An accelerating liquid drop, under the action of surface tension, is shown to be unstable to small disturbances above a first critical value of the Bond number. Both numerical and second-order asymptotic methods are employed in order to characterize the normal-mode response and the neutral-stable modes at larger values of the Bond number. The transient response of an initially spherical drop that is accelerated by the flow of an external gas is studied as an initial-value problem. A unified theory, that includes acceleration as well as aerodynamic effects, is presented in order to account for the complete dynamic range of Weber and Bond numbers. The results are compared with experimental observations that range from continuous vibration to irreversible aerodynamic distortion and unstable shattering.

A formal higher-order matching procedure is employed to obtain a second-order asymptotic solution, of the JWKB type, to a Legendre-like differential equation with a large parameter. The equation has two second-order poles and a first-order turning point. In addition to the usual nonuniformity, the second-order JWKB approximation exhibits a divergent integral at these points. Eigenfunctions and eigenvalues, valid to the second order of approximation, are found by simultaneously matching the latter approximation to a turning-point expansion and two boundary-layer expansions. The solutions, which heretofore have not been described, are appropriate to the neutral stable surface waves manifest by an accelerating liquid sphere.

No abstract