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

TL;DR: 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 in this paper.

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

TL;DR: In this paper, the authors pointed out that 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.

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.

3 citations

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TL;DR: In this paper , the effect of the contact angle on the generation position and focusing efficiency of annular focused jets between parallel plates was investigated, and a new calculation method for the jet focusing efficiency was proposed.

Abstract: Focused jets have been widely studied owing to the abundance of attractive flow phenomena and industrial applications, whereas annular focused jets are less studied. This study combines experiments, numerical simulations, and analytical modeling to investigate the effect of the contact angle on the generation position and focusing efficiency of annular focused jets between parallel plates. In the experiment, a pulsed laser generates a cavitation bubble inside the droplet, and the rapidly expanding cavitation bubble drives an annular-focused jet on the droplet surface. Changing the plate wettability creates different contact angles and droplet surface shapes between the droplet and plates, which modulates the position and focusing efficiency of the annular jet. Based on the jet singularity theory and by neglecting gravity, the derived formula for the jet position offset is found to depend only on the contact angle, which is in good agreement with the experimental and numerical simulation results. Combined with numerical simulations to analyze the flow characteristics of the droplets between the parallel plates, a new calculation method for the jet focusing efficiency is proposed. Interestingly, when the liquid surface radius is small, the focusing efficiency can be improved by adjusting the contact angle to make the jet position closer to the flat plate, whereas the same operation reduces the focusing efficiency when the radius is large. The study of annular jets can expand the scope of traditional jet research and has the potential to provide new approaches for applications such as high-throughput inkjet printing and liquid transfer.

3 citations

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TL;DR: In this article, 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.

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.

1 citations

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TL;DR: Chladni as mentioned in this paper 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.

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.

1,178 citations

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TL;DR: In this paper, a numerical method was proposed to solve the problem of balloon bubble collapse near a plane solid wall, using finite time steps and an iterative technique for applying the boundary conditions at infinity directly to the liquid at a finite distance from the free surface.

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.

890 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.

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.

232 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.

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.

217 citations

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TL;DR: In this article, the authors provided an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques.

Abstract: When two cylinders are counter-rotated at low Reynolds number about parallel horizontal axes below the free surface of a viscous fluid, the rotation being such as to induce convergence of the flow on the free surface, then above a certain critical angular velocity Ωc, the free surface dips downwards and a cusp forms. This paper provides an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques. Surface tension effects are included, but gravity is neglected. The solution is analytic for finite capillary number [Cscr ], but the radius of curvature on the line of symmetry on the free surface is proportional to exp (−32π[Cscr ]) and is extremely small for [Cscr ] [gsim ] 0.25, implying (in a real fluid) the formation of a cusp. The equation of the free surface is cubic in (x, y) with coefficients depending on [Cscr ], and with a cusp singularity when [Cscr ] = ∞.The influence of gravity is considered through a stability analysis of the free surface subjected to converging uniform strain, and a necessary condition for the development of a finite-amplitude disturbance of the free surface is obtained.An experiment was carried out using the counter-rotating cylinders as described above, over a range of capillary numbers from zero to 60; the resulting photographs of a cross-section of the free surface are shown in figure 13. For Ω Ωc, the downward-pointing cusp forms, and its structure shows good agreement with the foregoing theory.

182 citations