Papers Submitted
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 Oldroyd-B 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 power-law scaling as
$t \go \infty$. Based on the exact solution, we identify two
regimes of dynamical behavior called the weakly- and
strongly-viscoelastic limits. For the weakly-viscoelastic
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 strongly-viscoelastic 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 semi-dilute
(strongly-viscoelastic) polymer solutions. We find the exact
solution correctly predicts the time-dependence
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 stretch-to-coil transition of the polymer
molecules.