^{1,a)}

### Abstract

In this paper we analyze the final instants of axisymmetric bubble pinch-off in a low viscosityliquid. We find that both the time evolution of the bubble dimensionless minimum radius, , and of the dimensionless local axial curvature at the minimum radius, , are governed by a pair of two-dimensional Rayleigh-like equations in which surface tension,viscosity, and gas pressure terms need to be retained for consistency. The integration of the above-mentioned system of equations is shown to be in remarkable agreement with numerical simulations and experiments. An analytical criterion, which determines the necessary conditions for the formation of the previously reported tiny satellite bubbles, is also derived. Additionally, an estimation of the maximum velocity reached by the high speed Worthington jets ejected after bubble pinch-off, in the case axisymmetry is preserved down to the formation of the satellite bubble, is also provided.

I would like to thank C. Martínez-Bazán, M. Pérez-Saborid, and M. A. Fontelos for their useful suggestions. Special thanks are owed to A. Sevilla who, not only carefully proofread the manuscript but also suggested to include the axial curvature into Eq. (27). This research has been supported by the Spanish Ministry of Education and Science under Project No. DPI2005-08654-C04-02.

I. INTRODUCTION

II. THEORETICAL MODEL FOR THE FLOW NEAR THE MINIMUM RADIUS OF A COLLAPSING BUBBLE

III. NUMERICAL SIMULATIONS AND COMPARISON WITH THEORY

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 44.0
- Bernoulli's principle
- 22.0
- Bubble formation
- 16.0
- Surface tension
- 15.0
- Satellites
- 13.0

## Figures

(a) Numerical shapes of an axisymmetric bubble formed from a vertical underwater nozzle for . Note that, at a length scale of the order of , the shape of the bubble apparently forms two cones with a semiangle of close to the pinch-off point. (b) Detail of the bubble profiles—displaced in so that the minimum is located at —close to the pinch-off region. Note that the profiles become more symmetric with respect to the plane as pinch-off is approached. The values of in curves 1, 2, and 3 are, respectively, , , and . (c) Numerical bubble shapes represented in (b) as a function of the dimensionless stretched coordinates and . Note that the bubble shapes become more and more locally slender as pinch-off is approached. (d) In solid lines, full potential flow numerical simulations take into account the nozzle-bubble interaction and were previously represented in Figs. 1(b) and 1(c). In dashed lines, simplified potential flow numerical simulations in which an isolated bubble breaks symmetrically around the plane for a value of the Bond number, . Observe that there are no appreciable differences between the two types of simulations in the region near the minimum radius.

(a) Numerical shapes of an axisymmetric bubble formed from a vertical underwater nozzle for . Note that, at a length scale of the order of , the shape of the bubble apparently forms two cones with a semiangle of close to the pinch-off point. (b) Detail of the bubble profiles—displaced in so that the minimum is located at —close to the pinch-off region. Note that the profiles become more symmetric with respect to the plane as pinch-off is approached. The values of in curves 1, 2, and 3 are, respectively, , , and . (c) Numerical bubble shapes represented in (b) as a function of the dimensionless stretched coordinates and . Note that the bubble shapes become more and more locally slender as pinch-off is approached. (d) In solid lines, full potential flow numerical simulations take into account the nozzle-bubble interaction and were previously represented in Figs. 1(b) and 1(c). In dashed lines, simplified potential flow numerical simulations in which an isolated bubble breaks symmetrically around the plane for a value of the Bond number, . Observe that there are no appreciable differences between the two types of simulations in the region near the minimum radius.

Velocity at the minimum radius for a bubble in water, and two different gases (air, and , ). The details of the inviscid potential flow numerical simulations, represented in solid lines, will be provided in Sec. III. Note that, in agreement with Ref. 17 the radial velocity for is . Equation (6) is represented for different initial values of with dashed lines and differ from the numerics no matter how small the initial value of is.

Velocity at the minimum radius for a bubble in water, and two different gases (air, and , ). The details of the inviscid potential flow numerical simulations, represented in solid lines, will be provided in Sec. III. Note that, in agreement with Ref. 17 the radial velocity for is . Equation (6) is represented for different initial values of with dashed lines and differ from the numerics no matter how small the initial value of is.

Values of the local Weber number and of the local gas and liquid Reynolds numbers ( and , respectively) for the inviscid potential flow numerical simulations depicted in Fig. 2. Note that the evolution of the local Weber number is very similar to the experimental one reported in Fig. 46 of Ref. 17.

Values of the local Weber number and of the local gas and liquid Reynolds numbers ( and , respectively) for the inviscid potential flow numerical simulations depicted in Fig. 2. Note that the evolution of the local Weber number is very similar to the experimental one reported in Fig. 46 of Ref. 17.

(a) Dimensionless axial gas velocity profiles for different values of the local gas Reynolds number, ; (b) continuous line: dimensionless gas pressure gradient defined in Eq. (18) and given by Eq. (22); dotted line: , which proves to be a good approximation to the real solution.

(a) Dimensionless axial gas velocity profiles for different values of the local gas Reynolds number, ; (b) continuous line: dimensionless gas pressure gradient defined in Eq. (18) and given by Eq. (22); dotted line: , which proves to be a good approximation to the real solution.

Sketch of the geometry used for the symmetric type of simulations.

Sketch of the geometry used for the symmetric type of simulations.

In solid lines, the velocity at the minimum radius in the collapse of bubbles of two different gases in water [(a) air, and (b) , ]. The numerical computations have been performed using the inviscid symmetric code and the Bernoulli equation (35). Dashed lines: the theoretical result obtained integrating the two-dimensional Rayleigh equations (25) and (26) in the limits Re, with initial conditions for , , and the values of the numerical simulations when (a) , (b) . In (a) and (b) . The results of integrating Eqs. (25) and (26) in the limits Re, , and have also been included in each figure. While no appreciable differences between and are observed in the case of air, gas density plays a key role in the good agreement between theory and numerics in the case of .

In solid lines, the velocity at the minimum radius in the collapse of bubbles of two different gases in water [(a) air, and (b) , ]. The numerical computations have been performed using the inviscid symmetric code and the Bernoulli equation (35). Dashed lines: the theoretical result obtained integrating the two-dimensional Rayleigh equations (25) and (26) in the limits Re, with initial conditions for , , and the values of the numerical simulations when (a) , (b) . In (a) and (b) . The results of integrating Eqs. (25) and (26) in the limits Re, , and have also been included in each figure. While no appreciable differences between and are observed in the case of air, gas density plays a key role in the good agreement between theory and numerics in the case of .

Velocity at the minimum radius of an air bubble that collapses within liquids of different viscosities. The computations are performed using the symmetric code and the Bernoulli equation (40).

Velocity at the minimum radius of an air bubble that collapses within liquids of different viscosities. The computations are performed using the symmetric code and the Bernoulli equation (40).

The experimental time evolution of the bubble minimum radius is represented using dashed lines for two different values of the liquid viscosity (adapted from the experiments in Ref. 17, where it is reported that the exponents of the power law for the cases of the liquids of 4.2 and 21 cp are, respectively, 0.6 and 0.67). Solid lines: the resulting time evolution calculated through the integration of the two-dimensional Rayleigh-like equations (25) and (26).

The experimental time evolution of the bubble minimum radius is represented using dashed lines for two different values of the liquid viscosity (adapted from the experiments in Ref. 17, where it is reported that the exponents of the power law for the cases of the liquids of 4.2 and 21 cp are, respectively, 0.6 and 0.67). Solid lines: the resulting time evolution calculated through the integration of the two-dimensional Rayleigh-like equations (25) and (26).

(a) Comparison between the numerical computations depicted in Figs. 6 and 7 (solid lines) and the theoretical results obtained integrating the two-dimensional Rayleigh equations (25) and (26) using as initial conditions for , , and the values of the numerical simulations when oscillates between 120 and . Similarly, the comparison between numerics and theory in the case of is given in (b). In this case, the initial conditions for , , and are the values of the numerical simulation when . The instant at which , which is the condition for satellite formation, is indicated for each viscosity using a vertical arrow: (a) , (b) .

(a) Comparison between the numerical computations depicted in Figs. 6 and 7 (solid lines) and the theoretical results obtained integrating the two-dimensional Rayleigh equations (25) and (26) using as initial conditions for , , and the values of the numerical simulations when oscillates between 120 and . Similarly, the comparison between numerics and theory in the case of is given in (b). In this case, the initial conditions for , , and are the values of the numerical simulation when . The instant at which , which is the condition for satellite formation, is indicated for each viscosity using a vertical arrow: (a) , (b) .

(a) Satellite formation process in the case of air bubbles collapsing in a liquid with a viscosity of . In (b), the gas is and the material properties of the liquid are those of water.

(a) Satellite formation process in the case of air bubbles collapsing in a liquid with a viscosity of . In (b), the gas is and the material properties of the liquid are those of water.

(a) Sketch of the gas flow during the initial stages of bubble collapse, where it is expected that flow separation does not occur. (b) During the latest instants of bubble pinch-off, gas flow may separate and the stagnant gas pressure is not recovered from the exit of the tube to the main bubble.

(a) Sketch of the gas flow during the initial stages of bubble collapse, where it is expected that flow separation does not occur. (b) During the latest instants of bubble pinch-off, gas flow may separate and the stagnant gas pressure is not recovered from the exit of the tube to the main bubble.

Comparison between the results in Fig. 6 (, solid line) and those obtained integrating systems (B3) and (B4) for the same values of the initial conditions (dashed lines) and .

Comparison between the results in Fig. 6 (, solid line) and those obtained integrating systems (B3) and (B4) for the same values of the initial conditions (dashed lines) and .

## Tables

Physical properties of the different gases and liquid (water) considered.

Physical properties of the different gases and liquid (water) considered.

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