Page 33 - Bird R.B. Transport phenomena

18 Chapter 1 Viscosity and the Mechanisms of Momentum Transport

"vectors/ 7 Therefore we will refer to т as the viscous stress tensor (with components т,)
;
and IT as the molecular stress tensor (with components тг-). When there is no chance for
/;
confusion, the modifiers "viscous" and "molecular" may be omitted. A discussion of
vectors and tensors can be found in Appendix A.
The question now is: How are these stresses r /; related to the velocity gradients in
the fluid? In generalizing Eq. 1.1-2, we put several restrictions on the stresses, as follows:
• The viscous stresses may be linear combinations of all the velocity gradients:

dV k
Тц = — Sjt2/jLt/yjt/ -r- where i, , k, and / may be 1,2, 3 (1.2-3)
j
Here the 81 quantities /x,y W are "viscosity coefficients/ 7 The quantities x , x , x 3 in
2
x
the derivatives denote the Cartesian coordinates x, y, z, and v u v , v 3 are the same
2
as v , v yf v . z
x
• We assert that time derivatives or time integrals should not appear in the expres-
sion. (For viscoelastic fluids, as discussed in Chapter 8, time derivatives or time in-
tegrals are needed to describe the elastic responses.)
• We do not expect any viscous forces to be present, if the fluid is in a state of pure
rotation. This requirement leads to the necessity that r, be a symmetric combina-
;
tion of the velocity gradients. By this we mean that if / and; are interchanged, the
combination of velocity gradients remains unchanged. It can be shown that the
only symmetric linear combinations of velocity gradients are
dVj dVj\ (dv x dV y $V Z
dx dxJ \dX dy dZ
x
If the fluid is isotropic—that is, it has no preferred direction—then the coefficients
in front of the two expressions in Eq. 1.2-4 must be scalars so that

dy
We have thus reduced the number of "viscosity coefficients" from 81 to 2!
• Of course, we want Eq. 1.2-5 to simplify to Eq. 1.1-2 for the flow situation in Fig.
1.1-1. For that elementary flow Eq. 1.2-5 simplifies to т = A dvjdy, and hence the
ух
scalar constant A must be the same as the negative of the viscosity /JL.
• Finally, by common agreement among most fluid dynamicists the scalar constant
В is set equal to \i± - к, where к is called the dilatational viscosity. The reason for
writing В in this way is that it is known from kinetic theory that к is identically
zero for monatomic gases at low density.
Thus the required generalization for Newton's law of viscosity in Eq. 1.1-2 is then
the set of nine relations (six being independent):
dv j sv\ 2 (dv x dv y dv
+ {
"
+ ) ^ К \ + + f (L2 6)
Here Tjj = Tji, and i and; can take on the values 1, 2,3. These relations for the stresses in a
Newtonian fluid are associated with the names of Navier, Poisson, and Stokes. 2 If de-

C-L.-M.-H. Navier, Ann. Chimie, 19,244-260 (1821); S.-D. Poisson, /. Ecole Pohjtech., 13, Cahier 20,1-174
2
(1831); G. G. Stokes, Trans. Camb. Phil. Soc, 8,287-305 (1845). Claude-Louis-Marie-Henri Navier (1785-1836)
(pronounced "Nah-vyay," with the second syllable accented) was a civil engineer whose specialty was road
and bridge building; George Gabriel Stokes (1819-1903) taught at Cambridge University and was president
of the Royal Society. Navier and Stokes are well known because of the Navier-Stokes equations (see Chapter
3). See also D. J. Acheson, Elementary Fluid Mechanics, Oxford University Press (1990), pp. 209-212,218.