Dynamics and stability of a network of coupled drop<br/><br/>elements: Coarsening by capillarity

Tuesday, July 15, 2008 - 2:00pm - 2:10pm
EE/CS 3-180
Henrik van Lengerich (Cornell University)
A practical consequence of the breakup of a liquid jet by the
pinch-off singularity is the redistribution of volume. To the extent
that volume concentrates into drops in the streamwise direction,
pinch-off can lead to coarsening. The fundamental redistribution of
volume by surface tension can be understood in the absence of
pinch-off, however. We pose a simple model for the coarsening of
connected spherical-cap drops in the absence of pinch-off. Our study
shows that many properties of this simple model hold true for a
general class of coupled elements.

A system of N drops with pinned contact lines is coupled through a
network of conduits. The system coarsens in the sense that, as time
progresses, the volume becomes increasingly localized and ends up
primarily in a single 'winner' drop. Numerical simulations show that
the identity of the winner can depend discontinuously on the initial
condition and conduit network. This motivates a study of the
corresponding N-dimensional dynamical system. An analysis of the
system yields analytic expressions for the fixed points and their
energy stability, which depend only on the characteristic
pressure-volume response of each element. The dynamic stability is
shown to be identical to the energy stability, thus characterizing the
number of stable and unstable manifolds at each fixed point. To
determine which of the stable fixed points will be the winner,
separatrix manifolds of the attracting regions are found using a
method which combines local information from the eigenvectors at fixed
points with global information from invariant manifolds obtained from
symmetry. This method is used to explain phenomena observed in the
numerical simulations.

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