2D Clusters of Bubbles

Here are some examples of two-dimensional clusters of bubbles with some interesting properties.

Stable cluster with a bubble whose pressure is lower that the outside pressure

Chain ring cluster Each of the bubbles in the outer two rings has unit area. The central bubble has area 13.376 and pressure -0.056. The negative central pressure is made possible by the stiffness of the outer double ring of bubbles, which is enough stiffness to suppress modes of instability that would let bubbles collapse into the central bubble or deform the cluster elliptically. The stability was checked using the Surface Evolver's eigenvalue calculation capability. The Evolver datafile for the cluster shown is chainring.dmp.

Cluster between two parallel plates that is stable in compression

chain wall cluster The cluster shown has the plate separation at minimum energy. Eigenvalue analysis by the Evolver shows this is stable. In fact, it remains stable until the plate separation narrows enough that the internal triple junctions make contact with the plates. The Evolver datafile for the cluster shown is chainwall.dmp.

Stability of double chain in a flat torus

double chain To investigate the stiffness of a double chain of bubbles without the complication of walls or curvature, I made a chain in a flat torus (i.e. periodic boundary conditions) and adjusted the period to minimize the energy. Evolver's eigenvalue analysis shows the chain to be stable at all lengths tested (up to 64 bubbles). Therefore double chains will remain stable under some degree of compression.