We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven 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 ınfty\$. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly- and strongly-viscoelastic limits. 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 ınfty\$, observations of the filament thickness follow the Newtonian scaling \$1/t\$. The transition from viscoelastic to Newtonian scaling in the filament thickness is coupled to a stretch-to-coil transition of the polymer molecules.
%0 Journal Article
%1 citeulike:3048453
%A Smolka, Linda B.
%A Belmonte, Andrew
%A Henderson, Diane M.
%A Witelski, Thomas P.
%D 2005
%J European Journal of Applied Mathematics
%K 76d27-other-free-boundary-flows-hele-shaw-flows 76a10-viscoelastic-fluids
%N 6
%P 679--712
%R 10.1017/s0956792504005789
%T Exact solution for the extensional flow of a viscoelastic filament
%U http://dx.doi.org/10.1017/s0956792504005789
%V 15
%X We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven 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 ınfty\$. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly- and strongly-viscoelastic limits. 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 ınfty\$, observations of the filament thickness follow the Newtonian scaling \$1/t\$. The transition from viscoelastic to Newtonian scaling in the filament thickness is coupled to a stretch-to-coil transition of the polymer molecules.
@article{citeulike:3048453,
abstract = {We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven 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 \rightarrow \infty\$. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly- and strongly-viscoelastic limits. 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 \rightarrow \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.},
added-at = {2017-06-29T07:13:07.000+0200},
author = {Smolka, Linda B. and Belmonte, Andrew and Henderson, Diane M. and Witelski, Thomas P.},
biburl = {https://www.bibsonomy.org/bibtex/2a15e52cf7aef469a97100048617e1250/gdmcbain},
citeulike-article-id = {3048453},
citeulike-attachment-1 = {smolka_05_exact_33778.pdf; /pdf/user/gdmcbain/article/3048453/33778/smolka_05_exact_33778.pdf; 21e28c841a33332e374e69b4be3793245ac41efe},
citeulike-linkout-0 = {http://journals.cambridge.org/action/displayAbstract?fromPage=online\&aid=302338},
citeulike-linkout-1 = {http://dx.doi.org/10.1017/s0956792504005789},
day = 3,
doi = {10.1017/s0956792504005789},
file = {smolka_05_exact_33778.pdf},
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journal = {European Journal of Applied Mathematics},
keywords = {76d27-other-free-boundary-flows-hele-shaw-flows 76a10-viscoelastic-fluids},
number = 6,
pages = {679--712},
posted-at = {2008-07-28 08:35:56},
priority = {2},
timestamp = {2019-04-02T01:46:43.000+0200},
title = {{Exact solution for the extensional flow of a viscoelastic filament}},
url = {http://dx.doi.org/10.1017/s0956792504005789},
volume = 15,
year = 2005
}