# Absolute Instability of a Liquid Jet in a Coflowing Stream

Original Entry: Nick Chisholm, AP 225, Fall 2009

## General Information

Authors: A.S. Utada, A. Fernandez-Nieves, J.M. Gordillo, and D.A. Weitz

Publication: PRL 100, 014502 (2008)

## Summary

This paper analyzes a jet stream (in this particular case, they use deionized water) in a coflowing stream (a moving outer liquid, which in this case is poly(dimethylsiloxane) (PDMS) oil). As one can imagine, the jet flow will break up into droplets due to the surface-tension-driven Rayleigh-Plateau instability, which describes the disturbances caused by absolute and convective instabilities. Absolute instabilities are those which grow at a fixed spatial location, whereas convective instabilities are those that grow in amplitude as they are swept along by the flow. Usually, jet flows are convectively unstable (for example, the flow of tap water). Here, though, the authors present experimental results of jets in a coflowing stream that break up into drops due to absolute instability. The authors note that the formation of drops due to absolute instability is insensitive to external noise (since the drops form at a frequency intrinsic to the system), and as a result these drops are monodisperse (meaning same sized and shaped drops). These monodisperse drops are of great scientific interest, along with being technologically important (for example, in microfluidics, which require highly uniform drop formation).

## Soft Matter Discussion

It is interesting to discuss how one would create these jet flows of deionized water in the PDMS oil (the surface tension between the two is: $\gamma = 40 mN/m$). The authors create this jet flow by use of a capillary-based microfluidic device that consists of two co-axially aligned cylindrical capillaries housed within a larger square tube. One of the inner capillaries is tapered (and thus smaller in size), and this can be inserted into the second untapered tip (see the inset of Figure 1). Each fluid is driven with a syringe pump, and the jet is imaged by a high speed camera (see the main part of Figure 1). One should note that gravity does not play an important role in this system, since all the characteristic length scales are well below the capillary length.

Figure 1, taken from [1].

In order to create the flow, the water is pumped at a velocity large enough that the Weber number for the jet of water is greater than one at the tip. Alternatively, one could increase the outer flow rate (the flow rate of the outer liquid, in this case PDMS oil), which increases the capillary number of the outer fluid, $C_{out} = \frac{\eta_{out}u_{out}}{\sigma}$ where $\eta_{out}$ is the viscosity of the outer fluid and $u_{out}$ is the mean velocity of the outer fluid. The capillary number is similar to the Weber number, except that it is the ratio of viscous shear forces to surface tension forces [Note: viscous shear forces could also lead to flow, which seems intuitively clear].

As the jet increases in size, it develops a "remarkable" standing-wave-like oscillation that modulates the diameter of the drop. Eventually, these initially static undulations pulse radially and cause the diameter of the stream at the neck to modulate; this modulation is shown in Figure 2. Clearly, this modulation decreases the diameter towards zero, thus decreasing the Weber number to below unity at the exit of the tip, causing an absolute instability (which leads to dripping) at the tip. The authors conclude, then, that monodisperse drops arise only when the diameter of the flow is sufficiently increased so that the Weber number decreases to unity.

Figure 2, taken from [1].

One should take careful note of the underlying theme of this paper: control. A common trend in science is that control leads to innovative new technology. A very good example of this would be the development of the laser, which when it was originally developed, had no real use. As one now knows, lasers are an important part of our everyday lives (technology), healthcare, research and even defense mechanisms. Therefore, even though technological impact is already at least partially known for monodisperse drops, I suggest that there could potentially be more than we can currently imagine.

[Note: The authors present a linear stability analysis to verify their experimental results; however, this is rather dry, so I did not include a discussion of it here. I did not think it added to the understanding of the paper.]

## Reference

[1] A.S. Utada, A. Fernandez-Nieves, J.M. Gordillo, and D.A. Weitz, "Absolute Instability of a Liquid Jet in a Coflowing Stream," PRL 100, 014502 (2008)