FLUIDITY AND PLASTICITY

shearing stress. For convenience suppose that all
|fthe lamellae jff the one substance A have thfi same thickness
&i and 'that the laminae of the substance B have- the uniform
thickness s2, etc. Let the viscosities of the substances be 771,
772 • • - and the shearing stressJs per unit area pi, p% . . .
respectively; then if R is the distance between the horizontal
planes, the velocity of the moving surface is

_RP Rpi Rp%

H rji 772 '

where H is the viscosity of the mixture, and P is the average
shearing stress over the entire distance S.
But

T>O /w r>

JL o = pi$i "

hence

TT

= Rl

v\

S

Fig. 32.—Diagram to illus-

trate additive viscosities. But since Si/o is the fraction by volume
of the substance A present in the mixture,
which we may designate a, and similarly s2/S = b, etc.,

JT = am + fci,2 + . •. . (24)

This case is of particular interest in connection with emulsions
and many other poorly mixed substances. The formula tells
us that the viscosity of the mixture is the sum of the partial
viscosities of the components, provided that the drops of the
emulsion completely fill the capillary space through which the
flow is taking place.

Case EL Fluidities Additive—Fluid Mixtures.—If the larnelke
are arranged parallel to the direction of shear, as shown in Fig.
33, we have a constant shearing stress, so that

(24a)

v 2,

are the partial velocities as indicated in the

where
figure.
There are two different ways of defining the viscosity of a
mixture, and it becomes necessary for us to adopt one of these
before we proceed further.
1. If we measure viscosity with a viscorneter of the Coulomb