Bubble blowing by the numbers

It’s a sunny day, and the two of us are taking a stroll in the park. Children are playing with bubble wands. We see the bubbles shimmering brightly as they float up into a clear blue sky. Because we are physicists, we couldn’t help wondering whether the physics of blowing soap bubbles had been described in the literature. Our intuition was that, like most classical capillary phenomena, the common experience of bubble blowing would have been explained centuries ago. To our surprise, we found that the formation of those ephemeral and fascinating fluid structures had not been fully investigated. That gap in the literature whetted our appetite for an understanding of soap-bubble formation and led to months of experimental investigations and modeling work that we carried out with two PhD students, Louis Salkin and Alexandre Schmit.

Bubble machine learning

Every bubble blown with a commercial wand, as in figure 1, forms under different conditions. The average film thickness is not reproducible, and it varies locally due to evaporation and gravitational drainage. Moreover, after only a few bubbles have been blown, the film’s soapy water is exhausted and the film bursts. To ensure the reliability and reproducibility of our measurements, we designed a bubble machine. In brief, gravity causes a bubble solution placed in a tank above the setup to flow along a pair of 1-m-long vertical wires. A giant film forms between the wires when they are separated to a width $w$ of 1–15 cm. When soap solution reaches the bottom of the setup, it is pumped back up to the tank on top.

Unlike the stationary films of common commercial wands, the films generated by our bubble machine flow downward at a controlled speed in the range of 1–4 m/s. In addition, we can adjust the films’ thickness between 1 µm and 5 µm. Our artificial mouth for blowing bubbles is a nozzle of radius $R_0$ = 0.1–100 mm connected to a flow-control system and placed a distance $\delta$ up to 100 mm from the soap film. The nozzle emits a gas jet that hits the film at essentially normal incidence, as shown in figure 2a, which also defines some of the experimental parameters. A high-speed camera captures the jet impact. For all experimental conditions, bubbles such as those seen in the top photo of figure 2b form only when the gas speed exceeds a threshold $v_{\mathrm{c}}$. At lower gas speeds, the jet gently deforms the film and develops a dimple, as in the lower photo in the figure. As the speed of the jet increases toward the critical value, the dimple grows until a bubble pops out of the surface. In our study, we explored how the threshold speed changed as we varied experimental parameters.

Thanks to the bubble machine, we learned that, at least under the experimental conditions we considered, neither the thickness of the film nor its downward flow velocity affects the critical speed at which bubbles form. We were glad to obtain those results because they confirmed that the bubble machine is a reasonable laboratory setup for understanding the real-world blowing of soap bubbles with stationary films that span circular wands.

Gas inertia versus surface tension

Simple dimensional analysis shows that gravitational and viscous forces are not important in our experiments; bubble formation is governed by surface tension $\gamma$ and the force of blowing. Under those conditions, the formation of a dimple can be understood by way of Bernoulli’s principle, which describes the conversion of the kinetic energy of blowing to the potential energy of dimpling the fluid surface: ${\rho }_{\mathrm{g}}{v}_{\mathrm{g}}^2/2=\Delta p$. Here, ${\rho }_{\mathrm{g}}$ is the density of the blown gas and $v_{\mathrm{g}}$ is its speed. The right-hand term, $\Delta p$, is the pressure jump, or Laplace pressure, across the curved surface of the dimple. It is given by $\Delta p=4\kappa \gamma$, where $\kappa$ denotes the mean curvature of the surface. The factor of 4—instead of the usual 2 that arises when liquid drops are considered—accounts for the two air–liquid interfaces of a soap film. Note in particular that Bernoulli’s principle implies that the curvature of a dimple is an increasing function of the gas speed.

Our observations revealed that the shape of a dimple may be reasonably taken to be a spherical cap and that bubbles form when the radius of curvature $1/\kappa$ becomes comparable to the lesser of the jet radius and $w/2$. Hence for contained jets (cj)—that is, jets whose radius is less than $w/2$—placed at the film’s surface, the speed threshold for bubble formation is $v_{\mathrm{c}}^{\mathrm{cj}}(\delta =0)\approx \sqrt{8\gamma /{\rho }_{\mathrm{g}}{R}_0}$. That simple prediction can be expressed in the terms of the dimensionless Weber number, $W={\rho }_{\mathrm{g}}{v}_{\mathrm{g}}^2{R}_0/\gamma$, a parameter relevant to soap-bubble blowing and other phenomena that can be seen as a competition between inertial and tension forces. In that language, we’d say bubbles form once the Weber number reaches about 8, a prediction that matches experimental results obtained with both common bubble wands and the bubble machine.

Real-life bubble blowing is more complicated. For one thing, a person usually blows soap bubbles at a finite distance $\delta$ away from the film. Moreover, the blown jets are turbulent and have a complex velocity structure. However, an average over turbulent fluctuations yields a much simplified velocity profile across the jet that still provides a good description of the flow produced by someone blowing a bubble.

Before turning to velocity, we note that when a turbulent air jet penetrates air at rest, the jet assumes the shape of a cone with a universal opening angle of 23.6°. The size of the jet varies linearly with $\delta$ as $R(\delta )=R_0+\delta \tan (23.6°/2)\approx R_0+\delta /5$.

Once turbulent fluctuations are averaged away, the gas speed has a Gaussian distribution when plotted along the direction perpendicular to the jet propagation. Given that distribution, it is a straightforward task to estimate the average gas speed $v_{\mathrm{g}}(\delta )$ at $\delta$; namely, $v_{\mathrm{g}}(\delta )\approx v_{\mathrm{g}}/(1+\delta /5R_0)$. As in the $\delta =0$ scenario, bubble formation is governed by a competition between inertia and surface tension, and the same argument we advance above shows that the threshold velocity above which bubbles form is $v_{\mathrm{c}}\approx v_{\mathrm{c}}^{\mathrm{cj}}(\delta =0)\sqrt{1+\delta /5R_0}$ when $2R(\delta )<w$, and $v_{\mathrm{c}}\approx v_{\mathrm{c}}^{\mathrm{cj}}(\delta =0)\sqrt{2R_0/w}(1+\delta /5R_0)$, when $2R(\delta )>w$. Note that when $\delta \gg R_0$, the threshold velocity varies more strongly with $\delta$ when $2R(\delta )>w$ than it does when $2R(\delta )<w$. Figure 2b illustrates our theoretical results and shows that they match observations for various flow configurations.

Bubbles in the classroom

The experiments described in this Quick Study can be performed in the classroom without the need for sophisticated equipment. Students would need some wire, a solution for blowing bubbles, and a caliper. The transition between dimple formation and bubble creation is visible to the naked eye, so a camera is not essential. Likewise, there is no need to purchase a flowmeter for the measurements of the gas velocity. Instead, students can simply place the gas nozzle in a burette filled with water and then turn the burette upside down and submerge it in a pool of water. They could then use a stopwatch to measure the rate $q_{\mathrm{g}}$ at which the volume of gas emitted by the nozzle builds up over the water. That rate is related to the average gas velocity across the nozzle by $q_{\mathrm{g}}=v_{\mathrm{g}}(\pi R_0^2)$.

Supplemental materials

To watch dimple and bubble formation visit https://goo.gl/fTW7hV .