Abstract
Real foams can be viewed as geometrically well-organized dispersions of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the foam can be viewed now as a network of thin liquid films intersecting each other at the Plateau borders according to the celebrated Plateau’s laws.In this paper we estimate from below the surface area of a spherically bounded piece of a foam. Our main tool is a new version of the divergence theorem which is adapted to the specific geometry of a foam with special attention to its classical Plateau singularities.As a benchmark application of our results, we obtain lower bounds for the fundamental cell of a Kelvin foam, lower bounds for the so-called cost function, and for the difference of the pressures appearing in minimal periodic foams. Moreover, we provide an algorithm whose input is a set of isolated points in space and whose output is the best lower bound estimate for the area of a foam that contains the given set as its vertex set.
Highlights
A foam is a cell decomposition of the Euclidean 3-space R3 into a finite or infinite number of properly embedded, connected 3D chambers
The constant mean curvature of the surface of each face in a foam is proportional to the pressure difference between the two cells meeting along the face
Every foam is organized around two angles: Every vertex treats the foam like the center vertex that locates the 6 inner wings in a regular tetrahedron, and every edge organizes three of these wings to have equal angles 2π/3 between them, see Fig. 1
Summary
A foam is a cell decomposition of the Euclidean 3-space R3 into a finite or infinite number of properly embedded, connected 3D chambers. Our main objective is to determine a lower bound for the area of an extrinsic disc (a ‘spherical scoop’) of a foam centered at a given vertex point.
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