Hydrodynamical models for the chaotic dripping faucet

Abstract

We give a hydrodynamical explanation for the chaotic behaviour of a dripping faucet using the results of the stability analysis of a static pendant drop and a proper orthogonal decomposition (POD) of the complete dynamics. We find that the only relevant modes are the two classical normal forms associated with a Saddle-Node-Andronov bifurcation and a Shilnikov homoclinic bifurcation. This allows us to construct a hierarchy of reduced order models including maps and ordinary differential equations which are able to qualitatively explain prior experiments and numerical simulations of the governing partial differential equations and provide an explanation for the complexity in dripping. We also provide a new mechanical analogue for the dripping faucet and a simple rationale for the transition from dripping to jetting modes in the flow from a faucet.

‘Gobbling drops’: the jetting–dripping transition in flows of polymer solutions

“…This property of negligible upstream perturbation is a generic property of convective jetting flows, in contrast to 'pseudo-jet' flows, which are dominated by the upstream transfer of the tension force along the jet. Examples of the latter include fiber spinning (Pearson (1985)), coiling of viscous jets (Ribe et al (2006)), or slow periodic dripping (Coullet et al (2005)). In the following we consider a liquid drop attached to jet of uniform radius R 0 and uniform velocity U 0 (we incorporate in Appendix B the effects of gravitational acceleration and a slow axial variation in the radius of the jet).…”

Nonlinear Dynamics of a Flow-Focusing Bubble Generator: An Inverted Dripping Faucet

“…The interesting finding is that, in contrast to earlier work [8], our experimental observations indicate a dynamic similarity to a system that forms the topologically inverted version of our bubble generator -a dripping faucet. In our system gas bubbles into a stream of flowing liquid; in the faucet, liquid drips into ambient gas.The formation of droplets and bubbles is important to fluid dynamics in two classes of problems: the first deals with interfacial instabilities, and details the asymptotics of pinch off [9-12] of a single drop; the second tries to understand the mechanisms behind the generation of sequences of drops or bubbles [5,13,14]. Systems as simple as a leaky faucet [5,13-18], or a pressurized nozzle releasing gas into a tank of fluid [6,8,[19][20][21][22] provide archetypal examples of nonlinear dynamics.…”

Dripping and Jetting in Coflowing Liquid Streams

“…This behavior is familiar to anyone who has slowly increased the flow rate of water at a kitchen faucet. The dripping-to-jetting transition is sharp if the liquid viscosity is large compared to water [Ambravaneswaran et al (2004)], whereas the transition initially becomes chaotic [Clanet and Lasheras (1999); Shaw (1984); Ambravaneswaran et al (2000); Coullet et al (2005)] before jetting for lower viscosity fluids. The jets eventually break into drops due to the Rayleigh-Plateau instability [Plateau (1849); Rayleigh (1879)].…”