Abstract :
A feature characterizing partially inflated scientific balloons is a lobed structure surrounding the gas bubble. We present a mathematical model for such a balloon shape. We formulate a variational principal for the balloon, and seek to determine shapes that minimize its total energy subject to a volume constraint. The energy includes contributions due to film and load tendon strain, film and load tendon weight, and hydrostatic pressure. Wrinkling of the film is modeled via energy relaxation. We investigate the plausibility of a minimum energy selection principle for strained wrinkled balloon shapes with lobes. We find multiple solutions for the same set of conditions, demonstrating nonuniqueness. In solution space, these equilibrium configurations are local minimizers. The means by which a particular lobe pattern is “selected” in a real ascending balloon is dependent on a number of factors, some of which may be indeterminate (e.g., the distribution of the excess balloon material below the gas bubble). Nevertheless, with our approach, we are able to calculate partially inflated balloon equilibrium shapes that have many features of real ascent shapes, and, as a by-product of our solution process, we are able to obtain estimates of the film stress resultants.
