Page 31 - Bird R.B. Transport phenomena

16 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

§1.2 GENERALIZATION OF NEWTON'S LAW OF VISCOSITY
In the previous section the viscosity was defined by Eq. 1.1-2, in terms of a simple
steady-state shearing flow in which v x is a function of у alone, and v y and v z are zero.
Usually we are interested in more complicated flows in which the three velocity compo-
nents may depend on all three coordinates and possibly on time. Therefore we must
have an expression more general than Eq. 1.1-2, but it must simplify to Eq. 1.1-2 for
steady-state shearing flow.
This generalization is not simple; in fact, it took mathematicians about a century and a
half to do this. It is not appropriate for us to give all the details of this development here,
1
since they can be found in many fluid dynamics books. Instead we explain briefly the main
ideas that led to the discovery of the required generalization of Newton's law of viscosity.
To do this we consider a very general flow pattern, in which the fluid velocity may
be in various directions at various places and may depend on the time t. The velocity
components are then given by

v x = v (x, y, z, t); v y = v (x, y, z, t); v z = v (x, y, z, t) (1.2-1)
z
y
x
In such a situation, there will be nine stress components r /y (where / and / may take on
the designations x, y, and z), instead of the component r yx that appears in Eq. 1.1-2. We
therefore must begin by defining these stress components.
In Fig. 1.2-1 is shown a small cube-shaped volume element within the flow field,
each face having unit area. The center of the volume element is at the position x, y, z. At

-x,y,z

\

\

f 1
pS
z
(a) (b) (c)
Fig. 1.2-1 Pressure and viscous forces acting on planes in the fluid perpendicular to the three
coordinate systems. The shaded planes have unit area.

1 W. Prager, Introduction to Mechanics ofContinua, Ginn, Boston (1961), pp. 89-91; R. Aris, Vectors,
Tensors, and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J. (1962), pp. 30-34,
99-112; L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, London, 2nd edition (1987), pp. 44-45.
Lev Davydovich Landau (1908-1968) received the Nobel prize in 1962 for his work on liquid helium and
superfluid dynamics.