Minimum energy configurations of small bidisperse bubble clusters

Vaz, M. Fátima; Cox, Simon; Alonso, María Dolores

doi:10.1088/0953-8984/16/23/031

Cited by 11 publications

(7 citation statements)

References 20 publications

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“…The same expression gets all but thirteen values correct for the circular case and all but four values correct for free clusters. The expression for the perimeter derived in (9), with N p given by (12) and s = 6, works well here, indicating that the bubbles are quite regular. (This expression fails to describe the square and pentagonal data.)…”

Section: Hexagonal Boundarysupporting

confidence: 70%

“…Certainly, to extend the results presented here to higher N will require an improved algorithm. Similarly, relaxing the condition of monodispersity, to consider for example bidisperse clusters [9], leads to many more candidates and the likelihood of fully exploring the space of all good candidates decreases.…”

Section: Discussionmentioning

confidence: 99%

“…So, the honeycomb conjecture is proven, but there are many variations on the conjecture that are worthy of further exploration. One obvious direction is to relax the condition that all bubbles have equal area [8, §15.9] [9]. Another is to consider finite collections of N equal-area bubbles (clusters), which we do here.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

Cox¹,

Flikkema²

2010

Electron. J. Combin.

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Candidates to the least perimeter partition of various polygonal shapes into $N$ planar connected equal-area regions are calculated for $N\le 42$, compared to partitions of the disc, and discussed in the context of the energetic groundstate of a two-dimensional monodisperse foam. The total perimeter and the number of peripheral regions are presented, and the patterns classified according to the number and position of the topological defects, that is non-hexagonal regions (bubbles). The optimal partitions of an equilateral triangle are found to follow a pattern based on the position of no more than one defect pair, and this pattern is repeated for many of the candidate partitions of a hexagon. Partitions of a square and a pentagon show greater disorder. Candidates to the least perimeter partition of the surface of the sphere into $N$ connected equal-area regions are also calculated. For small $N$ these can be related to simple polyhedra and for $N \ge 14$ they consist of 12 pentagons and $N-12$ hexagons.

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Section: Hexagonal Boundarysupporting

confidence: 70%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

Cox¹,

Flikkema²

2010

Electron. J. Combin.

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“…For N = 2 regions in R 3 , the double bubble conjecture has been proved [8], and, in the plane, the extension of the honeycomb to two different areas (bidisperse) has led to conjectured solutions [9]. There has also been some experimental work that sought to correlate the frequency with which different configurations of bidisperse bubble clusters (which, to a good approximation, minimize their surface area [1]) were found with the least perimeter configuration [10].…”

Section: Introductionmentioning

confidence: 99%

Least perimeter partition of the disc into N bubbles of two different areas

Headley

Cox

2019

Eur. Phys. J. E

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We present conjectured candidates for the least perimeter partition of a disc into N ≤ 10 regions which take one of two possible areas. We assume that the optimal partition is connected, and therefore enumerate all three-connected simple cubic graphs for each N . Candidate structures are obtained by assigning different areas to the regions: for even N there are N/2 regions of one area and N/2 regions of the other, and for odd N we consider both cases, i.e. where the extra region takes either the larger or the smaller area. The perimeter of each candidate is found numerically for a few representative area ratios, and then the data is interpolated to give the conjectured least perimeter candidate for all possible area ratios. At larger N we find that these candidates are best for a more limited range of the area ratio.

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“…This last rule implies a minimal surface area property, leading to instabilities and topological rearrangements into larger clusters. In the literature, such situations are investigated in the framework of liquid foams (see [3][4][5]). Usual approaches for modelling and simulating these phenomena are essentially based on minimal surface area criteria (see [6,7]).…”

Section: Introductionmentioning

confidence: 99%

Direct 2D simulation of small gas bubble clusters: from the expansion step to the equilibrium state

Bruchon

Fortin

Bousmina

et al. 2006

Numerical Methods in Fluids

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International audienceA numerical strategy, based on an adaptive finite element method, is proposed for the direct two-dimensional simulation of the expansion of small clusters of gas bubbles within a Newtonian liquid matrix. The velocity and pressure fields in the liquid are first defined through the Stokes equations and are subsequently extended to the gas bubbles. The liquid-gas coupling is imposed through the stress exerted on the liquid by gas pressure (ruled by an ideal gas law) and by surface tension. A level set method, combined with a mesh adaptation technique, is used to track liquid-gas interfaces. Many numerical simulations are presented. The single bubble case allows to compare the simulations to an analytical model. Simulations of the expansion of small clusters are then presented showing the interaction and evolution of the gas bubbles to an equilibrium state, involving topological rearrangements induced by Plateau's rule

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Minimum energy configurations of small bidisperse bubble clusters

Cited by 11 publications

References 20 publications

The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

The Minimal Perimeter for $N$ Confined Deformable Bubbles of Equal Area

Least perimeter partition of the disc into N bubbles of two different areas

Direct 2D simulation of small gas bubble clusters: from the expansion step to the equilibrium state

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