Instability of an annular viscous liquid jet

A linear analysis has been carried out for the temporal instability of an annular viscous liquid jet moving in an inviscid gas medium, which includes three limiting cases of a round liquid jet, a gas jet and a plane liquid sheet. It is found that there exist two independent unstable modes, which become the well-known sinuous and varicose modes for plane liquid sheets as annular jet radii approach to infinity. Hence, they are named as para-sinuous and para-varicose. It is shown that an ambient gas medium always enhances the annular jet instability. The curvature effects in general increase the disturbance growth rate, and may not be neglected for the breakup process of an annular or conical liquid sheet. An annular jet with a sufficiently small thickness tends to break up much faster than the corresponding plane liquid sheet, in accordance with existing experimental observations. Liquid viscosity has complicated dual effects on the instability. It is also found that there exists a critical Weber number below which surface tension is the source of instability. Whereas above it, instability is suppressed by surface tension effect and is promoted by aerodynamic interaction between the liquid and gas phase. For the practical importance of large Weber numbers such as related to liquid atomization, the para-sinuous mode is always predominant.