In this chapter we carry out an analysis of the stability of a liquid jet into a gas or another fluid by using VPF. This instability may be driven by KH instability that is due to a velocity difference and a neck-down that is due to capillary instability. KH instabilities are driven by pressures generated by a dynamically active ambient flow, gas or liquid. On the other hand, capillary instability can occur in a vacuum; the ambient can be neglected. KH instability is included by a discontinuity of the velocity at a two-fluid interface. This discontinuity is inconsistent with the no-slip condition for Navier–Stokes studies of viscous fluids, but is consistent with the theory of potential flow of a viscous fluid. We start our study with an analysis of capillary instability.

Capillary instability of a liquid cylinder in another fluid

The study of this problem is especially valuable because it can be solved exactly and was solved by Tomotika (1935). This solution allows one to compute the effects of vorticity generated by the no-slip condition. The ES can be compared with irrotational solutions of the same problem. One effect of viscosity on the irrotational motion may be introduced by evaluation of the viscous normal stress at the liquid–liquid interface on the irrotational motions.