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 (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.

Linear stability analysis of capillary instabilities for concentric cylindrical shells

“…In addition, we show results for a number of situations that have not been previously studied. For the limiting case of a thin shell, we show a connection to the classic single-cylinder and flatplane results, consistent with a similar result for air-clad two-fluid jets (Chauhan et al 2000). In a three-fluid system, we exhibit an interesting case in which two growth modes at different wavelengths have the same effective growth rate, leading to competing breakup processes that we demonstrate with full 3-dimensional Stokes-flow simulations.…”

“…For example, we find identical results to Chauhan et al (2000) for the N = 3 case of a two-fluid compound jet surrounded by air (with negligible air density and viscosity). ).]…”

“…In this paper, we generalize previous linear stability analyses (Lord Rayleigh 1879, 1892Tomotika 1935;Stone & Brenner 1996;Chauhan et al 2000) of Plateau-Rayleigh (capillary) instabilities in fluid cylinders to handle any number (N ) of concentric cylindrical fluid shells with arbitrary thicknesses, viscosities, densities, and surface tensions. This analysis is motivated by the fact that experimental work is currently studying increasingly complicated fluid systems for device-fabrication applications, such as drawing of microstructured optical fibres with concentric shells of different glasses/polymers (Hart et al 2002;Kuriki et al 2004;Pone et al 2006;Abouraddy et al 2007;Sorin et al 2007;Deng et al 2008) or generating double emulsions (Utada et al 2005;Shah et al 2008).…”

“…Beyond the regime of a single cylindrical jet, Stone & Brenner (1996) analysed the three-fluid (N = 3) Stokes cylinder problem, but only for equal viscosities. Chauhan et al (2000) analysed the N = 3 case where the inner two fluids have arbitrary viscosities and the outermost fluid is inviscid gas, taking into account the full NavierStokes equations. Gunawan et al (2002Gunawan et al ( , 2004 considered an array of identical viscous cylinders (in a single row or in a triangular configuration).…”

Antibubbles and fine cylindrical sheets of air

“…Below, we discuss some possibilities in turn. Recent studies on linear stability of stationary compound cylinders are due to Chauhan et al (2000) and Liang et al (2011). Our system is isothermal, ruling out thermocapillary effects (Neitzel & Dell'Aversana 2002).…”