Instabilities in two-dimensional flower and chain clusters of bubbles

We assess the stability of simple two-dimensional clusters of bubbles relative to small displacements of the vertices, at fixed bubble areas. The clusters analysed are: (1) flower clusters consisting of a central bubble of area λ surrounded by N shells each containing n bubbles of unit area, (2) periodic chain clusters consisting of N “parallel” rows of n   bubbles of unit area and width ww. The energy and bubble pressures of the symmetrical, unbuckled clusters are found analytically as a function of λ   and ww for given N and n. Both types of clusters studied show a single energy minimum at a critical λm or wmwm. At the energy minimum for flower clusters, the pressure in the central bubble vanishes. The clusters show a symmetry-breaking buckling instability under compression at a critical λb or wbwb. The corresponding critical energy Eb was determined with the Surface Evolver software. While for N = 1 the conditions λb = λm, wb=wmwb=wm and Eb = Em hold, for N > 1 buckling requires further compression beyond the minimum, for which the energy increases with increasing compression (decreasing λ   or ww), and the excess pressure in the central bubble of the flower clusters becomes negative.