Immiscible fluid clusters in \({\mathbb{R}}^2\) and \({\mathbb{R}}^3\)
type: Item
instance of: scholarly article
title: Immiscible fluid clusters in \({\mathbb{R}}^2\) and \({\mathbb{R}}^3\)
zbMATH Open document ID: 0967.49026
author: Frank Morgan
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: The author considers a system of immiscible fluids \(F_1\), \(F_2\), \(\dots\), \(F_{m}\) in \({\mathbb R}^{n}\) for \(n = 2,3.\) The ambient space is represented by \(F_0\). The energy of each interface is defined to be proportional to the surface area with constant of proportionality \(a_{ij}>0\). A non-degeneracy condition \(a_{ik}<a_{ij}+a_{jk}\) is imposed. By a nice argument which he summarizes in Section 1.1, the author shows that the energy-minimizing cluster in the planar case consists of arcs of circles meeting at isolated points. The final section of the paper considers the problem in \({\mathbb R}^3\) and uses a compactness argument to generalize Taylor's classification of soap-bubble cluster singularities [\textit{J. E. Taylor}, Ann. Math. (2) 103, 489-539 (1976; Zbl 0335.49032)] to clusters of immiscible fluids with interface energies near 1.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 76M30
Mathematics Subject Classification ID: 58E12
zbMATH DE Number: 1548287
zbMATH Keywords: fluid clusters
zbMATH Keywords: regularity
zbMATH Keywords: immiscible fluids
zbMATH Keywords: surface area
reviewed by: Harold Parks
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132292
OpenAlex ID: W2020890591
DOI: 10.1307/MMJ/1030132292
description: scientific article; zbMATH DE number 1548287