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Kirchhoff–Plateau Problem

June 15, 2017

A few scientists have had the honor of having a physical law named after them. Quite a few examples come to mind. In physics, we have Newton's law of universal gravitation, Planck's law, Coulomb's law, Lenz's law, Snell's law, and Hooke's law. In chemistry, we have Avogadro's law, Raoult's law, and Boyle's law. Electrical engineers have Ohm's law, while materials scientists have Vegard's law and Fick's law of diffusion. Wikipedia lists many physical laws named after people.

It's a rare honor to have a single law named after you, but German physicist, Gustav Kirchhoff (1824-1887) has five eponymous laws.
• Kirchhoff's circuit laws - This law, beloved by electrical engineers, states that the algebraic sum of electrical currents at a node is zero. It's essentially a conservation law.

• Kirchhoff's law of thermal radiation - If a body is emitting and absorbing thermal radiation in thermodynamic equilibrium, its emissivity is equal to its absorptivity. One aspect of this law is that it's not possible to thermally radiate more energy than a black body at equilibrium.

• Kirchhoff equations of fluid dynamics - These equations describe how a rigid body will move in an ideal fluid.

• Kirchhoff's three laws of spectroscopy - These laws, subsequently explained by quantum mechanics, concern the emission and absorption spectra of gases.

• Kirchhoff's law of thermochemistry - This law relates the change in the enthalpy of a chemical reaction with temperature (dΔH/dT) to the difference in heat capacity between the products and reactants (ΔCp); that is, dΔH/dT = ΔCp.

Figure captionAn experimental physicist in his natural abode, the laboratory.

Here, Kirchhoff is using a spectrometer of his own design.

(An illustration from the History of Physics by Poul la Cour and Jacob Appel, 1896, via Wikimedia Commons.)

Kirchhoff, who in 1857, about the same time that many principles of electromagnetism were being discovered, found that a signal in an ideal conductor travels at the speed of light. It was Kirchhoff who coined the term "black body" in 1862, but it was as a student in 1845 that he discovered his circuit laws, a topic that became his doctoral dissertation.

At the University of Heidelberg (Heidelberg, Germany), he collaborated on spectroscopy with Robert Bunsen, inventor of the eponymous Bunsen burner, jointly discovering cesium and rubidium in 1861. There's a Bunsen–Kirchhoff Award for spectroscopy that's awarded by the German Working Group for Applied Spectroscopy.

Soap bubbles are an inexpensive way to entertain children, but they became useful in physics in the late 1940s. At that time, Nobel laureate, William Lawrence Bragg, and John Nye of the Cavendish Laboratory of Cambridge University produced arrays of floating soap bubbles to simulate atoms in crystals.[1] They produced these bubble rafts using a solution of glycerine (162 mL), oleic acid (15.2 mL in 50 mL of water), and 73 mL of a 10% solution of triethanolamine in water, the mixture diluted to attain a desired viscosity. From such a solution they created rafts of 100,000 or more sub-millimeter bubbles, and they used this simulation to investigate such things as grain boundaries and dislocations.[1]

Ordered array of soap bubbles (bubble raft)The beginning of good bubble raft.

Production of uniform soap bubbles is relatively easy using the mechatronics available today, but Bragg, and Nye devised an elegant way to form their bubble rafts in the 1940s.[1]

(Portion of a Wikimedia Commons image by Timothy Pilgrim.)

Soap bubble are spherical for the reason that the surface tension is minimized at each point. Likewise, soap films, such as those attached to the loop used to make bubbles, are minimal surfaces in which surface tension causes the films to minimize their surface area and energy. A soap film on a loop is flat, but a soap film on a contorted loop of wire will have a contorted shape that's a minimal surface for that shape.

Since the soap film is attached to the wire frame that holds it, a mathematical model should include properties of the wire frame, which may be flexible and deformable by the surface tension of the soap film. In 1859, Kirchhoff found that the same equations that govern the motion of a heavy top also describe a thin elastic rod in equilibrium. Since Belgian physicist, Joseph Plateau, was the first to conjecture that the soap film formed on the wire frame spans a minimum area, independent of the shape of the frame, the extended problem of the film surface for a flexible frame is called the Kirchhoff–Plateau problem.

A team of mathematicians from the Okinawa Institute of Science and Technology Graduate University (Onna, Japan) and the Università Cattolica del Sacro Cuore (Brescia,Italy) have looked more closely at the Kirchhoff–Plateau problem of the equilibrium shapes formed when a closed, flexible filament is spanned by a soap film. The filament was modeled as a Kirchhoff rod, and the soap film was modeled by its surface tension.[2-3]

This interesting problem has had a long history, with the first satisfactory solution for the rigid filament Plateau problem discovered by American mathematician Jesse Douglas (1897-1965), winner of the 1936 Fields Medal.[3] In 2015, Jenny Harrison of the University of California, Berkeley, and Harrison Pugh of the State University of New York at Stony Brook (Stony Brook, New York) provided a proof for the complicated case in which multiple soap films meet each other in a junction.[3]

Since the filament holding the soap film in the Kirchhoff-Plateau problem is flexible, the surface tension of the soap film can change its shape, sometimes substantially. For example, as shown in the figure, the same loop can form a swan- or star-shape depending on the film's surface tension.[3] While the filament in the Plateau problem is one-dimensional, the Kirchhoff-Plateau filament is actually a rod. While these filament rods are thin, they are still much thicker than the soap film, and the soap film can contact the rod at different points.[3]

Figure caption
Star, or swan? The same flexible filament can produce a different shape of the soap film for different surface tensions. (Okinawa Institute of Science and Technology Graduate University image.)

The Okinawa Institute mathematicians were able to capture all such physical effects in their mathematical model. The system adjusted to a least energy configuration, independently of the ratio of the competing forces between the surface tension of the soap film and the elastic response of the loop.[2-3] Says Giulio Giusteri, coauthor of the paper that describes these results in the Journal of Nonlinear Science, "Our solution of the Kirchhoff–Plateau problem brings beautiful mathematical results close to what happens in the physical world."[3]

Applications of this model might include new insights into how the shape of a protein determines its interaction with binding surface sites.[3] The Okinawa Institute team is now examining, via computer simulations, how their model can predict the behavior of physical systems.[3]

Figure captionThis soap film is an example of multiple film surfaces joining together, a condition first investigated by Harrison and Pugh (see above).

(Okinawa Institute of Science and Technology Graduate University image.)

References:

  1. Lawrence Bragg and J.F. Nye, "A Dynamical Model of a Crystal Structure," Proc. R. Soc. Lond. A., vol. 190, no. 1023 (September 9, 1947), pp. 474-481, doi:10.1098/rspa.1947.0089. See also the PDF file at MIT.
  2. Giulio G. Giusteri, Luca Lussardi, and Eliot Fried, "Solution of the Kirchhoff-Plateau Problem," Journal of Nonlinear Science, , vol. 27, no. 3 (June 2017), pp. 1-21, DOI: 10.1007/s00332-017-9359-4. This is an open access article with a PDF file available here.
  3. Greta Keenan, "Bursting the Bubble: Solution to the Kirchhoff-Plateau Problem," Okinawa Institute of Science and Technology Graduate University Press Release, March 31, 2017.

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