Math @ Duke

Publications [#14134] of Linda B. Smolka
Papers Submitted
 L. B. Smolka, A. Belmonte, D. M. Henderson, T. P. Witelski, Exact solution for the extensional flow of a viscoelastic filament,
European Journal of Applied Mathematics
(November, 2003)
(last updated on 2003/12/19)
Abstract: We solve the free boundary problem for the dynamics of a
cylindrical, axisymmetric viscoelastic filament
stretching in a purely extensional flow for the Upper
Convected Maxwell and OldroydB constitutive models.
Assuming the axial stress in the filament has a spatial
dependence provides the simplest coupling of viscoelastic
effects to the motion of the filament, and yields a closed
system of ODEs with an exact solution for the stretch rate
and filament thickness satisfied by both constitutive
models. This viscoelastic solution, which is a
generalization of the exact solution for Newtonian
filaments, converges to the Newtonian powerlaw scaling as
$t \go \infty$. Based on the exact solution, we identify two
regimes of dynamical behavior called the weakly and
stronglyviscoelastic limits. For the weaklyviscoelastic
case, corresponding to low Deborah numbers, the dynamics are
comparable to Newtonian behavior for all times and yield an
effective increase in the filament thickness relative to a
Newtonian fluid. In the stronglyviscoelastic case, initial
transient dynamics are not comparable to Newtonian behavior
and the effective filament thickness decreases with
increasing Deborah number. We compare the viscoelastic
solution to measurements of the thinning filament that forms
behind a falling drop for several semidilute
(stronglyviscoelastic) polymer solutions. We find the exact
solution correctly predicts the timedependence
of the filament diameter in all of the experiments. As $t
\go \infty$, observations of the filament thickness follow
the Newtonian scaling $1/\sqrt{t}$. The transition from
viscoelastic to Newtonian scaling in the filament thickness
is coupled to a stretchtocoil transition of the polymer
molecules.


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