Recovery of the Rayleigh capillary instability from slender 1-D inviscid and viscous models

Abstract

For either inviscid or viscous jets, Rayleigh proved cylindrical jets are linearly unstable due to surface tension of the interface, with instability precisely in all wavelengths greater than the jet circumference. As an alternative to linearized analysis, many past and present studies of surface tension-driven jet breakup are based on slender asymptotic 1-D models; here we clarify two issues regarding this approach. First, self-consistent, leading-order models of inviscid or viscous slender jets do not have a finite instability cutoff. Indeed, the inviscid 1-D equations exhibit unbounded exponential growth in the small scale limit, while the viscous counterparts bound the growth rate but remain unstable in all wavelengths. Second, one can recover a finite instability cutoff by extending the asymptotic analysis to higher order. The linearized growth rate corrections at each finite order arise as algebraic approximations to Rayleigh’s exact exponential rate. We explicitly match, at leading and subsequent order, the slender longwave expansion of the exact results with the linearized behavior of 1-D slender asymptotic equations.