were explored but N=64 was found to provide an adequate balance between resolution and computational effort. In fact, the results were little changed by higher values of N, suggesting that the calculation had converged. Complete details are given in Tjan (2007).

The evolution of ϕ is via the Bernoulli equation (2.5) with the partial derivative written as a material derivative, i.e. Embedded Image, yieldingEmbedded Image(3.2)which reduces to that of TP when At=1. (Note that (3.2) in TP contains a typographical error.)

Values of {r, z, ϕ} are updated using a fourth-order Runge–Kutta scheme, after which they are used to calculate the dipole strength by solving the Fredholm integral (2.6).

4. Results

Our problem is in terms of the three-dimensional parameter space (We, Fr, At), which is somewhat unwieldy, hence, to reduce the dimensionality, we hold Fr−1 constant and vary At and We. Specifically, we restrict our attention to the interplay between inertial and surface tension forces in the absence of gravity (i.e. Fr−1→0), while varying At over its full range, namely At∈[−1,+1]. TP, on the other hand, set At=1 and investigated the role of surface tension versus gravity over a range of Weber and Froude numbers. For reference, their phase diagram for At=1 and Fr−1>0 is presented in figure 1. It depicts regions where spherical drops, inverted tear-shaped drops and axisymmetric waves form; it also highlights a region, specifically for We−1<0.045, where surface tension is too weak to overcome numerical instability. The limit Fr−1→0 is not shown, but spherical drops occur in the range 0.045<We−1<0.105 and their size scales inversely with We−1. Once We−1>0.105, however, surface tension is sufficiently dominant to impede the evolving axisymmetric wave that precedes the drop, causing it to collapse upon itself without forming a drop.

Figure 1

Phase diagram of We−1 versus Fr−1 at At=1 from TP showing regions of instability, spherical drops, inverted gravity tear-shaped drops and axisymmetric standing gravity waves. Not shown is the double limit Fr−1→0, We−1→0 at which axisymmetric jets form.

(a) Fr−1→0, At∈[−1,+1]

Here, we find that spherical drops also form when At<1 for some We−1, as shown in a sequence of snapshots (for At=0.75) in figure 2. Accordingly, the size of the drop continues to scale inversely with the Weber number, although the maximum height attained by the drop is affected by At. But spherical drops do not form for all At and certainly not for all We−1 investigated. Indeed, as At is reduced from unity, which means that the density of the upper fluid is increasing relative to that of the lower fluid, topologies beyond those reported by TP are observed. A phase diagram is given in figure 3.

Figure 2

Snapshots of the interface with At=0.75, We−1=0.08 and Fr−1=0 showing the formation of a spherical drop. Here, the broad features are similar to the case We−1=0.08, Fr−1=0 and At=1 given by TP in so far as the size of the drop is concerned, which is comparable, whereas the maximum height attained by the drop is here lower. (a) t=0.00, (b) t=1.30, (c) t=2.60, (d) t=3.90, (e) t=5.20 and (f) t=6.50.

Figure 3

Phase diagram of At versus We−1 showing regions of instability, spherical drops, inverted capillary tear-shaped drops, pancake drops and axisymmetric standing capillary waves.

First, we observe for all At that an upper bound in We−1 exists beyond which no pinch off to drops is observed. Rather, in this range of We−1 and At, the interface oscillates as an axisymmetric wave about the mean elevation and is eventually damped. Snapshots of the free surface of a typical case are shown in figure 4. Such behaviour is reminiscent of a situation identified by TP where gravity acted to damp the axisymmetric wave, only here gravity is absent and oscillations are damped over time by surface tension. Thus, while TP observed axisymmetric gravity waves, we observed axisymmetric capillary waves. Furthermore, as the density differential is progressively reduced to the point that At becomes negative, i.e. the heavier fluid on top, the upper bound in We−1 is reduced. This suggests that although the level of surface tension required to prevent drop formation has reduced, there is, as we might expect physically, an increased inertial restraint from the upper fluid.

Figure 4

Snapshots of the evolution of the interface for At=0.5 and We−1=0.10, with Fr−1=0, in which no drop is formed in contrast to the case in figure 2, with At=0.75, where a spherical drop is formed. (a) t=0.08, (b) t=1.48, (c) t=2.88, (d) t=4.28, (e) t=5.68 and (f) t=7.08.

On lowering We−1 to just below the no-drop upper bound, we find for At∈(−1,+1) that tear-shaped, rather than spherical, drops form. Tear-shaped drops were also observed by TP, but the dynamics which gave rise to them is there different. Specifically, the evolving interface there collapsed onto itself under gravitational forces, whereas here, in the absence of gravity, the evolving interface is slowed jointly by inertial effects from the upper fluid coupled with surface tension, which eventually causes necking at the base of the tear-shaped drop. Snapshots of the free surface of such a case are depicted in figure 5. Here, there is no collapse, so we might say that while TP observed gravity tears, we observed surface tension or capillary tears.

Figure 5

Time evolution of the interface for We−1=0.10 and At=0.75, with Fr−1=0, showing the formation of an inverted tear-shaped drop due to inertial and interfacial effects. Note that the drop is not formed when the interface is moving down as is the case when tear-shaped drops form under gravity. (a) t=0.18, (b) t=1.58, (c) t=2.98, (d) t=4.38, (e) t=5.78 and (f) t=7.18.

Further reductions in We−1 expose, mainly for At>1, the formation of spherical drops and ultimately another class of drops not previously reported. For reasons evident in figure 6, we refer to this class as ‘pancake’ or ‘elongated’ drops. Here, surface tension would appear to have less influence than inertia to the extent that the upper-layer fluid grossly distorts the shape of the drop.

Figure 6

Time evolution of the interface for We−1=0.03 and At=0.50, with Fr−1=0, showing the formation of a pancake or elongated drop. (a) t=0.44, (b) t=1.04, (c) t=1.64, (d) t=2.24, (e) t=2.84 and (f) t=3.44.

Eventually, we encounter a lower bound in We−1 for all At∈[−1,1], below which surface tension is insufficient to overcome numerical instabilities and the calculation breaks down.

Finally, in viewing figure 3, we should point out that although the boundary between the formation of axisymmetric waves and tear-shaped drops is clearly defined, the interface between other classes is less clear. For example, tear-shaped drops become more spherical (or elongated at some At) in the vicinity of the boundary and there is a range of We−1 near the boundary in which they are not distinctly one class or the other. To that end, the boundaries (other than the axisymmetric wave one) indicated on figure 3 should be viewed as indicators of where transition occurs rather than rigid demarcations.

5. Scaling

Pinch-off is an example of a singularity which is formed in a finite time, the bifurcation being a topological singularity where the surface self-intersects at time t=t0. Of particular interest is how the system approaches this singularity and efforts to understand it began with an experimental study of a spherical pendant drop evolving from the end of a nozzle by Peregrine et al. (1990).

Mathematically, the self-intersection of the surface may be precisely phrased as r(s≠0)=0, where the radial coordinate vanishes for some location s≠0. To proceed, therefore, we define a radius rmin and track it as the surface evolves. Thus, let rmin be the radial coordinate of the point on the surface satisfying the two conditions dr/ds=0 and d2r/ds2>0, which decree that there will be a solution for rmin only after such time that the surface becomes vertical at some location.

TP found that rmin exhibits a power-law type singularity of the form (t0t)γ, where γ≈2/3. They further note, as do Keller & Miksis (1983), that such behaviour can be argued on dimensional grounds, on the assumption that the behaviour should, near the singularity time, be independent of initial conditions and thus that the relevant (dimensional) parameters are σ, ρ1, Embedded Image and Embedded Image0Embedded Image. From this set of parameters, the Buckingham pi-theorem dictates that only one dimensionless group can form and must itself be a universal constant, say Embedded Image. When further parameters play a role, a second density ρ2 and gravity g, for example, further dimensionless groups arise rendering Embedded Image a functional as (say) Embedded Image. Then, withEmbedded ImageEmbedded Imagewe have for all g thatEmbedded Image(5.1)Equation (5.1) thus recovers TP's expression in the double limit Embedded Image, at which they find Embedded Image. So of interest here is how non-zero values of Embedded Image affect Embedded Image. To that end, we study, in §5a,b, respectively, the limits Embedded Image, Embedded Image and Embedded Image, Embedded Image.

(a) Fr−1→0 with variable At and We

As is evident from (5.1), the pinch-off scaling law is not affected by the density ratio Embedded Image, although Embedded Image can affect the magnitude of Embedded Image. Our purpose here is to determine Embedded Image as a function of density ratio and, to expose that relationship, we exclude Embedded Image by working in the Fr−1→0 limit. In addition, because Embedded Image, we prefer, with no loss of generality, to use At as the independent variable (because At∈[−1,1] (2.4)) and plot Embedded Image, which we do in figure 7.

Figure 7

Plot of the functional Embedded Image with At in the zero-gravity limit Fr−1→0 for the observed classes of drops: square, spherical drops; plus, tear-shaped drops; diamond, pancake-shaped drops.

For each class of drops, we see that Embedded Image decreases monotonically with increasing At and recovers TP's universal constant Embedded Image only for spherical drops when At=1. Observe also that Embedded Image is lowest for spherical drops and successively higher, at a given At, for tear-shaped and elongated drops, although only tear-shaped drops occur over the full range of At (figure 3). Thus, in applying (5.1), we must be aware that Embedded Image is affected not only by the density ratio but also by the topology of the emerging drop.

(b) At=1 with variable We−1 and Fr−1

We turn now to the case Embedded Image and note that although TP explored the phase space for Embedded Image with variable We−1 and Fr−1, they explored the near singularity scaling relationship only in the Embedded Image limit. Here, we are interested in the role of gravity, so we set Embedded Image and explore Embedded Image. To ascertain this dependence, we plot Embedded Image against Embedded Image in figure 8 and for clarity depict spherical drops and tear-shaped drops separately, in figure 8a,b, respectively. As in TP's case, the r.h.s. of figure 8a,b indicates that Embedded Image is essentially independent of Embedded Image, indicating that in spite of gravity being present it plays no dynamical role in pinch off for either spherical or tear-shaped drops. However, as we found with the density ratio, gravity does influence the magnitude of Embedded Image, causing a significant departure from its zero-gravity single-layer universal value; indeed, we here find Embedded Image. The value of Embedded Image is also affected by the topology of the drops which form, being at the lower end for spherical drops and higher end for tear-shaped ones.

Figure 8

Plot of Embedded Image against Embedded Image, with At=1. (a) Spherical drops: We−1=0.05, Fr−1=(0.15, 0.3, 0.35, 0.375, 0.4) (plus, cross, star, square, filled square, respectively); We−1=0.05, Fr−1=(0.25, 0.275, 0.3) (circle, filled circle, triangle, respectively); We−1=0.05, Fr−1=0.2, filled triangle; We−1=0.05, Fr−1=(0.001, 0.01, 0.1) (inverted triangle, filled inverted triangle, diamond, respectively); We−1=0.05, Fr−1=0.09, filled diamond. (b) Tear-shaped drops: We−1=0.05, Fr−1=(0.5, 0.6, 0.65) (plus, cross, star, respectively); We−1=0.06, Fr−1=0.5, square; We−1=0.07, Fr−1=(0.25, 0.3, 0.40, 0.45) (filled square, circle, filled circle, triangle, respectively); We−1=0.08, Fr−1=(0.15, 0.17, 0.175, 0.2, 0.25, 0.325) (filled triangle, inverted triangle, filled inverted triangle, diamond, filled diamond, pentagon, respectively); We−1=0.09, Fr−1=(0.1, 0.2, 0.25) (filled pentagon, bold circle, bold circle filled first quadrant, respectively); We−1=0.10, Fr−1=(0.025, 0.05, 0.15) (bold circle filled fourth quadrant, bold circle filled first and fourth quadrants, bold circle filled third quadrant, respectively).

For Embedded Image, on the other hand, there would appear from figure 8 to be a noticeable gravitational influence, but our conclusion is that there is not. Rather, numerical uncertainties in deducing rmin and t0t as each approaches zero are compounded in evaluating rmin(t0t)−2/3, leading to specious results in the double limit. To pursue this further, we avoid evaluating the double limit and instead plot Π1 against Π2, as shown in figure 9. Here, there is less sensitivity as rmin and t0t approach zero and the data collapse about distinct straight lines of slope Embedded Image representative of the class of drops which form.

Figure 9

Plot of rmin versus Embedded Image after rescaling by Fr and We as suggested by Embedded Image and Embedded Image, with At=1. The data collapse into two distinct curves, according to whether a spherical or a tear-shaped drop is formed: plus, spherical; cross, tear shaped.

6. Remarks

We continue, in this work, a study by TP who consider the evolution of a free surface subject to an axisymmetric impulse in pressure. Their research was part of a study of haemorrhage in the lung caused by ultrasonic imaging and, in particular, to explore a non-thermal, non-cavitational damage mechanism. The mechanism they consider is built around the notion that ultrasound focused near a tissue–liquid interface acts to expel tiny droplets of blood or other fluids which then puncture the soft bubble-wrap-like air-filled sacs (alveolar) of the lung pleural surface. Moreover, they further show that droplets can indeed be ejected over the range of intensity levels, measured by a dimensional quantity denoted the mechanical index, employed in clinical ultrasonography. Of course, the interface is not always liquid–gas, rather the plural surface may at times be coated by a layer of mucus, and for that reason we here extend the work to include fluid–fluid interfaces with a broad range of density ratios. With that as background, however, we preferred not to restrict ourselves solely to the conditions associated with lung haemorrhage, but rather to look in general at impulsively generated waves and drops. Indeed, the phenomenon has the tantalizing prospect that it could be related to other physical scenarios such as ink-jet printing. Our findings also shed light on what might well be a ubiquitous finite-time singularity at drop pinch off, as we find that parameters other than those evident in zero gravity, liquid–air pinch-off, affect only the constant relating the necking radius to the time from pinch off, not the scaling law itself.

Acknowledgments

This work was supported by the National Institutes of Health grant R01-EB02641 (formerly HL58218). Our thanks to Jonathan Freund, Gustavo Gioia and Jonathan Higdon for their interest and helpful comments.

Footnotes

    • Received November 6, 2007.
    • Accepted January 7, 2008.

References

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