Breakup of Air Bubbles in Water: Memory and Breakdown of Cylindrical Symmetry

Abstract

Using high-speed video, we have studied air bubbles detaching from an underwater nozzle. As a bubble distorts, it forms a thin neck which develops a singular shape as it pinches off. As in other singularities, the minimum neck radius scales with the time until breakup. However, because the air-water interfacial tension does not drive breakup, even small initial cylindrical asymmetries are preserved throughout the collapse. This novel, non-universal singularity retains a memory of the nozzle shape, size and tilt angle. In the last stages, the air appears to tear instead of pinch.PACS numbers: 47.55.db, 47.55.df, 02.40.Xx The delightful tingling felt when drinking carbonated beverages, the glee of children blowing bubbles in a bathtub, and the importance of deep underwater fissures venting gasses into the oceans hint at the richness and significance of bubble formation in determining the texture and composition of our world. However, the process by which a bubble is formed is still full of surprises. A drop or bubble breaks up by forming a neck that thins to atomic dimensions, a process described as an approach towards a singularity where physical quantities such as stress or pressure grow infinitely large. Singularities often organize the overall dynamical evolution of nonlinear systems. Each symmetry in nature implies an underlying conservation law, so that the symmetries of the singularity associated with pinch-off naturally have important consequences for its dynamics. It was previously believed [1,2,3,4,5,6,7,8,9,10,11,12] that the pinching neck of any drop or bubble would become cylindrically (i.e. azimuthally) symmetric in the course of pinch-off. Recently, pinching necks of air in water were observed to lose cylindrical symmetry in the course of detachment [13,14].Here we show that this loss of symmetry is caused by a new form of memory in singular dynamics: even a small asymmetry in the initial conditions is preserved throughout bubble detachment. This novel singularity retains a memory of the nozzle shape, size and tilt angle. The asymmetry can be made so great that the air appears to tear. This symmetry breaking may be important in numerous applications [15,16,17], and for understanding other physical processes which are modeled as the formation of a singularity, such as star or black hole formation [18] and supernova explosions [19]. Thus our experimental observation of the breakdown of cylindrical symmetry in the air bubble demonstrates a new view of dynamical singularities that may be relevant even on a celestial scale.Singularities govern the dynamics in many familiar break-up events, such as the dispersal of oil drops into * Electronic address: nkeim@uchicago.edu vinegar during the making of a salad dressing, or the dripping of water from a leaky faucet. For many fluid pairs -for example, one viscous fluid breaking in a surrounding fluid of high viscosity [7,8,12] -the shape and dynamics of the pinching neck depend solely on the fluid parameters, as the breakup forgets its initial conditions on ...

Collapse and pinch-off of a non-axisymmetric impact-created air cavity in water

“…(typical values found are 0.54 − 0.60) (Burton et al 2005;Thoroddsen et al 2007;Keim et al 2006;Gordillo et al 2005;Bergmann et al 2006Bergmann et al , 2009b and theoretical studies have shown that the exponent indeed has a weak dependence on the logarithm of the remaining collapse time, approximating to 1 2 only asymptotically at the end (Gordillo & Perez-Saborid 2006;Eggers et al 2007;Gekle et al 2009b). Nonetheless, the full theoretical result lies remarkably close to a power law fit over many decades in time.…”

“…The first proposed model was a power law where the radius decreased proportionally to the square root of the remaining time until collapse, τ (Longuet- Higgins et al 1991;Oguz & Prosperetti 1993). Subsequent experimental and numerical studies consistently found the behaviour deviated slightly from that 1 2 power law (Burton et al 2005;Gordillo et al 2005;Keim et al 2006;Bergmann et al 2006;Thoroddsen et al 2007;Bergmann et al 2009b), generating doubts and starting a controversy about the universality of the phenomenon. Gordillo & Perez-Saborid (2006) and Eggers et al (2007) theoretically showed how the power law varies weakly as a function of τ due to a logarithmic correction.…”

“…If axial symmetry is broken by a small azimuthal perturbation, a truly universal system would be expected to converge to the same solution . However, through experiments and simulations of an air bubble disconnecting from an underwater nozzle, Keim, Møller, Zhang & Nagel (2006); Schmidt, Keim, Zhang & Nagel (2009) ;Turitsyn, Lai & Zhang (2009);and Keim (2011) recently showed that a slight azimuthal asymmetry can trigger vibrations that persist in time. The fact that a small perturbation is not smoothed out indicates that the system possesses memory of its initial conditions.…”

Bubbling in a co-flow at high Reynolds numbers

“…The formation of satellite drops and the description of the latest instants prior to pinch-off ͑see the review by Eggers 4 ͒, or the transition from the socalled jetting regime to the dripping regime, 5,6 are details extensively reported in the literature. However, although the generation of bubbles is a phenomenon as common as the drop formation, and of great relevance in a large number of industrial processes, our knowledge of the bubble formation process is still far from being fully complete despite the recent advances in the analytical and experimental description of the bubble pinch-off at low [7][8][9] and high [9][10][11][12][13] Reynolds numbers. For example, the most studied geometry, due its simplicity and importance in many chemical engineering applications, is the growth and subsequent pinch-off of a bubble from an orifice, or needle, placed at the bottom of a liquid pool.…”

Droplet formation and scaling in dense suspensions

“…In many cases, when the boundary stresses originate from surface tension, the framework of pure liquid detachment can be modified successfully and self-similar structures govern the breakup (19)(20)(21). However, in some cases the driving force comes from an alternative source, as in the case of bubble pinch-off (22)(23)(24), and self-similarity breaks down: To describe detachment, scaling laws must be used along with initial conditions.…”