Underwater bubble pinch-off: transient stretching flow
At the point of pinch-off of an underwater air bubble, the speed of water rushing in diverges. Previous studies that assumed radial flow throughout showed that the local axial shape is two smoothly connected, slender cones that transition very slowly (logarithmically) to a cylindrical segment. Our simulations show that even with initially radial flow, a transient vertical flow develops with comparable speeds. Bernoulli pressure draws water into the singularity region while incompressibility forces it away from the neck minimum, generating significant vertical flows that rapidly slenderize and symmetrize the collapse region. This transition is due to a different mechanism, occurring much faster than previously expected. Vertical flows dictate the neck shape evolution.
pacs:47.55.df, 02.40.Xx, 47.11.Hj
Introduction–Mathematical models of physical processes often predict the formation of a singularity. Initially smoothly distributed and finite physical quantities, such as velocity and pressure, diverge in a finite time. Examples include supernovae, gravitational collapse into black holes, and the pinch-off of a fluid drop. Most of the first-known singularities exhibited self-similarity and universality. Self-similarity indicates that the system, as it approaches the critical time is nearly identical at different times except for some rescaling by a function of time to singularity for example a system radius. Universality indicates that this function is independent of initial conditions and boundary conditions, so that every occurrence of a given type of singularity occurs exactly the same way. An example of a universal singularity is the pinch-off of a water drop in air, in which surface tension smooths out azimuthal shape vibrations Ting and Keller (1990); Bertozzi et al. (1994); Shi et al. (1994); Eggers (1997); Day et al. (1998). We focus on the opposite phenomenon, namely the pinching off of an air bubble underwater (Fig. 1). This commonplace phenomenon exhibits neither self-similarity nor universality, and its low energy makes it amenable to study using high-speed photography on a table-top setup Burton et al. (2005); Keim et al. (2006); Thoroddsen et al. (2007).
A nozzle submerged underwater blows a buoyant air bubble, which eventually detaches from the nozzle. Initially, surface tension dominates and the bubble neck shape is quadratic, with a slight asymmetry due to the hydrostatic pressure gradient (Fig. 1a). As the water rushes in faster, inertial forces overtake surface tension as the dominant driver, while viscosity continues to be negligible. The neck shape takes the form of a hyperbola of rotation Longuet-Higgins et al. (1991). As the system approaches the singularity, the two cones become more slender. Previous theory and simulation focused on the long-and-slender regime, where vertical flows are negligible a priori, so that individual vertical cross sections evolve independently Longuet-Higgins et al. (1991); Oguz and Prosperetti (1993); Gordillo et al. (2005); Eggers et al. (2007); Gekle et al. (2009). There, the approach to cylindrical is very slow (logarithmic in We focus on the transient regime where the vertical flows dominate the shape evolution and occur much faster. Finally, at small length scales, the singularity is pre-empted by one of two effects. One is airflow in the neck, which becomes important when the neck aspect ratio reaches or about 30 for air-water systems Gordillo and Fontelos (2007); Gekle et al. (2010). Second, if the neck is not perfectly axisymmetric, azimuthal vibrations are generated Keim et al. (2006); Schmidt et al. (2009); Turitsyn et al. (2009). As pinch-off proceeds, the amplitudes of the excited vibrations remain constant. When the neck radius shrinks down to the length scale of the vibrations, the sides contact before the void is filled.
These vibrations are what excludes the bubble pinch-off from being self-similar and universal. Analogous instances of memory-encoding vibrations arise, for example, in the implosion of shock fronts Whitham (1957) and implosion of spherical voids. All such systems have two commonalities: damping forces become negligible compared to inertia, and the flow is predominantly inward collapse. The second condition, specifically radial flow, was assumed in previous studies of azimuthal vibrations in bubble pinch-off Schmidt et al. (2009); Turitsyn et al. (2009), so the behavior could be quite different with vertical flow present.
In addition, we are interested in whether there is memory of the axial shape. This type of memory is present when water pinches off in a bath of viscous oil: the axial curvature persists until pinch-off Doshi et al. (2003). In that system, the viscosity dampens any azimuthal vibrations. Since the flow is purely radial, however, each cross section evolves independently. Incidentally, the collapse velocity is independent of height, and the curvature persists.
Previous theory and simulation has only studied air bubble pinch-off in long-and-slender regimes, but pinch-off in other regimes is relevant. Inviscid collapse of arbitrarily shaped voids is important, for example, in cavitation bubbles generated by complex flows, ultrasound, or laser pulses Lim et al. (2010); in hull breach; or in an object plunging into water Bergmann et al. (2009). The collapse of these voids is also controlled by inertia, but they are not necessarily in the long-and-slender limit, and significant vertical flows may be present. With that in mind, we simulate shapes with very squat and/or asymmetric cones under generic velocity profiles. Remarkably, no matter how exaggerated the initial conditions, strong vertical flows rapidly drive the shape to two slender, symmetric cones (assuming axisymmetry). This happens within observable time frames, before the singularity is pre-empted. Therefore, every axisymmetric cavity pinches off with the well known long-and-slender behavior.
Problem formulation–We use the fact that the inertia of the water inflow dominates viscosity, surface tension, and gravity in the limit the minimum radius Therefore, we set those ignorable terms to zero. The exterior velocity field is incompressible irrotational and decays to zero far away. We also assume that the air in the bubble is dynamically passive, with a uniform pressure whose value ensures constant bubble volume in time.
Being curl-free, the velocity can be described by a scalar potential, defined as Equating the relevant stresses on the interface gives a non-linear differential equation for first-order in time and space:
The kinematic condition, which says that surface elements are advected by determines how the interface evolves. Since the exterior flow is incompressible, i.e., Laplace’s equation holds. This allows to be solved everywhere in the exterior given on the surface.
We use the co-moving derivative in equation 1 to evolve for discrete fluid elements. Also, we only need on the interface, so we use the Green’s integral form for on the surface:
where is a fixed point on parameterizes the surface, and is the unit surface normal at pointing into the bubble Pozrikidis (1997).
With both governing equations defined on the interface, the formulation reduces to 2 dimensions. We further reduce to 1 dimension by assuming axisymmetry. At each time step, we use cubic splines to interpolate the discrete surface nodes . Then, for each we allow to take the place of in equation 2. We use Gaussian quadrature to perform the integral over segments connecting adjacent spline midpoints, with the first and last segments being only half splines. This gives the needed relationship for in terms of . Using we advect the nodes using and evolve using equation 1, completing the cycle.
In order to accurately resolve the pinch-off dynamics, we found it
important to use a node distribution scheme that maintains a gradual
variation in the spacing between node points and that continually
adds nodes in the neighborhood of the minimum (see Fig. 2).
At each moment, if the vertical distance between nodes
exceeds a maximum spacing in the
region where the neck radius is less than we
add a new node at the midpoint of that spline. After experimenting
with several functions for we found
allows the simulation to accurately track the dynamics
We begin the simulation by specifying an interface shape and velocity field. In general, there are an infinite number of each. Here, we examine generic smooth distributions. To expedite the computation, we do not prescribe a purely quadratic shape profile at but instead use the result from previous studies that a slender quadratic neck evolves into a hyperbola, and prescribe:
where is the opening angle of the cone, and is the radius of the end cap. We experimented with different smoothly-varying initial velocity fields, with corrections in the end caps to preserve bubble volume. One such field is given by the normal velocity on the surface: Incompressibility then determines the tangential component of the velocity. The is a correction term necessary for bubble volume conservation, where is the arc length along the surface from the neck minimum and is a constant. The parameter specifies the initial orientation of the vertical velocity in the neck region. If the vertical velocities are directed toward the midline, compressing the neck aspect ratio into a more squat shape. This configuration, though, is unstable. Very soon the vertical velocities near the midline flip to orient away from the midline and proceed to stretch the neck. If, on the other hand, the system is initialized with the vertical velocities point away from the midline and continue to point away, stretching the neck for the entire collapse. After their respective transients, however, both situations fall into the same dynamics, only differing by a time offset. Therefore, for the remainder of the paper we consider an initial radial flow, which corresponds to To specify a radial flow for up-down asymmetric shapes, we use an explicit radial flow with end-cap corrections:
Results–Small values of with radial flow correspond to the regime of the slender-body approximation employed in previous works. Our results (not shown here) agree quantitatively with their results in this regime. Here, we focus on large values of Since the bubble break-up experiment shows a slight up-down asymmetry, we first examine a severely up-down asymmetric shape. We produce such an initial state (Fig. 2a) by using different values of for the top and bottom portions of the bubble instead of prescribing a hydrostatic pressure gradient. The upper portion of the bubble is chosen to have the larger opening angle. We set the initial flow to be radial in the neck region. Figs. 2b-c show the shape evolution. Initially, the minimum shifts toward the smaller cone. As the minimum radius decreases, the upper portion of the bubble surface rolls while the lower portion more or less preserves the same profile. As a consequence, the profile near the minimum quickly approaches a symmetric shape. Simulations starting with various choices for up-down asymmetry yield the same qualitative outcome (not shown). We have found that an up-down asymmetric initial state always proceeds through a 3-stage evolution. First, it evolves into a symmetric profile as described. Second, the symmetric cones become slender near the pinch-off point. Finally, there is a slow transition to a cylinder. Hereafter we will focus on the second stage, which connects the asymmetry in experiment to prior works focusing on the last stage Gordillo et al. (2005); Eggers et al. (2007); Gekle et al. (2009).
Fig. 3 displays the evolution of a symmetric,
large initial opening-angle shape, initialized
with radial flow. Fig. 3b shows that
the shape quickly evolves into a nearly-cylindrical segment connecting
two cones. As the pinch-off proceeds and we zoom in close to the minimum,
the shape becomes long and slender. The effective half opening-angle
of the innermost profile in each sub-figure is (a)
We show the continuous evolution of the velocity distributions for by plotting the maximum of versus the aspect ratio (Fig. 6a, uppermost curve). Roman numerals indicate corresponding stages between Fig. 5 and Fig. 6a. Just after a stretching flow builds in strength before the shape has time to react (I). After reaching a peak vertical-to-radial ratio (II), the neck continues to stretch (III). The exterior flow decays slowly to radial implosion. Starting with different initial opening angles yields the same qualitative behavior. Smaller initial opening angles generate smaller maximum in the transient regime (I-II). Eventually all initial conditions enter the slender-body regime (IV) where we find the slow evolution towards a perfect cylinder described in previous works Gordillo et al. (2005); Eggers et al. (2007); Gekle et al. (2009).
Discussion–While our analysis has focused on axisymmetric dynamics, the conclusions should remain relevant when the neck shape has a slight azimuthal asymmetry since prior linear stability studies have shown that the asymmetry does not grow rapidly but instead persists as vibrations of fixed amplitude. In the opposite regime when the neck shape is strongly distorted from axisymmetry, experiments show the bubble neck rips itself apart in a series of violent, apparently discrete jerks Keim et al. (2006). The force balance controlling this break-up mode remains an open question.
Conclusions–We showed here that strong, transient vertical flows are the dominant effect in the transition of the neck shape from large to small cone angles. This transition happens as a distinct, transient phase, as opposed to the very slow, continuous transition predicted by previous studies assuming weak vertical coupling. The transition is effectively complete once the minimum radius has decreased by a factor of an easily observable window. Even if the initial cone angles are very large, the system just induces a stronger vertical flow, still guaranteeing a rapid transition. Moreover, if the vertical velocity is initially compressing, it quickly reorients to a strong stretching flow. All these things taken together indicate that the transition to small cone angles will occur before any cut-off length scale. After the magnitude of the vertical flow peaks relative to the radial flow, it soon becomes small in comparison. This guarantees that the system will reach the slender-body phase characterized by a logarithmically slow transition to cylindrical Gordillo et al. (2005); Eggers et al. (2007); Gekle et al. (2009), along with memory of azimuthal vibrations Keim et al. (2006); Schmidt et al. (2009); Turitsyn et al. (2009), both studied extensively. In sum, the radial-flow dominated implosion singularity controls the final stage of the dynamics even in situations with strong vertical coupling.
Acknowledgements.This work was supported by NSF No. CBET-0967282, NDSEG fellowship (DCH) and the Keck initiative for ultrafast imaging (University of Chicago). Monte Rinebold experimented with parameters to evolve up-down asymmetric shapes. We thank Justin Burton, Nathan Keim, Lipeng Lai, Sidney Nagel, and Laura Schmidt for discussions and encouragement. We also thank the anonymous referees for helpful input.
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- For the up-down asymmetric case, we define the current height of the neck minimum as and for (reverse superscripts for
- defined by
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