ScienceWeek

PHYSICS OF FLUIDS: ON DROP BREAKUP

The following points are made by P. Doshi et al (Science 2003 302:1185):

1) Underlying one of the common occurrences of daily life, the breakup of a liquid drop is a rich and beautiful phenomenon. As a drop divides, the neck connecting the different masses of fluid necessarily becomes arbitrarily thin with a curvature that grows without bound until molecular scales are reached. Because surface tension gives rise to a pressure proportional to the curvature, this pressure also diverges. Similar singularities, in which a physical quantity effectively diverges, occur in many different realms, ranging from the subatomic-nuclear fission (1) to celestial-star formation (2). The ubiquity, simplicity, and accessibility of drop breakup makes it ideal for studying divergent dynamical behavior that occurs elsewhere in nature.

2) Near such a singularity, the dynamics are normally governed by the proximity to the singularity, and the dynamics become universal so that all memory of initial and boundary conditions is lost. In such cases, the breakup becomes scale-invariant; after appropriate rescaling, drop shapes near the breakup can be superimposed at different times onto a single form, depending on only a few material parameters (3-5).

3) The authors report an important exception to this class of behavior: the breakup of a zero-viscosity drop inside an extremely viscous exterior fluid produces an unexpected nonuniversal form of singularity, in which the memory of the initial conditions persists throughout the breakup process. Axial structure imposed at the outset on large length scales remains as the thin neck collapses. The unusual character of this breakup suggests a novel and controllable method for producing submicrometer structures.

4) Because all classical fluids have a finite viscosity, it is important to understand the nature of the singularity when the interior viscosity is very small but nonzero. If the interior viscosity is sufficiently small, as it is for an air bubble in thick syrup, the zero-viscosity drop breakup dynamics persist down to the atomic scales. However, if the interior viscosity is large enough, or the exterior viscosity small enough, the singularity will be cut off. In this case, the large-scale shape of the drop assumes an unexpected appearance. The smooth profile is transformed into a long and thin thread, which can be less than 1 micron thick.

5) In summary: A low-viscosity drop breaking apart inside a viscous fluid is encountered when air bubbles, entrained in thick syrup or honey, rise and break apart. Experiments, simulations, and theory show that the breakup under conditions in which the interior viscosity can be neglected produces an exceptional form of singularity. In contrast to previous studies of drop breakup, universality is violated, so that the final shape at breakup retains an imprint of the initial and boundary conditions. A finite interior viscosity, no matter how small, cuts off this form of singularity and produces an unexpectedly long and slender thread. If exterior viscosity is large enough, however, the cutoff does not occur because the minimum drop radius reaches subatomic dimensions first.

References (abridged):

1. N. Bohr, Nature 143, 330 (1939)

2. S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, New York, 1967)

3. J. Eggers, Rev. Mod. Phys. 69, 865 (1997)

4. I. Cohen, S. R. Nagel, Phys. Fluids 13, 3533 (2001)

5. R. E. Goldstein, A. I. Pesci, M. J. Shelley, Phys. Rev. Lett. 70, 3043 (1993)

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