The shape and stability of a bubble at the axis of a rotating liquid

Abstract

The shape of a bubble of one liquid inside a denser body of liquid which rotate rigidly together is determined, the effect of gravity being neglected. When the anular velocity of the liquids is zero the bubble assumes a spherical form, and with increasing angular velocity the bubble flattens at the equator and the length increases. It is found that the length of the bubble is asymptotically proportional to the four-thirds power of the angular velocity. If the speed of the rotation is held constant and the volume is increased, then the bubble elongates, the radius approaches a limiting value, and the bubble length increases almost linearly with the volume. This result suggests a method whereby the interfacial surface tension can be measured.In the second part of the paper the stability of a long bubble subjected to small amplitude axisymmetric disturbances sinusoidal in the axial direction is investigated. The relation between the wave-number and angular velocity for neutral stability is elliptic. When account is taken of the decrease in the radius of the undisturbed bubble with increase in the angular velocity, it is found that the bubble is stable to all wave-lengths provided the radius attains at least 63% of the limiting value. A criterion is then found for the minimum length of the bubble consistent with stability.