Exact solution for the extensional flow of a viscoelastic filament

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 → ∞. 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 → ∞, 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.

Constant force extensional rheometry of polymer solutions

“…Analysis of the pendant drop experiment by Keiller [19] showed that both viscous and elastic forces contributed significantly to the thinning dynamics and complicated the use of this experiment as a true rheometer. The relatively low mass of a pendant drop (as compared to the weights used by Matta & Tytus [28] and in the current paper) also results in low imposed deformation rates; for example, the maximum accelerations achieved by Jones et al were only approximately ∼ 0.5 g. A recent detailed analysis of this problem for the Maxwell/Oldroyd-B model shows that as the tensile stresses grow and retard further acceleration of the pendant droplet and the elongating filament, the dynamics can in fact relax back towards a dominant Newtonian balance [43] which again limits the use of this geometry as an extensional rheometer.…”

“…The recent analysis of Smolka et al [43] highlights a key strength of the CFP technique suggested by Matta & Tytus: Applying a high enough initial force allows one to impose filament deformation rates d ln L/dt that are sufficiently fast to overcome the relaxation time λ and the relaxation processes in the elongating fluid filament. With Weissenberg numbers W i = λd ln L/dt ≫ 1 the constant force pull ultimately imposes an almost affine deformation on the microstructural elements in the polymeric sample towards their finite extensibility limit.…”

Charge screening effects on filament dynamics in xanthan gum solutions

“…A comparison of the interfacial motion of this filament for the 780 ppm xanthan gum solution with 0% ≤ [KCl] ≤ 0.047% to an analytic model shows strong quantitative agreement [34]. The model, which also provides the stretch rate in the filament, predicts De > 1 over a transient period while the filament initially stretches [34]. After this transient period, the stretch rate monotonically decreases so that eventually De < 1 [34].…”

“…The model, which also provides the stretch rate in the filament, predicts De > 1 over a transient period while the filament initially stretches [34]. After this transient period, the stretch rate monotonically decreases so that eventually De < 1 [34]. Thus we expect viscoelastic effects to be relevant to the filament motion during some initial time period in the experiment.…”

On the planar extensional motion of an inertially driven liquid sheet

“…[5][6][7][8] Frankel and Weihs, 5,6 applying their work to shaped charges, derived a time-dependent exact solution for the free surface motion of a Newtonian jet and examined the linear stability of this solution within the Navier-Stokes equations. The extensional motion of the unperturbed flow follows from the observation that the jet produced by a shaped charge increases linearly in the axial direction.…”