Page 34 - Bird R.B. Transport phenomena

§1.2 Generalization of Newton's Law of Viscosity 19

sired, this set of relations can be written more concisely in the vector-tensor notation of
Appendix A as
+
т = -/i(Vv + (Vv) ) + (|/Lt - K)(V • v)8 (1.2-7)
in which 5 is the unit tensor with components 6 /y/ Vv is the velocity gradient tensor with
components (d/dx)v jf (Vv) + is the "transpose" of the velocity gradient tensor with com-
ponents (d/dXj)Vj, and (V • v) is the divergence of the velocity vector.
The important conclusion is that we have a generalization of Eq. 1.1-2, and this gen-
eralization involves not one but two coefficients 3 characterizing the fluid: the viscosity /л
and the dilatational viscosity к. Usually, in solving fluid dynamics problems, it is not
necessary to know к. If the fluid is a gas, we often assume it to act as an ideal
monoatomic gas, for which к is identically zero. If the fluid is a liquid, we often assume
that it is incompressible, and in Chapter 3 we show that for incompressible liquids
(V • v) = 0, and therefore the term containing к is discarded anyway. The dilational vis-
4
cosity is important in describing sound absorption in polyatomic gases and in describ-
ing the fluid dynamics of liquids containing gas bubbles. 5
Equation 1.2-7 (or 1.2-6) is an important equation and one that we shall use often.
Therefore it is written out in full in Cartesian (x, y, z), cylindrical (г, 0, z), and spherical
(г, 0, ф) coordinates in Table B.I. The entries in this table for curvilinear coordinates are
obtained by the methods outlined in §§A.6 and A.7. It is suggested that beginning stu-
dents not concern themselves with the details of such derivations, but rather concen-
trate on using the tabulated results. Chapters 2 and 3 will give ample practice in doing
this.
In curvilinear coordinates the stress components have the same meaning as in Carte-
sian coordinates. For example, т in cylindrical coordinates, which will be encountered
п
in Chapter 2, can be interpreted as: (i) the viscous force in the z direction on a unit area
perpendicular to the r direction, or (ii) the viscous flux of z-momentum in the positive r
direction. Figure 1.2-2 illustrates some typical surface elements and stress-tensor compo-
nents that arise in fluid dynamics.
The shear stresses are usually easy to visualize, but the normal stresses may cause
conceptual problems. For example, T is a force per unit area in the z direction on a
ZZ
plane perpendicular to the z direction. For the flow of an incompressible fluid in the
convergent channel of Fig. 1.2-3, we know intuitively that v z increases with decreas-
ing z; hence, according to Eq. 1.2-6, there is a nonzero stress r zz = — 2jx{dv /dz) acting
z
in the fluid.
Note on the Sign Convention for the Stress Tensor We have emphasized in connection
with Eq. 1.1-2 (and in the generalization in this section) that r is the force in the posi-
yx
tive x direction on a plane perpendicular to the у direction, and that this is the force ex-
erted by the fluid in the region of the lesser у on the fluid of greater y. In most fluid
dynamics and elasticity books, the words "lesser" and "greater" are interchanged and
Eq. 1.1-2 is written as r yx = +/jL(dv /dy). The advantages of the sign convention used in
x
this book are: (a) the sign convention used in Newton's law of viscosity is consistent
with that used in Fourier's law of heat conduction and Fick's law of diffusion; (b) the
sign convention for т Х] is the same as that for the convective momentum flux pvv (see

3
Some writers refer to /л as the "shear viscosity," but this is inappropriate nomenclature inasmuch
as fi can arise in nonshearing flows as well as shearing flows. The term "dynamic viscosity" is also
occasionally seen, but this term has a very specific meaning in the field of viscoelasticity and is an
inappropriate term for /A.
4
L. Landau and E. M. Lifshitz, op. cit., Ch. VIII.
5
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press (1967), pp. 253-255.