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Atlanta, GA | Posted:
November 7, 2018 *

*-By John McCuan*

We are used to seeing a light object, like a beach ball, float on the surface of water while a heavy one, like a solid silver ball, sinks to the bottom (Fig.1-Fig.2). Over two-thousand years ago, based on similar observations, Archimedes proposed a simple and beautiful rule to determine which objects float, which objects sink, and how much liquid will be displaced by a floating object. He asserted that everything should be determined by relative densities.

Archimedes might be surprised to see this green plastic ball (Fig. 3-Fig. 5) which sinks to the bottom if pushed below the surface but also floats on the surface of the water if it is gently released there. The framework needed to understand the behavior of a “heavy” floating ball like this one was introduced by the mathematician Carl Friedrich Gauss in 1830. He applied his ideas about minimizing energy to the geometrical and analytical concepts of surface tension and contact angle introduced by Thomas Young and Pierre Simone Laplace in 1805 and 1806.

Nevertheless, theoretical verification of the possibility of a heavy floating object like the green ball was first obtained by Rajat Bhatnagar and Robert Finn of Stanford University in 2006. To obtain their result various simplifications were made. One of those simplifications was to assume the liquid bath was infinite in extent with the walls of the container infinitely far away. John McCuan of the School of Mathematics has been interested in floating objects in laterally bounded containers since about the same time. In 2013 he was able, along with Ray Treinen of Texas State University, to analyze the energy landscape for problems that include the green ball floating in a finite cylindrical container as in the photo above. They showed, in particular, that if such a ball, floating on the surface of the water is pushed downward, the energy of the system will increase at first, eventually reaching a single maximum, at which point, as the ball moves lower, the energy of the system decreases and eventually the ball slips below the surface and sinks.

While relaxing the assumption of an infinite sea on which the ball floats, McCuan and Treinen introduced an additional symmetry assumption, effectively requiring the ball to be constrained to a frictionless vertical wire through its center keeping the ball in the middle of a circular cylindrical container. The characterization of parameters (density, surface tension versus gravity, the size of the ball relative to that of the container, and adhesion properties) for which a floating ball will remain in the center without the guide-wire is still a major open problem.

Buoyed up by some success, McCuan and Treinen attempted to characterize the equilibrium configurations (maxima and minima of the Gauss energy) for balls like the beach ball with density lower than that of the liquid. They were able to obtain a number of results, but they were also in for a big surprise. The natural expectation would be that for the light ball there is a unique equilibrium (energy minimum) with the energy increasing monotonically as the ball is pushed downward (and constrained to the center) in a cylindrical container. This is true for a beach ball in, say, a swimming pool. Sometimes, however, for certain collections of parameters, the energy will, in fact, increase but then decrease to another local minimum before increasing as the ball is submerged. (See chart, first image)

*Note: For purposes of illustration the figure is neither to scale nor accurately proportioned.*

There are several consequences of this 2018 discovery. One is that a ball floating in a cylinder need not have a unique floating height; the ball may rest at equilibrium in two different positions. If, for example, the ball is positioned as on the left, it will remain there, but if the ball is manually moved to the position on the right, it will also float in position there. Such a ball in a cylinder might be used as a two position switch. Furthermore, the phenomenon first encountered with the heavy green ball is not isolated to the heavy floating ball. Even with a light floating ball, the observed floating configuration can depend on where one positions the ball initially. The only known instances of this behavior for a light ball occur when the ball fits within the cylinder leaving only a small gap (several one hundredths of a millimeter) between the ball and the wall, so the phenomenon would likely never have been discovered without considering the case of laterally bounded containers.

- 1. It was about 200 years between the time a mathematical framework describing floating objects (including capillarity and adhesion energies) was proposed and the time it was actually used with any success to describe floating objects.

Part of the groundwork for this kind of application of the theory was laid in McCuan’s 2007 paper which adapts the framework of Gauss to situations which allow floating. Previous to this, force phenomena such as buoyancy were viewed as separate from capillary equilibrium theory. McCuan showed all conditions for equilibrium (including various generalized force equations) follow from the basic approach of Gauss.

- 2. An essential difference between the analysis of floating objects (say balls) based on Archimedes’ principle and that based on capillarity is that in the former the liquid surface is assumed to be a flat plane, while in the latter the geometric shape of the liquid surface can be curved and plays a central role. Sometimes the liquid surface surrounding a floating ball can be so far from a plane that it bends back over itself as suggested by the exaggerated figure below.[PP] Several results in the paper of McCuan and Treinen (2013) give conditions under which this cannot happen. They show, for example, that if the ball is too heavy (dense) or the ball is too small, then such “folding over” is not possible. Also, if the ball is too light and the adhesion of the liquid with the ball is too small (resulting in an angle between the liquid and the ball measured within the liquid which is too big), then, again, folding over is not possible.
- 3. Another factor in the recent progress on problems like this (in spite of interest in them from antiquity) is the new capability to numerically analyze the model equations.
- 4. One approach (and perhaps the only approach) to understanding when a floating ball will remain centered in the container (rather than move to the side) requires an extension of McCuan’s 2007 first variation formula to the second variation of energy. In some instances (experimentally) when the outer edge of the liquid interface is higher than the edge on the ball, and the ball is heavy, the ball will stay in the center. Similarly, when the outer edge is lower than the inner edge, then a heavy ball will tend to the side. These observations can be reversed for a light ball. These experimentally observed conditions are (first of all) far from a mathematical analysis; it is very unlikely that they capture the entire range of possibilities.
- 5. Most of the known results are for a system which is simplified in dimension. Mathematically, we are really considering (in the drawings above for example) a two dimensional problem which can be viewed as treating an infinite log (extending directly out of the paper) floating in a trough. It seems likely that all equilibria for this simplified problem can be identified/classified within the next decade. A similar time frame applies to the spherical ball in a cylindrical container as indicated in the photographs. Some fundamental advance, like obtaining a second variational formula for energy as mentioned in the previous point will be necessary for understanding/classifying the conditions characterizing central floating versus moving to the side.

References:

250 B.C. Archimedes, *On floating bodies*

1805 Thomas Young, *An essay on the cohesion of fluids*, Philos. Trans. R. Soc. Lond. 95[PP]

1806 Pierre Simone Laplace, *Mécanique céleste*

2006 Raj Bhatnagar and Robert Finn, *Equilibrium configurations of an infinite cylinder in an unbounded fluid*. Phys. Fluids 18 no. 4

2007 John McCuan, *A variational formula for floating bodies*, Pac. J. Math. 231 no. 1

2009 John McCuan, *Archimedes’ principle revisited*, Milan J. Math. 77

2013 John McCuan and Ray Treinen, *Capillarity and Archimedes’ principle of flotation*, Pacific J. Math. 265 no 1

2018 John McCuan and Ray Treinen, *On floating equilibria in a laterally finite container*, SIAM J. Appl. Math. 78 no. 1