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Canonical Soap Film Problem

In: Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences

Author

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  • David J. Wollkind

    (Washington State University, Department of Mathematics)

  • Bonni J. Dichone

    (Gonzaga University, Department of Mathematics)

Abstract

In this chapter, the canonical Plateau problem of determining the shape of the soap film formed between two concentric circular wire rings of possibly different radii, as a function of the axial distance between the rings, is considered. Since that shape is a catenoid of revolution, which is a minimal surface determined by a calculus of variations approach, the concepts of a catenary curve, surface area integrals, and the initial treatment of the Calculus of Variations are all introduced as pastoral interludes. Then the behavior of the soap film as the distance of separation slowly increases is examined until that surface ruptures and occupies the two concentric circular rings alone. Given that this critical distance requires a determination of the envelope of a one-parameter family of curves, that concept is presented as a pastoral interlude as well. The problem deals with the closely related Calculus of Variations brachistochrone example of determining the shape for the curve which provides the shortest time of descent of a bead sliding along it acted upon by gravitational force alone.

Suggested Citation

  • David J. Wollkind & Bonni J. Dichone, 2017. "Canonical Soap Film Problem," Springer Books, in: Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences, chapter 0, pages 81-98, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-73518-4_4
    DOI: 10.1007/978-3-319-73518-4_4
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