Abstract
Shape and stability of a bubble embedded in a Rankine's combined vortex is investigated theoretically. Assuming that the gas inside the bubble changes polytropically, we introduce a variational function with a Lagrangian multiplier. Steady solutions are obtained from the condition of vanishing first variation and their stability is studied by looking at the sign of second variation. In general, a bubble stretches axially as the vortex becomes stronger, but a further increase of vortex strength makes the bubble to shrink to a small sphere, if the pressure inside the bubble is constant and its radius is small compared with the vortex radius. Any bubble is stable against asymmetric disturbances. Stability against axisymmetric disturbances depends, mainly, on the value of the polytropic exponent. Bubbles that are filled with a gas of constant temperature is stable, whereas those filled by a constant pressure gas are unstable irrespective of the vortex strength.
