A lower bound for the area of Plateau foams

Journal of inequalities and applications

article

Vicent Gimeno¹  Steen Markvorsen²  José M. Sotoca³ 

[1] Departament de Matemàtiques-IMAC, Universitat Jaume I;Technical University of Denmark;Departament de Llenguatges I Sistemes Informátics-INIT, Universitat Jaume I

关键词: Foams;    Bubbles;    Density;    Pressure;    Comparison geometry;   

DOI  :  10.1186/s13660-020-02362-4

学科分类：电力

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY