Georgia Institute of Technology College of Sciences

Volume X Contents

• Welcome from the Chair
• Benefits Add Up for Undergrads in SoM REU Programs
• Seven REUs Planned for Summer 2018
• Georgia Tech Hosts Annual High School Competition
• Libby Taylor Feature: Georgia Tech Undergrad Takes Home AWM Math Prize
• Annual TA Student Award Winners
• Recent Graduates Give Advice to Incoming Freshmen
• Geometric Group Theory Gets an Informal Take from Tech Professor
• Donor Awards
• PhD Program
• Events
• Awards
• Featured Article: Researchers Determine Routes of Respiratory Infection Disease Transmission on Aircraft
• Members of SoM at the Helm of National Research Programs
• Discrete Math/Combinatorics Moves Up to No. 2 in US News Graduate School Rankings
• Faculty Profiles
• Teaser: New Frontiers Beckon Math and Biology in Multi-million Dollar NSF-Simons Project
• ProofReader Article Picked Up by Notices of American Math Society
• SoM Professor Called to Give Expert Testimony in Jury Selection Case

Please see the Proofreader page on our website or click here to view a .pdf of the new ProofReader. 

Please send comments to Sal Barone at comm@math.gatech.edu, with subject line "ProofReader".

Mysteries of Floating

-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.

Other “fun” facts:

• 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

Congratulations go to Dan Margalit and Chongchun Zeng, who have been named American Mathematical Society (AMS) Fellows.

A special issue of the journal Discrete and Continuous Dynamical Systems-A has been dedicated to Prof. Rafael de la Llave.

The issue 38-12 of the journal Discrete and Continuos Dynamical Systems-A contains the proceedings of the international conference LLAVEFEST, which was celebrated June, 2017 in Barcelona. The conference was devoted to the interface of dynamics and partial differential equations and applications.

The goal of the conference was to present new advances in different aspects on Dynamical Systems and Partial Differential Equations.

There were 151 participants in attendance.

Topics covered included:
-  Dynamical systems and ergodic theory
-  Global dynamics in Hamiltonian systems
-  KAM theory
-  Arnol'd diffusion
-  PDEs and their applications
-  Lattice systems
-  Action-minimizing orbits and measures
-  Invariant manifold theory
-  Hyperbolic systems
-  Renormalization group methods

The main goal of the conference was bringing together many researchers from different disciplines, who presented high level talks. The conference also served as a celebration of Prof. de la Llave 60th birthday.

Several of the presentations in the conference have been written up, refereed for correctness and relevance, and gathered in a special volume of Discrete and Continuous Dynamical Systems-A. 

Prof. de la Llave and Prof. C. Zeng have been editors of the journal for several years.

Conference website:

http://www.crm.cat/en/Activities/Curs_2016-2017/Pages/C_FIDDS.aspx

Preface of Llavefest:

http://aimsciences.org//article/doi/10.3934/dcds.201812i#FullText

Discrete and Continuous Dynamical Systems-A Issue 38-12:

http://aimsciences.org/journal/1078-0947/2018/38/12

Episode 9 of ScienceMatters' Season 1 stars Dan Margalit.  Listen to the podcast or read the transcript here. 

Dan Margalit is a professor in the School of Mathematics. 

Margalit's research area is topology. He studies the properties of shapes that persist even when the shapes are stretched or bent. 

For example, two metal rings that are linked stay linked even if you bend or stretch the metal. A typical question in topology is the following: Someone hands you two rings made of metal; if you are allowed to bend and stretch the metal, can you pull the rings apart or not? 

Most of Margalit's research in topology is about surfaces. The surface could be that of a ball or a donut. Surfaces are central in mathematics. They can describe the possible motions of a robot arm or all the possible solutions of a polynomial.

Margalit's particular research is on the symmetries of surfaces. Some symmetries of surfaces are easy to understand. But when bending and stretching are allowed, the symmetries are more challenging.

For Margalit, "mathematics is important because it describes the world in a beautiful and coherent way. Even the most far-fetched and abstract mathematical ideas can make their way into everyday life."

In Episode 9, Margalit talks about the beauty of mathematics and offers advice to overcome "math phobia."

Take a listen at sciencematters.gatech.edu.

Enter to win a prize by answering the question for Episode 9: 

According to Episode 9, what group of people can’t tell the difference between a coffee cup and a donut?

Submit your entry by 11 AM on Monday, Oct. 22, at sciencematters.gatech.edu. 

Elizabeth Ann “Libby” Peck earned two degrees from Georgia Tech: a B.S. in applied mathematics in 1975, and -- from the H. Milton Stewart School of Industrial and Systems Engineering -- an M.S. in industrial engineering in 1976. For over 40 years she applied the knowledge she learned from Tech in building mathematical models to answer myriad questions of the Coca-Cola Company—from supply chains to strategic infrastructure to delivery routes.

Libby was the first woman to use mathematical models for supply chain analysis at the Coca-Cola Company. Often, she was the only woman among male colleagues working on global problems. By standing up to defend her work vigorously and completing projects with assiduous diligence, she proved herself equal to the best of the men around her.

Read a Q&A with Peck in which she talks about her roles at Coca-Cola, how her ISyE degree prepared her for her work there, and her memories of being a student at Georgia Tech: https://b.gatech.edu/2RaZaiT.