Page 258 - Modelling in Transport Phenomena A Conceptual Approach

238 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

8.1 MOMENTUM TRANSPORT

Momentum per unit mass, by definition, is the fluid velocity and changes in velocity
can result in momentum transport. For fully developed flow' through conduits,
velocity variations take place in the direction perpendicular to the flow since nG
slip boundary conditions must be satisfied at the boundaries of the conduit. This
results in the transfer of momentum in the direction perpendicular to the flow
direction.
The inventory rate equation for momentum at the microscopic level is called
the equation of motion. It is a vector equation with three components. For steady
transfer of momentum without generation, the conservation statement for momen-
tum reduces to

(Rate of momentum in) - (Rate of momentum out) = 0 (8.1-1)

When there is no generation of momentum, this implies that both pressure and
gravity terms are zero. Hence, flow can only be generated by the movement of
surfaces enclosing the fluid and the resulting flow is called Couette flow. We will
restrict our analysis to cases in which the following assumptions hold:

1. Incompressible Newtonian fluid,

2. One-dimensional2 , fully developed laminar flow,

3. Constant physical properties.

The last assumption comes from the fact that temperature rise as a result of viscous
dissipation during fluid motion, i.e., irreversible degradation of mechanical energy
into thermal energy, is very small and cannot be detected by ordinary measuring
devices in most of the cases. Hence, for all practical purposes the flow is assumed
isothermal.

8.1.1 Plane Couette Flow

Consider a Newtonian fluid between two parallel plates that are separated by a
distance B as shown in Figure 8.1. The lower plate is moved in the positive
z-direction with a constant velocity of V while the upper plate is held stationary.

Fully developed flow means there is no variation of velocity in the axial direction, In this way,
the flow development regions near the entrance and exit are not taken into consideration.
One-damensional flow indicates that there is only one non-zero velocity component.