Page 26 - Bird R.B. Transport phenomena

Chapter 1

Viscosity and the Mechanisms

of Momentum Transport

§1.1 Newton's law of viscosity (molecular momentum transport)

§1.2 Generalization of Newton's law of viscosity
§1.3 Pressure and temperature dependence of viscosity
§1.4° Molecular theory of the viscosity of gases at low density
§1.5° Molecular theory of the viscosity of liquids
§1.6° Viscosity of suspensions and emulsions
§1.7 Convective momentum transport

The first part of this book deals with the flow of viscous fluids. For fluids of low molecu-
lar weight, the physical property that characterizes the resistance to flow is the viscosity.
Anyone who has bought motor oil is aware of the fact that some oils are more "viscous"
than others and that viscosity is a function of the temperature.
We begin in §1.1 with the simple shear flow between parallel plates and discuss how
momentum is transferred through the fluid by viscous action. This is an elementary ex-
ample of molecular momentum transport and it serves to introduce "Newton's law of vis-
cosity" along with the definition of viscosity /л. Next in §1.2 we show how Newton's law
can be generalized for arbitrary flow patterns. The effects of temperature and pressure
on the viscosities of gases and liquids are summarized in §1.3 by means of a dimension-
less plot. Then §1.4 tells how the viscosities of gases can be calculated from the kinetic
theory of gases, and in §1.5 a similar discussion is given for liquids. In §1.6 we make a
few comments about the viscosity of suspensions and emulsions.
Finally, we show in §1.7 that momentum can also be transferred by the bulk fluid
motion and that such convective momentum transport is proportional to the fluid density p.

§1.1 NEWTON'S LAW OF VISCOSITY (MOLECULAR
TRANSPORT OF MOMENTUM)

In Fig. 1.1-1 we show a pair of large parallel plates, each one with area A, separated by a
distance У. In the space between them is a fluid—either a gas or a liquid. This system is
initially at rest, but at time t = 0 the lower plate is set in motion in the positive x direc-
tion at a constant velocity V. As time proceeds, the fluid gains momentum, and ulti-
mately the linear steady-state velocity profile shown in the figure is established. We
require that the flow be laminar ("laminar" flow is the orderly type of flow that one usu-
ally observes when syrup is poured, in contrast to "turbulent" flow, which is the irregu-
lar, chaotic flow one sees in a high-speed mixer). When the final state of steady motion

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