Question
How large a velocity gradient would be needed to make an O–W emulsion with
droplets of 1mm diameter in plane hyperbolic laminar flow, assuming the effective
interfacial tension to be 0:01 N?m^1? Is the resulting gradient likely to be achievable
in practice?
Answer
About 2? 106 s^1.
11.3.3 Turbulent Flow
Turbulent flow is characterized by the presence ofeddies(vortices, whirls),
which means that the local flow velocityugenerally differs from its time-
average valueuu. The local velocity fluctuates in a chaotic manner, and the
average difference betweenuanduuequals zero. However, the root-mean-
square velocity difference,
u^0 :hðuuuÞ^2 i^1 =^2 ð 11 : 7 Þis finite. Its value generally depends on direction, but for large Re and small
length scale, the turbulence can be regarded as beingisotropic, implying that
u^0 does not significantly depend on direction. This condition is often more or
less fulfilled at the scale of drops or bubbles during agitation.
Kolmogorov theorygives relations for the effects of isotropic turbulent
flow. These relations are in fact scaling laws, but most constants in the
equations are of order unity. The flow shows a spectrum of eddy sizes (l).
The largest eddies have the highest value ofu^0. They transfer their kinetic
energy to smaller eddies, which have a smalleru^0 value but a larger velocity
gradient (u^0 /l). Small eddies have thus a high specific kinetic energy; they are
calledenergy bearing eddies, sizele. The local velocity near such an eddy
depends on its size and is given by
u^0 ðleÞ&e^1 =^3 le^1 =^3 r^1 =^3 ð 11 : 8 ÞHereris the mass density of the liquid, andeis thepower density. The latter
is defined as the amount of energy dissipated per unit volume of liquid per
unit time (i.e., in J?m^3 ?s^1 ¼W?m^3 ).