Page 49 - Bird R.B. Transport phenomena

34 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

For emulsions or suspensions of tiny droplets, in which the suspended material may un-
dergo internal circulation but still retain a spherical shape, the effective viscosity can be
considerably less than that for suspensions of solid spheres. The viscosity of dilute emul-
sions is then described by the Taylor equation-? 2

±Щ 0.6-5)
Mo + Mi
in which /л } is the viscosity of the disperse phase. It should, however, be noted that
surface-active contaminants, frequently present even in carefully purified liquids, can ef-
13
fectively stop the internal circulation; the droplets then behave as rigid spheres.
For dilute suspensions of charged spheres, Eq. 1.6-1 may be replaced by the Smolu-
chowski equation™

in which D is the dielectric constant of the suspending fluid, k the specific electrical con-
e
ductivity of the suspension, £ the electrokinetic potential of the particles, and R the parti-
cle radius. Surface charges are not uncommon in stable suspensions. Other, less well
understood, surface forces are also important and frequently cause the particles to form
loose aggregates. 4 Here again, non-Newtonian behavior is encountered. 15

§1.7 CONVECTIVE MOMENTUM TRANSPORT

Thus far we have discussed the molecular transport of momentum, and this led to a set of
quantities ir X], which give the flux of /-momentum across a surface perpendicular to the /
direction. We then related the тт Х] to the velocity gradients and the pressure, and we
found that this relation involved two material parameters JJL and к. We have seen in §§1.4
and 1.5 how the viscosity arises from a consideration of the random motion of the mole-
cules in the fluid—that is, the random molecular motion with respect to the bulk motion
of the fluid. Furthermore, in Problem 1C.3 we show how the pressure contribution to тг /у
arises from the random molecular motions.
Momentum can, in addition, be transported by the bulk flow of the fluid, and this
process is called convective transport. To discuss this we use Fig. 1.7-1 and focus our atten-
tion on a cube-shaped region in space through which the fluid is flowing. At the center
of the cube (located at x, y, z) the fluid velocity vector is v. Just as in §1.2 we consider
three mutually perpendicular planes (the shaded planes) through the point x, y, z, and
we ask how much momentum is flowing through each of them. Each of the planes is
taken to have unit area.
The volume rate of flow across the shaded unit area in (a) is v . This fluid carries
x
with it momentum pv per unit volume. Hence the momentum flux across the shaded
area is v p\; note that this is the momentum flux from the region of lesser x to the region
x

12
G. I. Taylor, Proc. Roy. Soc, A138,41-48 (1932). Geoffrey Ingram Taylor (1886-1975) is famous for
Taylor dispersion, Taylor vortices, and his work on the statistical theory of turbulence; he attacked many
complex problems in ingenious ways that made maximum use of the physical processes involved.
13 V. G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J. (1962), Chapter
8. Veniamin Grigorevich Levich (1917-1987), physicist and electrochemist, made many contributions to
the solution of important problems in diffusion and mass transfer.
14 M. von Smoluchowski, Kolloid Zeits., 18,190-195 (1916).
15 W. B. Russel, The Dynamics of Colloidal Systems, U. of Wisconsin Press, Madison (1987), Chapter 4;
W. B. Russel, D. A. Saville, and W. R. Schowalter, Colloidal Dispersions, Cambridge University Press
(1989); R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press (1998).