Doughnut-shaped soap bubbles


Deison Préve and Alberto Saa

Departamento de Matemática Aplicada, Universidade Estadual de Campinas, 13083-859 Campinas, SP, Brazil

Abstract


Soap bubbles are thin liquid films enclosing a fixed volume of air. Since the surface tension is typically assumed to be the only responsible for conforming the soap bubble shape, the realized bubble surfaces are always minimal area ones. Here, we consider the problem of finding the axisymmetric minimal area surface enclosing a fixed volume V and with a fixed equatorial perimeter L. It is well known that the sphere is the solution for V = L 3 /6π 2 , and this is indeed the case of a free soap bubble, for instance. Surprisingly, we show that for V < αL 3 /6π 2 , with α ≈ 0.21, such a surface cannot be the usual lens-shaped surface formed by the juxtaposition of two spherical caps, but rather a toroidal surface. Practically, a doughnut-shaped bubble is known to be ultimately unstable and, hence, it will eventually lose its axisymmetry by breaking apart in smaller bubbles. Indisputably, however, the topological transition from spherical to toroidal surfaces is mandatory here for obtaining the global solution for this axisymmetric isoperimetric problem. Our result suggests that deformed bubbles with V < αL 3 /6π 2 cannot be stable and should not exist in foams, for instance.

PACS numbers: 47.55.D-, 47.55.db, 47.55.df



For the full arXiv text, click here. For the animation illustrating the transition from spherical to toroidal surfaces, click here.

Lens
Donut

FIG. 1: Top: lens-shaped surface of minimal area with perimeter L and volume V = αL 3 /6π 2 , with α ≈ 0.21. No stable lens-shaped surface with V < αL 3 /6π 2 should exist. Bottom: doughnut-shaped surface of minimal area with perimeter L and V = αL 3 /6π 2 . Axisymmetric minimal area surfaces with V < αL 3 /6π 2 are necessarily of this type. No doughnut-shaped minimal area surfaces exist with V > αL 3 /6π 2 . In both cases, the angle θ is the internal angle of the surface at the equatorial perimeter.



Fig.1


FIG. 2: Solid red line: solution for (12) with boundary condition f (ρ min ) = 0 for d ≈ 1.6235, which corresponds to λρ min ≈ 0.6197, λρ ∗ ≈ 2.2491, and λρ max ≈ 2.6197. The doughnut-shaped solution depicted in 1(b) is obtained by the revolution around the vertical axis of the closed curve formed by the solution and its reflection on the horizontal axis (dashed blue line) for ρ min ≤ ρ ≤ ρ ∗ . This particular value of d corresponds to the toroidal solution with maximal enclosed volume. The doughnut-shaped surface is regular everywhere except on the equatorial perimeter ρ = ρ ∗ .


Fig. 2


FIG. 3: Area × Volume diagram for axisymmetric minimal area surface with fixed equatorial perimeter L = 2π. The solid red line corresponds to the doughnut-shaped solutions, with the arrows indicating the direction of increasing d. The dashed green line corresponds to the lens-shaped solutions. In the detail, the region corresponding to the topological transition. The maximum volume for the doughnut-shaped solution is V ≈ 0.869, corresponding to the case depicted in Fig. 2. An animation illustrating the transition from spherical to toroidal surfaces is available at [2].


Fig. 3

FIG. 4: Dihedral angle θ between the tangent planes at the equatorial perimeter as a function of the enclosed volume V for axisymmetric minimal area surfaces with equatorial perimeter L = 2π. The solid red line corresponds to the doughnut-shaped solutions, with the arrows indicating the direction of increasing d, while the lens-shaped solutions are the dashed green line. Notice that for d → ∞, V → 0 and the dihedral angle tends to 180 degrees. The doughnut-shaped surface in this limit tends to the usual torus of circular section.
Handle
FIG. 5: Sphere with a toroidal handle: by shrinking the handle smaller radius, we have a surface with fixed external perimeter L, enclosing a volume V < αL 3 /6π 2 , and with area A smaller than our axisymmetric toroidal surface. Axisymmetric surfaces are not global solutions for the problem.