Page 28 - Bird R.B. Transport phenomena

§1.1 Newton's Law of Viscosity (Molecular Transport of Momentum) 13

should not refer to Eq. 1.1-2 as a "law/' since Newton suggested it as an empiricism —
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the simplest proposal that could be made for relating the stress and the velocity gradi-
ent. However, it has been found that the resistance to flow of all gases and all liquids
with molecular weight of less than about 5000 is described by Eq. 1.1-2, and such fluids
are referred to as Newtonian fluids. Polymeric liquids, suspensions, pastes, slurries, and
other complex fluids are not described by Eq. 1.1-2 and are referred to as non-Newtonian
fluids. Polymeric liquids are discussed in Chapter 8.
Equation 1.1-2 may be interpreted in another fashion. In the neighborhood of the
moving solid surface at у = 0 the fluid acquires a certain amount of x-momentum. This
fluid, in turn, imparts momentum to the adjacent layer of liquid, causing it to remain in
motion in the x direction. Hence x-momentum is being transmitted through the fluid in
the positive у direction. Therefore r may also be interpreted as the flux of x-momentum
yx
in the positive у direction, where the term "flux" means "flow per unit area." This interpre-
tation is consistent with the molecular picture of momentum transport and the kinetic
theories of gases and liquids. It also is in harmony with the analogous treatment given
later for heat and mass transport.
The idea in the preceding paragraph may be paraphrased by saying that momentum
goes "downhill" from a region of high velocity to a region of low velocity—just as a sled
goes downhill from a region of high elevation to a region of low elevation, or the way
heat flows from a region of high temperature to a region of low temperature. The veloc-
ity gradient can therefore be thought of as a "driving force" for momentum transport.
In what follows we shall sometimes refer to Newton's law in Eq. 1.1-2 in terms of
forces (which emphasizes the mechanical nature of the subject) and sometimes in terms
of momentum transport (which emphasizes the analogies with heat and mass transport).
This dual viewpoint should prove helpful in physical interpretations.
Often fluid dynamicists use the symbol v to represent the viscosity divided by the
density (mass per unit volume) of the fluid, thus:
v = p/p (1.1-3)
This quantity is called the kinematic viscosity.
Next we make a few comments about the units of the quantities we have defined. If
we use the symbol [=] to mean "has units of," then in the SI system r XJX [=] N/m 2 = Pa,
v [ = ] m/s, and у [=] m, so that
x
l
д = —тЛ —г^ [ = ] Pa)[(m/s)(m )] l = Ра • s (1.1-4)
(
\dy)
since the units on both sides of Eq. 1.1-2 must agree. We summarize the above and also
give the units for the c.g.s. system and the British system in Table 1.1-1. The conversion
tables in Appendix F will prove to be very useful for solving numerical problems involv-
ing diverse systems of units.
The viscosities of fluids vary over many orders of magnitude, with the viscosity of
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air at 20°C being 1.8 X 10~ Pa • s and that of glycerol being about 1 Pa • s, with some sili-
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cone oils being even more viscous. In Tables 1.1-2,1.1-3, and 1.1-4 experimental data are
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A relation of the form of Eq. 1.1-2 does come out of the simple kinetic theory of gases (Eq. 1.4-7).
However, a rigorous theory for gases sketched in Appendix D makes it clear that Eq. 1.1-2 arises as the
first term in an expansion, and that additional (higher-order) terms are to be expected. Also, even an
elementary kinetic theory of liquids predicts non-Newtonian behavior (Eq. 1.5-6).
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A comprehensive presentation of experimental techniques for measuring transport properties can be
found in W. A. Wakeham, A. Nagashima, and J. V. Sengers, Measurement of the Transport Properties of Fluids,
CRC Press, Boca Raton, Fla. (1991). Sources for experimental data are: Landolt-Bornstein, Zahlemverte und
Funktionen, Vol. II, 5, Springer (1968-1969); International Critical Tables, McGraw-Hill, New York (1926);
Y. S. Touloukian, P. E. Liley, and S. С Saxena, Thermophysical Properties of Matter, Plenum Press, New York
(1970); and also numerous handbooks of chemistry, physics, fluid dynamics, and heat transfer.