Préve Deison, Saa Alberto
Departamento de Matemática Aplicada, IMECC-UNICAMP, 13083-859 Campinas, São Paulo, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):042402. doi: 10.1103/PhysRevE.92.042402. Epub 2015 Oct 9.
Soap bubbles are thin liquid films enclosing a fixed volume of air. Since the surface tension is typically assumed to be the only factor 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 is 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.
肥皂泡是包裹着一定体积空气的薄液膜。由于通常假定表面张力是决定肥皂泡形状的唯一因素，所以实际形成的泡表面总是具有最小面积的。在此，我们考虑这样一个问题：寻找一个轴对称的最小面积表面，它包裹着固定体积(V)且具有固定的赤道周长(L)。众所周知，当(V = L^3 / 6\pi^2)时，球体就是该问题的解，例如自由肥皂泡的情况确实如此。令人惊讶的是，我们发现当(V < \alpha L^3 / 6\pi^2)（其中(\alpha\approx0.21)）时，这样的表面不是由两个球冠并置形成的常见透镜状表面，而是一个环形表面。实际上，人们知道甜甜圈形状的气泡最终是不稳定的，因此它最终会通过分裂成更小的气泡而失去其轴对称性。然而，无可争议的是，对于这个轴对称等周问题，从球形表面到环形表面的拓扑转变在这里是获得全局解所必需的。我们的结果表明，例如在泡沫中，体积(V < \alpha L^3 / 6\pi^2)的变形气泡不可能是稳定的，也不应存在。
