Temporal instability of compound threads and jets
Authors: CHAUHAN A. , MALDARELLI C. , PAPAGEORGIOU D. T. , RUMSCHITZKI D. S. 
Published: July 2002

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 (kmax). 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 kmax by a factor of approximately 2. However, for these
changes in the system ratios, the growth rate (smax) and the ratio of the amplitude
of the outer to the inner interface (Amax) for the fastest growing wave only change
marginally, with Amax 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, kmax
again changes by a factor of 2, but Amax and smax vary more significantly with large
amplitude ratios for low density ratios. The amplitude ratio of the stretching mode
at the maximally growing wave (Amax) 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 kmax 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.