Calculus of Variations and Geometric Measure Theory

G. Bevilacqua

Symmetry break in the eight bubble compaction

created by bevilacqua on 16 Dec 2021
modified on 13 Apr 2022


Accepted Paper

Inserted: 16 dec 2021
Last Updated: 13 apr 2022

Journal: Mathematics in Engineering
Year: 2022

ArXiv: 2007.15399 PDF


Geometry and mechanics have both a relevant role in determining the three-dimensional packing of 8 bubbles displyaed in a foam structure. We assume that the spatial arrangement of bubbles obeys a geometrical principle maximizing the minimum mutual distance between the bubble centroids. The compacted structure is then obtained by radially packing the bubbles under constraint of volume conservation. We generate a polygonal tiling on the central sphere and peripheral bubbles with both flat and curved interfaces. We verify that the obtained polyhedra is optimal under suitable physical criteria. Finally, we enforce the mechanical balance imposing the constraint of conservation of volume. We find an anisotropy in the distribution of the field of forces: surface tensions of bubble-bubble interfaces with normal oriented in the circumferential direction of bubbles aggregate are larger than the ones with normal unit vector pointing radially out of the aggregate. We suggest that this mechanical cue is key for the symmetry break of this bubbles configuration.