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Philosophical Magazine Volume 83, 2003 - Issue 11
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Original Articles

Minimal perimeter for N identical bubbles in two dimensions: Calculations and simulations

S. J. Cox Department of Physics, Trinity College, Dublin 2, Ireland
,
F. Graner Laboratoire de Spectrométrie Physique, Unité Mixte de Recherche associée au CNRS 5588 et Université Grenoble I, BP 87, St Martin d'Hères Cedex, 38402, France
,
FÁtima Vaz Instituto de Ciência e Engenharia de Materiais e Superficies and Departamento de Engenharia de Materiais, Instituto Superior Técnico, Avenida Rovisco Pais, Lisboa, 1049-001, Portugal
,
C. Monnereau-Pittet Department of Physics, Trinity College, Dublin 2, Ireland
&
N. Pittet Department of Physics, Trinity College, Dublin 2, Ireland
Pages 1393-1406 | Published online: 15 Nov 2010

Citations (23)

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M. Fátima Vaz. (2008) Liquid foams: an introduction. Philosophical Magazine Letters 88:9-10, pages 627-636.
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M. A. Fortes , F. Morgan & M. Fátima Vaz. (2007) Pressures inside bubbles in planar clusters. Philosophical Magazine Letters 87:8, pages 561-565.
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S. J. Cox. (2006) Calculations of the minimal perimeter for deformable cells of equal area confined in a circle. Philosophical Magazine Letters 86:9, pages 569-578.
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A. Heppes & F. Morgan . (2005) Planar clusters and perimeter bounds. Philosophical Magazine 85:12, pages 1333-1345.
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M. A. Fortes & P. I. C. Teixeira . (2004) Helmholtz and Gibbs free energies of liquid foams. Philosophical Magazine Letters 84:11, pages 737-743.
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Weizhu Bao, Harald Garcke, Robert Nürnberg & Quan Zhao. (2022) A structure‐preserving finite element approximation of surface diffusion for curve networks and surface clusters. Numerical Methods for Partial Differential Equations 39:1, pages 759-794.
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Feng Xie, Zhiduo Liu, Changxing Wang, Da Chen, Wei Zhu, Xiamen Li, Qinglei Guo & Gang Wang. (2022) Graphene quantum dots assisted synthesis of highconcentration nitrogen doped graphene for infrared photodetectors. Diamond and Related Materials 121, pages 108774.
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Shu Zhang, Jinbo Pang, Qilin Cheng, Feng Yang, Yu Chen, Yu Liu, Yufen Li, Thomas Gemming, Xiaoyan Liu, Bergoi Ibarlucea, Jiali Yang, Hong Liu, Weijia Zhou, Gianaurelio Cuniberti & Mark H. Rümmeli. (2021) High‐performance electronics and optoelectronics of monolayer tungsten diselenide full film from pre‐seeding strategy. InfoMat 3:12, pages 1455-1469.
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Giulia Bevilacqua. (2021) Symmetry break in the eight bubble compaction. Mathematics in Engineering 4:2, pages 1-24.
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Élie Bretin, Roland Denis, Jacques-Olivier Lachaud & Édouard Oudet. (2019) Phase-field modelling and computing for a large number of phases. ESAIM: Mathematical Modelling and Numerical Analysis 53:3, pages 805-832.
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Dong Wang, Andrej Cherkaev & Braxton Osting. (2019) Dynamics and stationary configurations of heterogeneous foams. PLOS ONE 14:4, pages e0215836.
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Jinbo Pang, Rafael G. Mendes, Pawel S. Wrobel, Michal D. Wlodarski, Huy Quang Ta, Liang Zhao, Lars Giebeler, Barbara Trzebicka, Thomas Gemming, Lei Fu, Zhongfan Liu, Juergen Eckert, Alicja Bachmatiuk & Mark H. Rümmeli. (2017) Self-Terminating Confinement Approach for Large-Area Uniform Monolayer Graphene Directly over Si/SiO x by Chemical Vapor Deposition . ACS Nano 11:2, pages 1946-1956.
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. 2016. Geometric Measure Theory. Geometric Measure Theory 235 254 .
Sebastián Sensale, Washington Jones & R. Ernesto Blanco. (2014) Does osteoderm growth follow energy minimization principles?. Journal of Morphology 275:8, pages 923-932.
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John L. Moore, Guoxiang Li & Susannah M. Porter. (2014) Chancelloriid sclerites from the Lower Cambrian (Meishucunian) of eastern Yunnan, China, and the early history of the group. Palaeontology 57:4, pages 833-878.
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S. J. Cox, F. Morgan & F. Graner. (2013) Are large perimeter- minimizing two-dimensional clusters of equal-area bubbles hexagonal or circular?. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469:2149, pages 20120392.
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Denis Weaire, Steven T. Tobin, Aaron J. Meagher & Stefan Hutzler. 2012. Foam Engineering. Foam Engineering 5 26 .
Robert Nürnberg. (2009) Numerical simulations of immiscible fluid clusters. Applied Numerical Mathematics 59:7, pages 1612-1628.
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W. Nowicki & G. Nowicka. (2008) Spontaneous processes and stable structures in some three-phase cellular fluids. The European Physical Journal E 25:2, pages 153-161.
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M.A. Fortes, M. Fátima Vaz, S.J. Cox & P.I.C. Teixeira. (2007) Instabilities in two-dimensional flower and chain clusters of bubbles. Colloids and Surfaces A: Physicochemical and Engineering Aspects 309:1-3, pages 64-70.
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Frank Morgan. (2007) Colloquium : Soap bubble clusters . Reviews of Modern Physics 79:3, pages 821-827.
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QiCheng Sun. (2007) The mechanism of pattern formation in the developing drosophila retina. Science in China Series C: Life Sciences 50:1, pages 120-124.
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M F Vaz, S J Cox & M D Alonso. (2004) Minimum energy configurations of small bidisperse bubble clusters. Journal of Physics: Condensed Matter 16:23, pages 4165-4175.
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