  Prior modal stability analysis (Kojima et al., Phys. Fluids, vol. 27, 1984)
predicted that a rising or sedimenting droplet in a viscous fluid is stable in
the presence of surface tension no matter how small, in contrast to
experimental and numerical results. By performing a non-modal stability
analysis, we demonstrate the potential for transient growth of the interfacial
energy of a rising droplet in the limit of inertialess Stokes equations. The
predicted critical capillary numbers for transient growth agree well with those
for unstable shape evolution of droplets found in the direct numerical
simulations of Koh & Leal (Phys. Fluids, vol. 1, 1989). Boundary integral
simulations are used to delineate the critical amplitude of the most
destabilizing perturbations. The critical amplitude is negatively correlated
with the linear optimal energy growth, implying that the transient growth is
responsible for reducing the necessary perturbation amplitude required to
escape the basin of attraction of the spherical solution.
