Research Article: Dynamics and stationary configurations of heterogeneous foams

Date Published: April 29, 2019

Publisher: Public Library of Science

Author(s): Dong Wang, Andrej Cherkaev, Braxton Osting, Timon Idema.

http://doi.org/10.1371/journal.pone.0215836

Abstract

We consider the variational foam model, where the goal is to minimize the total surface area of a collection of bubbles subject to the constraint that the volume of each bubble is prescribed. We apply sharp interface methods to develop an efficient computational method for this problem. In addition to simulating time dynamics, we also report on stationary states of this flow for ≤ 21 bubbles in two dimensions and ≤ 17 bubbles in three dimensions. For small numbers of bubbles, we recover known analytical results, which we briefly discuss. In two dimensions, we also recover previous numerical results, computed using other methods. Particular attention is given to locally optimal foam configurations and heterogeneous foams, where the volumes of the bubbles are not equal. Configurational transitions are reported for the quasi-stationary flow where the volume of one of the bubbles is varied and, for each volume, the stationary state is computed. The results from these numerical experiments are described and accompanied by many figures and videos.

Partial Text

We consider the model for a d-dimensional foam (d = 2, 3) comprised of n bubbles, {Ωi}i=1n, each with a prescribed volume, Hd(Ωi)=Vi, that arrange themselves as to minimize the total surface area,
minHd(Ωi)=ViHd-1(∪i=1n∂Ωi).(1)

In this section, we review some relevant previous results in two and three dimensions.

In this section, we discuss computational methods for the foam model problem (1). Here, the goal is to find interfaces between adjacent bubbles such that the total interfacial area is minimal with the constraint that the volume of each bubble is fixed. To design a numerical algorithm for (1), the first consideration is the method to represent the interfaces between bubbles. For contrast, we review several choices before describing the method used in the present work.

In this paper, we considered the variational foam model (1), where the goal is to minimize the total surface area of a collection of bubbles subject to the constraint that the volume of each bubble is prescribed. Sharp interface methods together with an approximation of the interfacial surface area using heat diffusion leads to (9), which can be efficiently solved using the auction dynamics method developed in [27]. This computational method was then used to simulate time dynamics of foams in two- and three-dimensions; compute stationary states of foams in two- and three-dimensions; and study configurational transitions in the quasi-stationary flow where the volume of one of the bubbles is varied and, for each volume, the stationary state is computed. The results from these numerical experiments are described and accompanied by many figures and videos.

 

Source:

http://doi.org/10.1371/journal.pone.0215836

 

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