STABILITY OF A PRESSURIZED ELLIPSOIDAL BALLOON

A pressurized ellipsoidal balloon may bifurcate into different shapes depending on its precise shape. For a rugby-shaped balloon, there exists a threshold ratio of the axes in the Z- and R- directions, above which the balloon tends to bifurcate into a pear shape. Otherwise, the pear shape is impossible and when the balloon is slender enough, it may bulge out locally in a symmetric manner more like a tube. However, for a pumpkin-shaped balloon, bifurcation into a pear shape is always possible. In this paper, by using an energy criterion, we determine the stability properties of the primary and bifurcated solutions under pressure control and volume control,respectively. The total energy of the equilibrium state and its disturbed state are calculated, and the di erence between these two states is used to evaluate the stability of current state. Our analyses indicate that under pressure control, both primary and bifurcated solutions that exist on the descending branch of the pressure versus volume curve are unstable, but under volume control, the bifurcated solution is always stable whenever it appears while the primary solution is only stable when there does not exist any bifurcated solution. However, the primary solutions that exist on the two ascending branches are always stable.