Shape bifurcation of a pressurized ellipsoidal balloon

It is well-known that for most spherical and cylindrical rubber balloons the pressure versus volume curve associated with uniform inflation both has an N-shape, but their shape bifurcation has different characters: whereas a spherical balloon tends to bifurcate into a pear shape through localized thinning near one of the poles, a cylindrical balloon would always bulge out locally in a symmetric manner. To understand the connection between these two different bifurcation behaviors, we study in this paper the shape bifurcation of an ellipsoidal balloon which becomes a spherical balloon when the three axes are identical, and approximates a cylindrical balloon when one axis is much larger than the other two axes. The ellipsoidal shape is obtained by rotating an ellipse about one of its axes, that gives rise to two possibilities: a rugby shape or a pumpkin shape. It is shown that for a rugby-shaped balloon, there exists a threshold axes ratio below which the slender ellipsoidal balloon behaves more like a tube and bifurcation into a pear shape becomes impossible, whereas for a pumpkin-shaped balloon bifurcation into a pear shape is always possible.