Page 47 - Bird R.B. Transport phenomena

32 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

Newton's law of viscosity (Eq. 1.1-2 or 1.2-7) with two modifications: (i) the viscosity /л is
replaced by an effective viscosity ix and (ii) the velocity and stress components are then
eif/
redefined (with no change of symbol) as the analogous quantities averaged over a vol-
ume large with respect to the interparticle distances and small with respect to the dimen-
sions of the flow system. This kind of theory is satisfactory as long as the flow involved
is steady; in time-dependent flows, it has been shown that Newton's law of viscosity is
inappropriate, and the two-phase systems have to be regarded as viscoelastic materials. 1
The first major contribution to the theory of the viscosity of suspensions of spheres was
that of Einstein. 2 He considered a suspension of rigid spheres, so dilute that the move-
ment of one sphere does not influence the fluid flow in the neighborhood of any other
sphere. Then it suffices to analyze only the motion of the fluid around a single sphere,
and the effects of the individual spheres are additive. The Einstein equation is

in which /л is the viscosity of the suspending medium, and ф is the volume fraction of
0
the spheres. Einstein's pioneering result has been modified in many ways, a few of
which we now describe.
For dilute suspensions of particles of various shapes the constant § has to be replaced by
a different coefficient depending on the particular shape. Suspensions of elongated or
3 4 5 6
flexible particles exhibit non-Newtonian viscosity. ' ' '
For concentrated suspensions of spheres (that is, ф greater than about 0.05) particle in-
teractions become appreciable. Numerous semiempirical expressions have been devel-
oped, one of the simplest of which is the Mooney equation 7

in which ф is an empirical constant between about 0.74 and 0.52, these values corre-
0
sponding to the values of ф for closest packing and cubic packing, respectively.

1
For dilute suspensions of rigid spheres, the linear viscoelastic behavior has been studied by
H. Frohlich and R. Sack, Proc. Roy. Soc, A185,415-430 (1946), and for dilute emulsions, the analogous
derivation has been given by J. G. Oldroyd, Proc. Roy. Soc, A218,122-132 (1953). In both of these
publications the fluid is described by the Jeffreys model (see Eq. 8.4-4), and the authors found the relations
between the three parameters in the Jeffreys model and the constants describing the structure of the two-
phase system (the volume fraction of suspended material and the viscosities of the two phases). For
further comments concerning suspensions and rheology, see R. B. Bird and J. M. Wiest, Chapter 3 in
Handbook of Fluid Dynamics and Fluid Machinery, J. A. Schetz and A. E. Fuhs (eds.), Wiley, New York (1996).
Albert Einstein (1879-1955) received the Nobel prize for his explanation of the photoelectric effect,
2
not for his development of the theory of special relativity. His seminal work on suspensions appeared in
A. Einstein, Ann. Phys. (Leipzig), 19, 289-306 (1906); erratum, ibid., 24, 591-592 (1911). In the original
publication, Einstein made an error in the derivation and got ф instead of \ф. After experiments
showed that his equation did not agree with the experimental data, he recalculated the coefficient.
Einstein's original derivation is quite lengthy; for a more compact development, see L. D. Landau and
E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 2nd edition (1987), pp. 73-75. The mathematical
formulation of multiphase fluid behavior can be found in D. A. Drew and S. L. Passman, Theory of
Multicomponent Fluids, Springer, Berlin (1999).
H. L. Frisch and R. Simha, Chapter 14 in Rheology, Vol. 1, (F. R. Eirich, ed.), Academic Press, New
3
York (1956), Sections II and III.
E. W. Merrill, Chapter 4 in Modern Chemical Engineering, Vol. 1, (A. Acrivos, ed.), Reinhold, New
4
York (1963), p. 165.
5
E. J. Hinch and L. G. Leal, /. Fluid Mech., 52, 683-712 (1972); 76,187-208 (1976).
W. R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon, Oxford (1978), Chapter 13.
6
7
M. Mooney, /. Coll. Sci., 6,162-170 (1951).