Page 262 - Modelling in Transport Phenomena A Conceptual Approach

242 CHAPTER 8. STEADY MSCROSCOPIC BUAiVCES WITHOUT GEN.

For the problem at hand, the simplification of the velocity components is shown
in Figure 8.4. Since v, = vz(r) and Vr = ve = 0, Table C.2 in Appendix C indicates
that the only non-zero shear-stress component is TpZ. Therefore, the components
of the total momentum flux axe given by
dvz
-
Trz = Trz + (P'Vz) Vr = Trz = - (8.1-17)
dr
nex =Tez+(pvz)ve=O (8.1-18)
7Czz = Tzz + (pv,) v, = pv: (8.1-19)

One-dimensional flow
v,.=ve=O vz = vz e, z, t)
I

Fully developed flow
hz/az = 0 vz = VZ (r)

Figure 8.4 Simplification of the velocity components for Couette flow in a
concentric annulus.

For a cylindrical differential volume element of thickness Ar and length Az, as
shown in Figure 8.3, &. (8.1-1) is expressed as

( n,, 2mAr + nrz I,2nrAr) - [ n,, I++Az 2nrAr
1,
4- TrZlr+Ar 2n(r + Ar)Az] = 0 (8.1-20)
Dividing Eq. (8.1-20) by 2nArAa and taking the limit as Ar -+ 0 and Az --t 0
gives

(8.1-22)