Physics, and the bursting of a bubble (2)

A bursting bubble. Source: New Scientist

A continuation from here:

Physics, and the bursting of a bubble

Knowing that the cause of movement is surface tension, it’s tempting for us to conduct a quantitative analysis. After all, quantitative analysis is usually considered more “useful”, as it allows us to apply physical theory to resolve real-life problems.

In reality, this question was first answered by William Ranz in 1959, and his work was published in the Journal of Applied Physics. Below, we will retrace his analytical process and make a preliminary attempt to solve this problem.

Depiction of forces acting on the bubble. Source: Pandit & Davidson, Hydrodynamics of Thin Liquid Films

Ranz began by considering a basic question: how did surface tension contribute to the movement of the rim? Of course, one could naturally analyze this problem step-by-step: considering the force acting on each part of the rim, and calculating the acceleration of the rim under the effects of these forces.

However, this is quite inconvenient; the acceleration of the rim is not along a straight line but along a more complex, spherical path. In addition, we’re unsure of how the rim behaves during this entire process; calculating the acceleration will likely require us to compute the mass of the changing rim, which is an even more difficult problem.

When faced with these types of complex problems, physicists usually avoid this kind of detailed analysis. Instead, they try to rethink the problem from a broader perspective; i.e., they consider the complex movements and interactions as a whole, attempting to find some sort of simple balance or state of equilibrium.

By noting the spherical symmetry, Ranz found this by considering the total energy of the bubble; i.e., he used a “balance of energy”, converting this physical problem into a simple mathematical problem, hence resolving the velocity of the rim. These were the specific steps he took:

1. Using the conservation of energy: Conservation of energy is a very simple but extremely useful tool in applied physics problems. The law of conservation of energy states that the total energy of an isolated system remains constant; put simply, if we consider some objects with some energy, then if no energy is gained or lost to the surroundings, the total energy of the objects doesn’t change. This leads to a very easy balance; when the objects are isolated from their surroundings, energy can only be converted from one form into another, and summing these forms of energy will always give a constant value. In our case, if we neglect the presence of air resistance and gravity, then the bubble can be considered an isolated system.
2. Determining the types of energy present: Although there is no exchange of energy between the bubble and its surroundings, we still have to consider how energy changes from one form to another within the bubble. There are several main forms of energy: potential energy, kinetic energy, thermal energy, nuclear energy, and chemical energy. However, it’s quite obvious that many of these forms of energy do not change during the rupture process, hence we only consider potential and kinetic energy.
3. Evaluating the contributions of each type of energy: Initially, since the film is stationary, the energy of the system is entirely stored in the form of surface energy, a type of potential energy due to the extension of the surface under tension. When the rim of the film begins to move, the surface area of the film decreases. This means that potential energy is lost during the rupture process. Since surface energy is given by the product of surface tension and surface area (explanation), this loss of potential energy is proportional to the decrease in surface area.

A force F is required to increase the surface area by a distance l. This force is proportional to the surface tension by definition. Source: M. Durand, Mechanical Approach to Surface Tension

1. Resolving a mathematical problem: Combining the above considerations, we can obtain a differential equation regarding the balance of different forms of energy in the system. When the rim begins to move, the surface energy of the film is continuously converted without loss into the kinetic energy of the moving rim. Hence, we obtain the differential equation

This was the analytical process used by Ranz, and at the time, it was considered a major breakthrough in the study of surface fluid dynamics. After all, nobody expected the phenomenon to be so simple that a single mathematical expression could explain it accurately. However, as we will see later, there is a crucial mistake present in his reasoning which renders the result inaccurate. We’ll discuss this in the next article.