Page 346 - Modelling in Transport Phenomena A Conceptual Approach

326 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

1. Incompressible Newtonian fluid.

2. Onedimensional, fully developed laminar flow,

3. Constant physical properties.

9.1.1 Flow Between Parallel Plates
Consider the flow of a Newtonian fluid between two parallel plates under steady con-
ditions as shown in Figure 9.1. The pressure gradient is imposed in the z-direction
while both plates are held stationary.

I------&+

Figure 9.1 Flow between two parallel plates.

Velocity components are simplified according to Figure 8.2. Since v, = v,(x)
and v, = vy = 0, Table C.l in Appendix C indicates that the only non-zero shear-
stress component is T,~. Hence, the components of the total momentum flux are
given by

dvz
TZZ = T,z + (PV,)V, =TZZ = -/l- (9.1-3)
dx
Tyz = Tyz + (PV,) vy = 0 (9.1-4)
Trs = 71% + (PV,) v, = PV,2 (9.1-5)
The pressure, on the other hand, may depend on both x and z. Therefore, it is
necessary to write the x- and z-components of the equation of motion.

x-component of the equation of motion
For a rectangular differential volume element of thickness Ax, length Az and width
W, as shown in Figure 9.1, Eq. (9.1-2) is expressed as

(PI, - P~,+A,) W AZ + pg W AX AZ = 0 (9.1-6)