Studies of complexity in fluid systems Page: 4 of 10
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09/26/01 10:41 FAX 773 702 2142 Univ Research Adm lj003
describing the interface, so the dynamics should be self similar. Peregrine refined this
idea and applied it to their experiments on falling water drops: They point out that for low
viscosity fluids there is a range of scales where the thickness of the fluid neck is much
larger than the viscous length scale, but much smaller than the length scale where energy
is fed into the system. Over this range of scales, the self--similarity hypothesis might be
expected to hold.
These considerations suggest that a breaking fluid interface should be described
by a similarity solution to the governing hydrodynamic equations. Under the auspices of
the DOE Kadanoff et al. performed the first study that succeeded in constructing a
similarity solution for a breaking fluid thread in the rupture of a drop in the two
dimensional Heleshaw cell. This initial success was followed up with the identification
of further similarity solutions for the Heleshaw cell as well as the discovery of a similarity
solution for three dimensional droplet fission for fluid threads. In this case the similarity
solution is unstable to finite amplitude perturbations, with the critical amplitude for
instability approaching zero at the singularity. At the high viscosities where this
similarity solution is relevant, the droplet shape is long and slender.
Nagel and collaborators have also studied the question whether similar scaling
ideas can explain the interfacial shapes occurring during the breakup of low viscosity
fluids, such as a water drop. Based on the above discussion, this translates into the
question of whether the self similar singularities in the equations of inviscid
hydrodynamics describe experiments when the thread thickness is larger than the viscous
length scale? To answer this question, we have studied in detail the characteristics of a
water drop falling from a nozzle both before and after a fission event. Through a
combination of experiments, numerical simulations and theory we argued that the scaling
hypothesis fails for this problem. In the absence of viscosity, this instability actually
causes a singularity in the curvature of the interface before rupture occurs .
Sedimentation at Contact Lines
When a spilled drop of coffee dries on a solid surface, it leaves a dense, ring-like
stain along the perimeter. The coffee---initially dispersed over the entire drop---becomes
concentrated into a tiny fraction of it. Such ring deposits are commonplace wherever
drops containing dispersed solids evaporate on a surface. Thus ring deposits influence
printing, washing and coating processes. They provide a potential means to write or
deposit a fine pattern onto a surface. Dupont, Nagel and collaborators have investigated
the causes of such deposits. We ascribe the deposition to a previously unexplored form
of capillary flow: the contact line of the drying drop is pinned so that liquid evaporating
from the edge must be replenished by liquid from the interior. The resulting outward
flow can carry virtually all the dispersed material to the edge. This mechanism predicts a
distinctive power-law growth of the ring mass with time---a law independent of the
particular substrate, carrier fluid or deposited solids. We have verified this law by
microscopic observations of colloidal fluids.
Our initial observations show that rings form for a wide variety of substrates,
dispersed materials (solutes), and carrier liquids (solvents), as long as (1) the solvent
meets the surface at a nonzero contact angle, (2) the contact line is pinned to its initial
position, and (3) the solvent evaporates. In addition, we found that mechanisms
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Nagel, Sidney R. Studies of complexity in fluid systems, report, June 12, 2000; United States. (https://digital.library.unt.edu/ark:/67531/metadc734776/m1/4/: accessed May 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.