Canonical Soap Film Problem

David J. Wollkind () and Bonni J. Dichone
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David J. Wollkind: Washington State University, Department of Mathematics
Bonni J. Dichone: Gonzaga University, Department of Mathematics

Chapter Chapter 4 in Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences, 2017, pp 81-98 from Springer

Abstract: 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.

Keywords: Soap Film; Pastoral Interlude; Catenary Curve; Lateral Surface Area; Concentric Circular Rings (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-73518-4_4

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DOI: 10.1007/978-3-319-73518-4_4

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