Page 450 - Modelling in Transport Phenomena A Conceptual Approach

430 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

'L, +v

Figure 10.1 Unsteady Couette flow between parallel plates.

Postulating v, = v,(t, x) and v, = vy = 0, Table C.l in Appendix C indicates that
the only non-zero shear-stress component is T,,. Therefore, the components of the
total momentum flux are expressed as

T,, = T,, + (pv,)v, = T,, = -p- av, (10.1-1)
ax
TU% = 79% + (pv,) vy = 0 ( 10.1.2)
T%% = T,, + (pv,) v, = pv; (10.1-3)

The conservation statement for momentum is expressed as
Rate of Rate of ) = ( Rate of momentum
( momentum in ) - ( momentum out accumulation ) (10.1-4)
For a rectangular differential volume element of thickness Ax, length AZ and width
W, as shown in Figure 10.1, &. (10.1-4) is expressed as

Dividing Eq. (10.1-5) by WAX Az and taking the limit as Ax + 0 and AZ -+ 0
gives

8% - lim TzS Ix - Tz& ],+Ax + lim %%I% - T%%l%+A% (10.1-6)
- Ax+O Ax A%-0 AZ

av, ax,, ar,,
at
p-=---- dx aZ (10.1-7)