Dimensionless Numbers in Microfluidics - Microfluidics for Biotechnology - page 5

Figure1.2 Reynolds numbers: (a) Reynolds number in a channel and (b) Reynolds number around

a spherical particle.

V R depending on the
specific configuration, and the capillary pressure γ / R , one immediately sees that the
capillary number is the ratio of the viscous forces to the capillary forces. Let us il-
lustrate this remark by two examples.
In flow focusing devices (FFD), the capillary number is a relevant criterion to
predict liquid thread breakup [Figure 1.3(a)]. In this case, the capillary number is
built on the shear rate (or elongation rate) [7]. As the filament stretches and thins
down, the capillary number decreases and when it becomes less than a critical value
Ca crit ~ 0.1 to 0.01, the surface tension forces break the liquid filament into drop-
lets, minimizing the interfacial area (hence the surface energy); this phenomenon is
usually called the Rayleigh-Plateau instability [8, 9].
In a plug flow, the friction of the plugs with the walls is a function of the capil-
lary number of the liquid of the plugs [Figure 1.3(b)] [10].
The Weber number is used to predict the disruption of an interface under the
action of strong inertial forces. More specifically, the Weber number is the ratio of
inertial forces to surface tension forces

Remarking that the shear stress can be written η , ηγ η � � ,

2 2

ρ V R ρ V

We = =

R (1.11)
The numerator corresponds to a dynamic pressure ρV 2 and the denominator to a
capillary pressure γ / R . A strong surface tension maintains the droplet as a unique

γ γ /

Figure1.3 (a) Capillary number for an elongating liquid thread in a flow focusing device (FFD) and

(b) capillary number for plug flow in a tube.