3 results
An asymptotic theory for the linear stability of a core–annular flow in the thin annular limit
- E. Georgiou, C. Maldarelli, D. T. Papageorgiou, D. S. Rumschitzki
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- Journal:
- Journal of Fluid Mechanics / Volume 243 / October 1992
- Published online by Cambridge University Press:
- 26 April 2006, pp. 653-677
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We study the linear stability of a vertical, perfectly concentric, core–annular flow in the limit in which the gap is much thinner than the core radius. The analysis includes the effects of viscosity and density stratification, interfacial tension, gravity and pressure-driven forcing. In the limit of small annular thicknesses, several terms of the expression for the growth rate are found in order to identify and characterize the competing effects of the various physical mechanisms present. For the sets of parameters describing physical situations they allow immediate determination of which mechanisms dominate the stability. Comparisons between the asymptotic formula and available full numerical computations show excellent agreement for non-dimensional ratio of undisturbed annular thickness to core radius as large as 0.2.
The expansion leads to new linear stability results (an expression for the growth rate in powers of the capillary number to the $\frac{1}{3}$ power) for wetting layers in low-capillary-number liquid–liquid displacements. The expression includes both capillarity and viscosity stratification and agrees well with the experimental results of Aul & Olbrich (1990).
Finally, we derive Kuramoto–Sivashinsky-type integro-differential equation for the later nonlinear stages of the interfacial dynamics, and discuss their solutions.
The double layer–capillary stability of an annular electrolyte fluid surrounding a dielectric-fluid core in a tube
- E. Georgiou, D. T. Papageorgiou, C. Maldarelli, D. S. Rumschitzki
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- Journal:
- Journal of Fluid Mechanics / Volume 226 / May 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 149-174
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In this paper we examine the linear stability of an annular film surrounding a dielectric-fluid core in a tube in the presence of double layers of charges at the film core and at the film–tube interfaces, when the fluid-fluid interface is of low tension. In the absence of electrostatic forces, the surface tension force arising from the circumferential curvature destabilizes, and that from the axial curvature stabilizes the system. The competition is such that waves larger than the unperturbed interface circumference are unstable and those shorter are stable. For charged layers in the film, two cases are examined: (i) double-layer repulsion where the volume charge density is everywhere of the same sign and (ii) double-layer attraction where the diffusive layers next to the film interfaces are of opposite signs. In the first case, double-layer repulsion and surface tension lowering stabilize the destabilizing action of the circumferential component of the surface tension force, and a window of stability can exist. In the case of double layers of opposite signs, double-layer attraction destabilizes the system, and growth rates larger than those caused by pure capillarity can arise. Finally, for the case of a core bounded by an infinite electrolyte, surface tension lowering stabilizes the destabilizing action of the circumferential component of the surface tension force and destabilizes the longitudinal one, although the magnitudes of these effects may differ. As a result the thread can become unstable to waves shorter than the interface circumference.
Temporal instability of compound threads and jets
- A. CHAUHAN, C. MALDARELLI, D. T. PAPAGEORGIOU, D. S. RUMSCHITZKI
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- Journal:
- Journal of Fluid Mechanics / Volume 420 / 10 October 2000
- Published online by Cambridge University Press:
- 17 October 2000, pp. 1-25
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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.