Droplet Breakup in Emulsions

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number 

This definition is suitable for axisymmetric droplet shapes or for small deformations. The aspect ratio of the droplet is then L/B = (1- -D)/(1 — D). 

Taylor predicted that for small deformations, the droplet deformation D in a steady flow should follow the prediction 

This formula applies to planar extensional flow as well as to shear, if the shear rate y in Eq. (9-11) is replaced by 2e, where is the extension rate. Taylor predicted that droplet breakup should occur when the viscous stresses that deform the droplet overwhelm the surface tension forces that resist deformation this occurs when D reaches a value Db given approximately by 

Bentley and Leal have measured droplet shapes and critical conditions for droplet breakup over a wide range of capillary numbers, viscosity ratios, and flow types. The flow type is conveniently controlled in an apparatus called a four-roll mill, in which a velocity field is generated by the rotation of four rollers in a container of liquid (see Fig. 1-15). By varying the rotation rate of one pair of rollers relative to that of a second pair, velocity fields ranging from planar extension to nearly simple shear can be produced near the stagnation point. 

---