Temporal instability analysis of inviscid compound jets falling under gravity

Compound liquid jets can be used in a variety of industrial applications ranging from capsule production in pharmaceutics to enhance printing methods in ink-jet printing. An appreciation of how instability along compound jets can lead to breakup and droplet formation is thus critical in many fields in science and engineering. In this paper, we perform a theoretical analysis to examine the instability of an axisymmetric inviscid compound liquid jet which falls vertically under the influence of gravity. We use a long-wavelength, slender-jet asymptotic expansion to reduce the governing equations of the problem into a set of one-dimensional partial differential equations, which describe the evolution of the leading-order axial velocity of the jet as well as the radii of both the inner and the outer interfaces. We first determine the steady-state solutions of the one-dimensional model equations and then we perform a linear temporal instability analysis to obtain a dispersion relation, which gives us useful inf...