# Mechanism for Flow-Rate Controlled Breakup in Confined Geometries:A Route to Monodisperse Emulsions

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Original entry: Pratomo Putra (Tom) Alimsijah, APPHY 226, Spring 2009

# Mechanism for Flow-Rate Controlled Breakup in Confined Geometries:A Route to Monodisperse Emulsions

Piotr Garstecki, Howard A. Stone, and George M. Whitesides, PRL 94, 164501, 2005

## Soft Matter Keywords

Curvature, capillary number, capillary wave, interfacial forces

## Abstract

In their previous paper, the authors showed that the volume of the bubbles formed in a Flow Focusing Device (FFD) is proportional to the pressure (p) applied to the gas stream and inversely proportional to the product of the flow rate q and viscosity of the fluid. In this paper, the authors explain how this scales and demonstrates how geometrical confinement can be used to stabilize the interface between the dispersed and continuous phase.

## Soft Matter Discussion

Fig. 1:Schematic of a Flow Focusing Device (FFD)

In a FFD such as the one shown above, most of the experiments done are mainly at low capillary numbers (Ca < 10^-2) since the characteristic dimensions of the devices used is 100micron.

The time evolution of the process is characterized by the minimal width of the collapsing neck and the axial curvature at the point of minimum width. The axial curvatures are found by fitting the axial profile with a parabola ($\kappa =\frac{d^{2}w}{dz^{2}}$).

This is summarized below:

Fig. 2: (a) Evolution of the minimal width and the axial curvature. (b-g) Evolution of the shape of the dispersed-phase

Compared to a classic instability in an unbounded gaseous thread ($u_{inert}\approx \left( \frac{\gamma }{\rho L} \right)^{\frac{1}{2}},u_{visc}\approx \frac{\gamma }{\mu }$)

the results for the collapse speeds in an FFD device is 1 to 3 orders of magnitude smaller. The authors first postulate the slower collapse to an earlier result by Hammond. However, Hammond’s results depend on the value of the interfacial tension while the authors do not observe such dependence.

The authors then argued that this is caused by the pressure drop across the droplet. At low capillary number, the interfacial forces dominate the shear stresses, thus the shear stresses exerted on the interface of the emerging droplet are not sufficient to distort it significantly. Thus, the droplet blocks almost the entire cross-section of the main channel. This leads to an increase of pressure upstream of the emerging droplet which leads to the squeezing of the neck of the immiscible thread. This displacement has to occur at a rate proportional to q since the squeezing is caused by the hydrodynamic static pressure. This is in agreement of the experiments done.

They also observed that there is a critical value of capillary number (Ca ~ 10^-2) above which shear stresses start to play an important role. In this regime (Ca>10^-2), the system operates in a similar manner to the dripping regime in an unbounded, co-flowing liquid. It is also worth noting that this behavior is very different from the classic capillary instability of an inviscid thread in an unbounded liquid. Unlike the latter which equilibriates its shape at the speed of a capillary wave, the collapse sequence of the dispersed-phase in an FFD goes through a sequence of equilibria parametrized by the volume enclosed in the gas-liquid interface.