# 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[1]

### Brief Summary

This letter deals with the chaotic behaviour of a microfluidic bubble generator (the device reported in [2]). 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 deviced is pictured in micrographs in figures 1 and 2. Figure 2 shows different higher-order regimes of bubble production; the figure letter corresponds to the flow rates demarcated in figure 3. The study of the chaotic behaviour of these types of devices is crucial in order to implement them as emulsion makers, for colloid production and in droplet-based microfluidics among many other applications.

### Capillarity Phenomena

The main aim of this letter is to pinpoint the physical reason for the bifurcations observed in a microfluidic bubble generator (see [3] 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>
- <math>Ca_{flow}=u\mu / \gamma</math>; <math>Ca_{inter}=u_{inter}\mu / \gamma</math>
- <math>We_{flow}=\rho u^2 L/ \gamma</math>; <math>We_{inter}=\rho u_{inter}^2 L/ \gamma</math>

the "inter" subscript refers to velocities of the interface, that is <math>u_{inter}=fl_{or}</math> where <math>l_{or}</math> is the length of the nozzle orifice and <math>f</math> is the frequency of bubble generation. Even though the Reynolds number has an upper limit of ~100, this is not high enough to start expecting turbulent flow. However, inertial effects are not negligible. Specifically, the Weber number (<math>We_{flow}</math>) ranges from <math>10^{-4} - 10^2</math>, which is far larger than the range of the capillary number (<math>10^{-2} - 10^{-1}</math>). This suggests that there should a regime where inertial effects are dominated by interfacial forces (low We) to the opposite where inertial forces become important but not to the point where turbulence occurs.

To further solidify the importance of inertial effects, the authors compute the so-called "critical" capillary and Weber numbers based on <math>u_{interCR}=l_{or}f_{CR}</math> where <math>f_{CR}</math> are the bubble generation frequencies at bifurcations. The capillary number was found to lie within a rather large range (<math>10^-2 - 1</math>) whereas the critical Weber number resided in a very small range from 0.3 to 0.15. This appears to be a very strong argument for inertial forces inducing the chaotic behaviour of the device. In a naive manner, this is expected since the nonlinearities of fluid mechanics are inertial in nature (namely the navier stokes equation at modest Re). Though the context of this paper is somewhat more complex, it is at least comforting that the chaotic behaviour is the result of inertial effects. It might be somewhat perplexing if it were the other way around!