The axisymmetric pinch-off of an inviscid drop of density ρ1 immersed in an ambient inviscid fluid of density ρ2 is examined over a range of the density ratio D=ρ2/ρ1. For moderate values of D, time-dependent simulations based on a boundary-integral representation show that inviscid pinch-off is asymptotically self-similar with both radial and axial length scales decreasing like τ2/3 and velocities increasing like τ−1/3, where τ is the time to pinch-off. The similarity form is independent of initial conditions for a given value of D. The similarity equations are solved directly using a modified Newton’s method and continuation on D to obtain a branch of similarity solutions for 0⩽D⩽11.8. All solutions have a double-cone interfacial shape with one of the cones folding back over the other in such a way that its internal angle is greater than 90°. Bernoulli suction due to a rapid internal jet from the narrow cone into the folded-back cone plays a significant role near pinching. The similarity solutions are linearly stable for 0⩽D⩽6.2 and unstable to an oscillatory instability for D⩾6.2. Oscillatory behavior is also seen in the approach to self-similarity in the time-dependent calculations. Further instabilities are found as D increases and the steady solution branch is lost at a stationary bifurcation at D=11.8.
We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we show that the minimum radius of the cavity scales like h0 ∝ t ′α , where t ′ is the time from collapse. The exponent α very slowly approaches a universal value according to α = 1/2 + 1/(4 − ln(t ′ )). Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single non-trivial scaling exponent. Our predictions are confirmed by numerical simulations. PACS numbers: Valid PACS appear hereOver the last decade, there has been considerable progress in understanding the pinch-off of fluid drops, described by a set of universal scaling exponents, independent of the initial conditions [1,2]. The driving is provided for by surface tension, the value of the exponents depend on the forces opposing it: inertia, viscosity, or combinations thereof. Bubble collapse appears to be a special case of an inviscid fluid drop breaking up inside another inviscid fluid, which is a well studied problem [3,4,5]: the minimum drop radius scales like h 0 ∝ t ′2/3 , where t ′ = t 0 − t and t 0 is the pinch-off time. Thus, huge excitement was caused by the results of recent experiments on the pinch-off of an air bubble [6,7,8,9,10], or the collapse of a cavity [11] in water, which resulted in a radically different picture, in agreement with two earlier studies [12,13]. As demonstrated in detail in [10], the air-water system corresponds to an inner "fluid" of vanishing inertia, surrounded by an ideal fluid.Firstly, the scaling exponent α was found to be close to 1/2, (typical values reported in the literature are 0.56 [9] and 0.57 [10]), which means that breakup is much faster than in the fluid-fluid case, and surface tension must become irrelevant as a driving force. Secondly, the value of α appeared to depend subtly on the initial condition [11], and was typically found to be larger than 1/2. This raised the possibility of an "anomalous" exponent, selected by a mechanism as yet unknown. To illustrate the qualitative appearance of the pinch-off of a bubble, in Fig. 1 we show a temporal sequence of profiles, using a full numerical simulation of the inviscid flow equations [5]. We confine ourselves to axisymmetric flow, which experimentally is found to be preserved down to a scale of a micron [10], provided the experiment is aligned carefully [9].The only existing theoretical prediction [7,11,15] is based on treating the bubble as a (slightly perturbed) cylinder [12,13]. This leads to the exponent being 1/2 with logarithmic corrections, a result which harks back to the 1940's [16]. Our numerics, to be reported below, are inconsistent with this result. Moreover, a cylinder is not a particularly good description of the actual profiles (cf. Fig. 1), as has been remarked before [9]. In this Letter, we present a systematic expansion in the slenderness of the cavity, ...
Cavitation and bubble dynamics have a wide range of practical applications in a range of disciplines, including hydraulic, mechanical and naval engineering, oil exploration, clinical medicine and sonochemistry. However, this paper focuses on how a fundamental concept, the Kelvin impulse, can provide practical insights into engineering and industrial design problems. The pathway is provided through physical insight, idealized experiments and enhancing the accuracy and interpretation of the computation. In 1966, Benjamin and Ellis made a number of important statements relating to the use of the Kelvin impulse in cavitation and bubble dynamics, one of these being ‘One should always reason in terms of the Kelvin impulse, not in terms of the fluid momentum…’. We revisit part of this paper, developing the Kelvin impulse from first principles, using it, not only as a check on advanced computations (for which it was first used!), but also to provide greater physical insights into cavitation bubble dynamics near boundaries (rigid, potential free surface, two-fluid interface, flexible surface and axisymmetric stagnation point flow) and to provide predictions on different types of bubble collapse behaviour, later compared against experiments. The paper concludes with two recent studies involving (i) the direction of the jet formation in a cavitation bubble close to a rigid boundary in the presence of high-intensity ultrasound propagated parallel to the surface and (ii) the study of a ‘paradigm bubble model’ for the collapse of a translating spherical bubble, sometimes leading to a constant velocity high-speed jet, known as the Longuet-Higgins jet.
In this paper we examine the dynamics of an initially stable bubble due to ultrasonic forcing by an acoustic wave. A tissue layer is modelled as a density interface acted upon by surface tension to mimic membrane effects. The effect of a rigid backing to the thin tissue layer is investigated. We are interested in ultrasound contrast agent type bubbles which have immediate biomedical applications such as the delivery of drugs and the instigation of sonoporation. We use the axisymmetric boundary integral technique detailed in Curtiss et al. (J. Comput. Phys., 2013, submitted) to model the interaction between a single bubble and the tissue layer. We have identified a new peeling mechanism whereby the re-expansion of a toroidal bubble can peel away tissue from a rigid backing. We explore the problem over a large range of parameters including tissue layer depth, interfacial tension and ultrasonic forcing.
Cavitation occurs around dental ultrasonic scalers, which are used clinically for removing dental biofilm and calculus. However it is not known if this contributes to the cleaning process. Characterisation of the cavitation around ultrasonic scalers will assist in assessing its contribution and in developing new clinical devices for removing biofilm with cavitation. The aim is to use high speed camera imaging to quantify cavitation patterns around an ultrasonic scaler. A Satelec ultrasonic scaler operating at 29 kHz with three different shaped tips has been studied at medium and high operating power using high speed imaging at 15,000, 90,000 and 250,000 frames per second. The tip displacement has been recorded using scanning laser vibrometry. Cavitation occurs at the free end of the tip and increases with power while the area and width of the cavitation cloud varies for different shaped tips. The cavitation starts at the antinodes, with little or no cavitation at the node. High speed image sequences combined with scanning laser vibrometry show individual microbubbles imploding and bubble clouds lifting and moving away from the ultrasonic scaler tip, with larger tip displacement causing more cavitation.
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