Temporal instability of compound threads and jets

Abstract

Compound threads and jets consist of a core liquid surrounded by an annulus of a second immiscible liquid. Capillary forces derived from axisymmetric disturbances in the circumferential curvatures of the two interfaces destabilize cylindrical base states of compound threads and jets (with inner and outer radii R₁ and aR₁ respectively). The capillary instability causes breakup into drops; the presence of the annular phase allows both the annular- and core-phase properties to influence the drop size. Of technological interest is breakup where the core snaps first, and then the annulus. This results in compound drops. With jets, this pattern can form composite particles, or if the annular fluid is evaporatively removed, single drops whose size is modulated by both fluids.

This paper is a study of the linear temporal instability of compound threads and jets to understand how annular fluid properties control drop size in jet breakup, and to determine conditions which favour compound drop formation. The temporal dispersion equation is solved numerically for non-dimensional annular thicknesses a of order one, and analytically for thin annuli (a – 1 = ε [Lt ] 1) by asymptotic expansion in ε. There are two temporally growing modes: a stretching mode, unstable for wavelengths greater than the undisturbed inner circumference 2πR₁, in which the two interfaces grow in phase; and a squeezing mode, unstable for wavelengths greater than 2πaR₁, which grows exactly out of phase. Growth rates are always real, indicating that in jetting configurations disturbances convect downstream with the base velocity. For order-one thicknesses, the growth rate of the stretching mode is higher for the entire range of system parameters examined. The drop size scales with the wavenumber of the maximally growing wave (k_max). We find that for the dominant stretching mode and a = 2, variations from 0.1 to 10 in the ratios of the annulus to core viscosity, or the tension of the outer surface to that of the inner interface, can result in changes in k_max by a factor of approximately 2. However, for these changes in the system ratios, the growth rate (s_max) and the ratio of the amplitude of the outer to the inner interface (A_max) for the fastest growing wave only change marginally, with A_max near one. The system appears most sensitive to the ratio of the density of the annulus to the core fluid. For a variation between 0.1 and 10, k_max again changes by a factor of 2, but A_max and s_max vary more significantly with large amplitude ratios for low density ratios. The amplitude ratio of the stretching mode at the maximally growing wave (A_max) indicates whether the film or core will break first. When this ratio is near one, linear theory predicts that the core breaks with the annulus intact, forming compound drops. Except for low values of the density ratio, our results indicate that most system conditions promote compound drop formation.

For thin annuli, the growth rate disparity between modes becomes even greater. In the limit ε → 0, the squeezing growth rate is roughly proportional to ε² while the stretching mode growth rate is roughly proportional to ε⁰ and asymptotes to a single jet with radius R₁ and tension equal to the sum of the two tensions. Thus, in this limit the growth rate and k_max are independent of the film density and viscosity. The amplitude ratio of the stretching mode becomes equal to one for all wavenumbers; so thin films break as compound drops. Our results compare favourably with previously published measurements on unstable waves in compound jets.