Physical Chemistry of Foods

given size. IfZD> 4 ZC, the drop is not disrupted at all. The magnitude ofD
remains small and the slightly elongated drop starts to rotate (see also
Figure 5.3). This is because the time needed for deformation of the drop,
which is given by

tdef&

drop viscosity

external stress

¼

ZD

ZCC

ð 11 : 6 Þ

will be longer than the time needed for half a rotation, given byp=C:
ZD=ZCC¼p=CyieldsZD¼pZC, close to the critical value.

Note The part after the second equal sign in Eq. [11.6] only

applies in simple shear flow. In plane hyperbolic flow, 2ZCshould be

inserted in the denominator: see below.

Also ifZD 5 ZCbreakup is difficult. AtZD=ZC¼ 10
^4 , which is about
the magnitude in most foams, Wecrwould be as large as 30. At such a small
viscosity ratio, the bubble or drop is deformed into a long thread before
breaking. However, for some protein surfactants, the surface layer of the
drop can be stagnant (Section 10.8.3) and then the drop can presumably
break at a smaller Weber number.
Inelongational flow, there is no internal rotation in the drop, and it
becomes elongated, irrespective of the viscosity ratio. If We is high enough,
the drop attains a slender shape (see Figure 11.7b), in which it is subject to
Rayleigh instability (see Figure 10.23b) and thereby breaks into a number of
small drops. As depicted in Figure 11.8, curvea¼1, Wecris always smaller
for elongational than for simple shear flow. This is because of the absence of
rotational flow inside the drop and because the effective viscosity in
elongational flow is larger than the shear viscosity (by a factor of two for a
Newtonian liquid in plane hyperbolic flow).
The flow pattern can beintermediatebetween simple shear and pure
elongation, and such a situation is very common in stirred vessels and
comparable apparatus at Re <Recr. The extent to which elongation
contributes to the velocity gradient can be expressed in a simple parameter,
here calleda, that can vary between 0 and 1. As shown in Figure 11.8, a little
elongation suffices to reduce markedly the magnitude of Wecrand allows
breakup at a higher viscosity ratio.
The results in Figure 11.8 have been obtained for (large) single drops
in precisely controlled flow at constant conditions. They agree well with
theoretical predictions. Results on average droplet size of emulsions (ZC
fairly high) obtained in a colloid mill reasonably agree with theory.