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Exact solution for the extensional flow of a viscoelastic filament

Published online by Cambridge University Press:  03 May 2005

LINDA B. SMOLKA
Affiliation:
The W. G. Pritchard Fluid Mechanics Laboratory, Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA email: smolka@bucknell.edu Current address: Department of Mathematics, Bucknell University, Lewisburg, PA, 17837, USA.
ANDREW BELMONTE
Affiliation:
The W. G. Pritchard Fluid Mechanics Laboratory, Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
DIANE M. HENDERSON
Affiliation:
The W. G. Pritchard Fluid Mechanics Laboratory, Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
THOMAS P. WITELSKI
Affiliation:
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA email: smolka@bucknell.edu

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.

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Papers
Copyright
2004 Cambridge University Press

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