Non-Linear Dynamics of a Flow-Focusing Bubble Generator: An Inverted Dripping Faucet
Physical Review Letters [0031-9007] P. Garstecki, M.J. Fuerstman, G. M. Whitesides (2005) vol:94 iss:23 pg:234502
Brief Summary). Briefly, a gaseous thread is forced through a small orifice while being squeezed by an outer liquid flow; the thread then becomes energetically unstable which results in a bubble being pinched off. Upon varying various parameters of the system such as the gas pressure and liquid flow rate, several bifurcations in the bubble producing behaviour are observed.
The main aim of this letter is to pinpoint the physical reason for the bifurcations observed in a microfluidic bubble generator (see  for a description of the operation of the device and the initial report of chaotic behaviour). Figure 3 shows the main result: as the fluid flow rate is increased, the diameter of the bubbles decreases until a first bifurcation at <math>Q=1.06 \mu L/s</math>. At this point, the device produces bubbles of two distinct radii sequentially so that one cycle results in the production of both sizes of bubbles. This weak "period-2" behaviour quickly disappears in what is termed a "frequency halving" - the two radii converge and the bubble generation returns to "period-1" behaviour. Another bifurcation occurs at <math>Q=2.8 \mu L/s</math>, which begins a series of further period-doublings. Above the seemingly chaotic branching region just below<math>Q=3.45 \mu L/s</math>, the device appears to stabilize into period-3 oscillations. In order to explain these non-linear behaviours, the authors proceed to list the dimensionless quantities which compare the magnitude of the various forces at play.
- <math>Re=\rho u L / \mu</math>