The stability of a rotating liquid drop

Abstract

In this paper, the stability of a rotating drop held together by surface tension is investigated by an appropriate extension of the method of the tensor virial. Consideration is restricted to axisymmetric figures of equilibrium which enclose the origin. These figures form a one-parameter sequence; and a convenient parameter for distinguishing the members of the sequence is Σ = ρΩ^2a^3/8T, where Ω is the angular velocity of rotation, a is the equatorial radius of the drop, ρ is its density, and T is the interfacial surface tension. It is shown that Σ <=slant 2.32911 (not 1 + surd2 as is sometimes supposed) if the drop is to enclose the origin. It is further shown that with respect to stability, the axisymmetric sequence of rotating drops bears a remarkable similarity to the Maclaurin sequence of rotating liquid masses held together by their own gravitation. Thus, at a point along the sequence (where Σ = 0.4587) a neutral mode of oscillation occurs without instability setting in at that point (i.e. provided no dissipative mechanism is present); and the instability actually sets in at a subsequent point (where Σ = 0.8440) by overstable oscillations with a frequency Ω. The dependence on Σ of the six characteristic frequencies, belonging to the second harmonics, is determined (tables 3 and 4) and exhibited (figures 3 and 4).