Temporal instability of compound threads and jets

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 destab ilize cylindrical base states of compound threads and jets (with inner and outer radii R-1 and aR(1) respectively). The capillary instability causes b reakup into drops; the presence of the annular phase allows both the annula r- and core-phase properties to influence the drop size. Of technological i nterest is breakup where the core snaps first, and then the annulus. This r esults in compound drops. With jets, this pattern can form composite partic les, or if the annular fluid is evaporatively removed, single drops whose s ize is modulated by both fluids. This paper is a study of the linear temporal instability of compound thread s and jets to understand how annular fluid properties control drop size in jet breakup, and to determine conditions which favour compound drop formati on. The temporal dispersion equation is solved numerically for non-dimensio nal annular thicknesses a of order one, and analytically for thin annuli (a - 1 = epsilon much less than 1) by asymptotic expansion in epsilon. There are two temporally growing modes: a stretching mode, unstable for wavelengt hs greater than the undisturbed inner circumference 2 pi R-1, in which the two interfaces grow in phase; and a squeezing mode, unstable for wavelength s greater than 2 pi aR(1), which grows exactly out of phase. Growth rates a re always real, indicating that in jetting configurations disturbances conv ect downstream with the base velocity. For order-one thicknesses, the growt h rate of the stretching mode is higher for the entire range of system para meters 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 viscosi ty, 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 t hese changes in the system ratios, the growth rate (s(max)) and the ratio o f the amplitude of the outer to the inner interface (A(max)) for the fastes t growing wave only change marginally, with A(max) near one. The system app ears 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 r atios for low density ratios. The amplitude ratio of the stretching mode at the maximally growing wave (A(max)) indicates whether the film or core wil l break first. When this ratio is near one, linear theory predicts that the core breaks with the annulus intact, forming compound drops. Except for lo w values of the density ratio, our results indicate that most system condit ions promote compound drop formation. For thin annuli, the growth rate disparity between modes becomes even great er. In the limit epsilon --> 0, the squeezing growth rate is roughly propor tional to epsilon(2) while the stretching mode growth rate is roughly propo rtional to epsilon(0) and asymptotes to a single jet with radius R-1 and te nsion equal to the sum of the two tensions. Thus, in this limit the growth rate and k(max) are independent of the him density and viscosity. The ampli tude 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.