# Effect of liquid depth on dynamics and collapse of large cavities generated by standing waves

Abstract: The effect of fluid depth on the collapse of large cavities generated by over-driven axisymmetric gravity waves in a 10 cm diameter cylindrical container has been studied. At a large fluid depth in a viscous glycerine–water solution, the collapse of the cavities is inertia dominant at the initial phase with the time-dependent cavity radius (rm) obeying rm ∝ τ1/2; τ = t − t0 being the time remaining for collapse, with t0 being the time at collapse. However, enhanced damping at a low liquid depth turns the late stage of the transition into the viscous regime (rm ∝ τ) at some critical depth beyond which a singular collapse (transition from non-pinch-off and pinch-off collapse) is impossible. At a shallow depth, the change in cavity radius follows a flip of the power law, i.e., rm ∝ τ at the initial stage of collapse followed by a transition to rm ∝ τ1/2, suggesting a viscous–inertial transition. For fluids with relatively lower viscosity but similar surface tension, here water, a smoother cavity with damped parasitic waves at a small liquid depth collapses at a smaller radius. The surface jet velocity due to the collapse of the cavity monotonically decreases with the decrease in the depth, whereas in the case of water, it increases with the depth reaching a maximum at a critical depth followed by a decrease again. The self-similarity, exhibited by the cavity up to the critical depth, is lost due to the axial movement restriction by the bottom wall.

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01 Jan 1998-

Abstract: As pointed out to us by Mr T. Heath, the following printing errors can be quite misleading when using the formulas in the paper to obtain eigenfrequencies and damping rates to compare with experiments:
in (A 13) 1 should read −1 on the right-hand side;
in (A 22) and (A 26) Ω20 should read Ω−20;
in (A 25) the factor Ω40 must be omitted on the right-hand side.
When revising again the printed version of the paper, we discovered several additional misprints:
A factor C was omitted in the first two integrals in the expression for J2, immediately following equation (2.9).
The sign of the second expression for I1 in (2.23) should be changed.
The expression (W0Wz +3WW0z)z=0 should read 2(W0Wz +WW0z)z=0 in equation (2.24).
The expression W0(1, z)W0z(1, z) in (2.26) should read W0(r, 0)W0z(r, 0).
None of the misprints above affect the results of the paper, which were obtained with the correct expressions.

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3 citations

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Abstract: Jet droplets from bubble bursting are determined by a limited parametrical space: the liquid properties (surface tension, viscosity, and density), mother bubble size and acceleration of gravity. Thus, the two resulting parameters from dimensional analysis (usually, the Ohnesorge and Bond numbers, Oh and Bo) completely define this phenomenon when both the trapped gas in the bubble and the environment gas have negligible density. A detailed physical description of the ejection process to model both the ejected droplet radius and its initial launch speed is provided, leading to a scaling law including both Oh and Bo. Two critical values of Oh determine two limiting situations: one (Oh$_1$=0.038) is the critical value for which the ejected droplet size is minimum and the ejection speed maximum, and the other (Oh$_2$=0.0045) is a new critical value which signals when viscous effects vanish. Gravity effects (Bo) are consistently introduced from energy conservation principles. The proposed scaling laws produce a remarkable collapse of published experimental measurements collected for both the ejected droplet radius and ejection speed.

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Abstract: 1. The beautiful series of forms assumed by sand, filings, or other grains, when lying upon vibrating plates, discovered and developed by Chladni, are so striking as to be recalled to the minds of those who have seen them by the slightest reference. They indicate the quiescent parts of the plates, and visibly figure out what are called the nodal lines. 2. Afterwards M. Chladni observed that shavings from the hairs of the exciting violin bow did not proceed to the nodal lines, but were gathered together on those parts of the plate the most violently agitated, i. e. at the centres of oscillation. Thus when a square plate of glass held horizontally was nipped above and below at the centre, and made to vibrate by the application of a violin bow to the middle of one edge, so as to produce the lowest possible sound, sand sprinkled on the plate assumed the form of a diagonal cross; but the light shavings were gathered together at those parts towards the middle of the four portions where the vibrations were most powerful and the excursions of the plate greatest.

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1,089 citations

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Abstract: Vapor bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. The method uses finite time steps and features an iterative technique for applying the boundary conditions at infinity directly to the liquid at a finite distance from the free surface. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)^1/2 where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For Δp/ρ = 10^6 (cm/sec)^2 ~ 1 atm./density of water the jet had a speed of about 130 m/sec in the first case and 170 m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapor are not important.

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789 citations

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TL;DR: This paper reports a theoretical and experimental study of the generation of a singularity by inertial focusing, in which no break-up of the fluid surface occurs, and predicts that the surface profiles should be describable by a single universal exponent.

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Abstract: Finite-time singularities—local divergences in the amplitude or gradient of a physical observable at a particular time—occur in a diverse range of physical systems. Examples include singularities capable of damaging optical fibres and lasers in nonlinear optical systems1, and gravitational singularities2 associated with black holes. In fluid systems, the formation of finite-time singularities cause spray and air-bubble entrainment3, processes which influence air–sea interaction on a global scale4,5. Singularities driven by surface tension have been studied in the break-up of pendant drops6,7,8,9 and liquid sheets10,11,12. Here we report a theoretical and experimental study of the generation of a singularity by inertial focusing, in which no break-up of the fluid surface occurs. Inertial forces cause a collapse of the surface that leads to jet formation; our analysis, which includes surface tension effects, predicts that the surface profiles should be describable by a single universal exponent. These theoretical predictions correlate closely with our experimental measurements of a collapsing surface singularity. The solution can be generalized to apply to a broad class of singular phenomena.

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200 citations

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Abstract: The weakly nonlinear, weakly damped response of the free surface of a liquid in a vertical circular cylinder that is subjected to a simple harmonic, horizontal translation is examined by extending the corresponding analysis for free oscillations. The problem is characterized by three parameters, α, β, and d/a, which measure damping, frequency offset (driving frequency–natural frequency), and depth/radius. The asymptotic (t↑∞) response may be any of: (i) harmonic (at the driving frequency) with a nodal line transverse to the plane of excitation (planar harmonic); (ii) harmonic with a rotating nodal line (non-planar harmonic); (iii) a periodically modulated sinusoid (limit cycle); (iv) a chaotically modulated sinusoid. It appears, from numerical integration of the evolution equations, that only motions of type (i) and (ii) are possible if 0.30 < d/a < 0.50, but that motions of type (iii) and (iv) are possible for all other d/a in some interval (or intervals) of β if α is sufficiently small. Only motion of type (i) is possible if α exceeds a critical value that depends on d/a.

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169 citations

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TL;DR: It is shown that when water droplets gently impact on a hydrophobic surface, the droplet shoots out a violent jet, the velocity of which can be up to 40 times the drop impact speed.

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Abstract: We show that when water droplets gently impact on a hydrophobic surface, the droplet shoots out a violent jet, the velocity of which can be up to 40 times the drop impact speed. As a function of the impact velocity, two different hydrodynamic singularities are found that correspond to the collapse of the air cavity formed by the deformation of the drop at impact. It is the collapse that subsequently leads to the jet formation. We show that the divergence of the jet velocity can be understood using simple scaling arguments. In addition, we find that very large air bubbles can remain trapped in the drops. The surprising occurrence of the bubbles for low-speed impact is connected with the nature of the singularities, and can have important consequences for drop deposition, e.g., in ink-jet printing.

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168 citations