View source for Dripping to Jetting Transitions in Coflowing Liquid Streams

Original entry:  Scott Tsai,  APPHY 226,  Spring 2009

"Dripping to Jetting Transitions in Coflowing Liquid Systems"
Andrew S. Utada, Alberto Fernandez-Nieves, Howard A. Stone, and David A. Weitz

Physical Review Letters 99, 094502 (2007)

== Soft Matter Keywords ==
Droplet, jetting, dripping, surface tension, viscous shear, drop formation, microfluidics. 

== Overview ==
Refer to abstract of paper

== Soft Matter Examples ==
[[Image:Paper3_fig1.jpg|thumb|right|300px|'''Fig.1''']] 
When the flow rate of a kitchen tap increases slowly, drops initially grow and break off when the body force due to gravity acting on the drop overcomes the surface tension forces keeping the drop attached to the water in the tap. This is called "dripping". For viscous fluids, as the flow rate reaches a critical level, the liquid becomes a jet and drops at the tip of the jet break off when the inertial forces of the water overcomes the surface tension forces. For low viscosity fluids, the jets form Rayleigh-Plateau  instability and break off. Anytime when a jet forms before breaking off into droplets, it is called "jetting". 

When the same problem is applied to co-flowing liquid-liquid microfluidic devices, two different behaviors occur.

'''1. q<sub>out</sub> is larger than q<sub>in</sub> (Fig. 1a)'''

In this case, the viscous shear forces of the outer fluid are what drive the forming drop foward. At small q<sub>in</sub> and q<sub>out</sub>, dripping occurs (Fig. 1b) as the balance of surface tension and viscous shear force from the outside fluid dictates the size of the break-off drop. As q<sub>in</sub>/q<sub>out</sub> becomes smaller, a jet forms and the break off point departs from the nozzle of the inner fluid. The jet develops undulations driven by the Rayleigh-Plateau instability.

When dripping occurs, the size of the droplet can be estimated by a force balance between viscous shear force of the outside fluid and the surface tension of the inner fluid, <math>\mu_{out} u_{out} d_{out} \propto \gamma_{in} d_{tip}</math>. When jetting occurs, the drop diameter can be determined by finding out the volume conateined in one wavelength of the Rayleigh-Plateau instability, <math> \pi d_{jet}^2 \lambda / 4</math>. This volume is approximately the same as the volume of the drop. So <math>d_{drop} \approx 2 d_{jet}</math>.

These predictions compare favorably with experimental results (Fig. 2)
[[Image:Paper3_fig2.jpg|thumb|right|300px|'''Fig.2''']] 
'''2. q<sub>out</sub> is smaller than q<sub>in</sub> (Fig. 1d)'''

In this case, it is the inertia of the inner fluid that drives the formed Rayleigh-Plateau instability and the formed drop. From experiments, it is seen that drop formation includes three steps: advection and necking at the end of the jet, drop pinchoff, and jet retraction. The longest part of the three is the advection and necking of the jet. So the overall time scale will be dominated by the time it takes to advect and neck. Here, the velocity of the inner fluid is much larger than advection velocity, so the dominating time scale will be determined by the advection distance divided by the advection velocity. The advection distance is usually <math>\approx d_{drop}</math>, and the advection velocity is just u<sub>out</sub>. This leads to <math> d_{drop} \approx (q_{in}/u_{out})^{1/2} </math>. 

These predictions also compare favorably with experimental results, which predict <math> d_{drop} \approx (q_{in}/u_{out})^{0.46} </math> (Fig. 3).
[[Image:Paper3_fig3.jpg|thumb|right|300px|'''Fig.3''']]