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Title: Gobbling Drops: Jetting/Dripping Transition in Flows of Macromolecular Solutions
Authors: Bico, Jose
Clasen, Christian
Entov, Vladimir
McKinley, Gareth #
Issue Date: Aug-2004
Host Document: Proceedings of the 12th International Conference on Experimental Mechanics pages:1-6
Conference: 12th international Conference on Experimental Mechanics edition:12 location:Polytecnico di Bari Italy date:29 August - 2 September 2004
Abstract: Gobbling: physical picture
Experiments with thin jets of dilute polymer solutions at flow rates corresponding to a transition from dripping to jet flow reveal a peculiar breakup pattern of a thin jet ending with large end drop (Fig. 1). The jet appears to be stationary; the drop first grows while remaining almost at rest. Eventually it becomes much “thicker” than the jet, begins to accelerate and then detaches. This process repeats itself with the formation of a new end drop. Since a beads-on-string pattern (characteristic for polymeric jets) develops on jet, the process looks like a gobbling of these beads by a greedy end drop (hence the title).

Experiments
Experiments were performed with water and several dilute polymer and surfactant solutions. Thin jets were expelled vertically downward or horizontally from standard nozzles of different diameters. Critical flow rates corresponding to a transition from jetting to dripping are in reasonable agreement with literature data for water; for polymer solutions, they are much smaller, while the size of drops grows dramatically as the flow rate approaches the critical value. At the same time, the flow becomes evidently non-steady; the jet length and the end drop size experience significant variations in time. Frame-by-frame computer-aided analysis of videoimages produced by a high-speed camera (such as shown in Fig. 1a) reveals the main features of the phenomenon. Characteristic results are presented in Fig. 1b, that shows traces of the end drop and asperities corresponding to emerging capillary instabilities on the jet and serves to determine directly the jet velocity.
It could be shown that in the developing gobbling regime the end drop moves slowly back and forth up to detachment, while the primary jet is effectively unaffected by this motion. This observation serves as a starting point for an elementary dynamical model of the phenomenon.


Fig.1. (a): Phenomenon of gobbling: large end drop at the end of thin jet of 100 ppm PAA solution, prior and after detachment; (b): Fragment of XLT-diagram; traces of the end drop (circles), asperities on the impinging drop (points), (c) Critical flow rates vs nozzle radius.

Model
The model is based upon mass and momentum balances of the end drop and the assumption that the jet may break due to capillary forces if the flight time becomes sufficiently large. The simple fact that outgoing drops carry with them some momentum implies existence of a minimum flow rate. Taking gravity into account, and neglecting rheological, i.e. viscous or elastic stresses, the critical rate is estimated as:

; , (1)

where, Qc0 is the critical flow rate in the absence of gravity,  is the surface tension,  is the fluid density, R0, and l are jet radius and length to breakup. In Fig. 1c experimentally measured critical flow rates for a polyacrylamide solution (circles) are compared to predicted values of the critical flow rates. Points are values of Qc0, with gravity neglected, crosses are values of Qc evaluated according to Eq. 1 with l/R0 = 20 ; the line is the best fit of the experimental data with a linear dependence. The fact that the elementary estimate provides a quite reasonable approximation for experimentally observed critical flow rates in the case of polymer solution implies that the transition phenomenon is primarily controlled by the requirement of positive downstream momentum flux. Secondly, it means that for dilute solutions considered, the contribution of the rheological stress into the overall momentum balance is secondary, and the transition is primarily controlled by interplay between inertia and capillarity with some contribution of gravity at larger jet diameters. A significant difference between critical conditions of water and dilute polymer solutions is explained qualitatively by quite different distances of the jet breakup. A dynamical model of gobbling is based on the same mass and momentum balances, expressed as a set of o.d.e.'s for mass, position and velocity of the drop, and accounting for the mass and momentum influx from the jet. The critical detachment distance is a parameter of the model. The model predicts the existence of a critical flow rate and periodic oscillations of drop mass, velocity and position in the super-critical regime. The amplitude and period of the oscillations grow with approach to the critical rate, and become small at ~ 1.5 of the critical velocities. The theory predicts that the relative size of the end drop increases with a decreasing nozzle diameter. This explains why the gobbling phenomenon is best observed for very thin jets.
Publication status: published
KU Leuven publication type: IC
Appears in Collections:Soft Matter, Rheology and Technology Section
# (joint) last author

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