Parallel Flow in a Straight Channel

Consider steady flow between two infinitely broad parallel plates as shown in Fig. 25.2.

Flow is independent of any variation in z direction, hence, z dependence is gotten rid of and Eq. (25.11) becomes

FIG 25.2 Parallel flow in a straight channel

(25.12)

The boundary conditions are at y = b, u = 0; and y = -b, u = O.

• From Eq. (25.12), we can write

or

• Applying the boundary conditions, the constants are evaluated as

and

So, the solution is

(25.13)

which implies that the velocity profile is parabolic.

Average Velocity and Maximum Velocity

• To establish the relationship between the maximum velocity and average velocity in the channel, we analyze as follows
  
  At y = 0,                          ; this yields

(25.14a)

On the other hand, the average velocity,

or

Finally,

(25.14b)

So,    or

(25.14c)

• The shearing stress at the wall for the parallel flow in a channel can be determined from the velocity gradient as

Since the upper plate is a "minus y surface", a negative stress acts in the positive x direction, i.e. to the right.

• The local friction coefficient, C_f is defined by

	(25.14d)

where is the Reynolds number of flow based on average velocity and the channel height (2b).

• Experiments show that Eq. (25.14d) is valid in the laminar regime of the channel flow.
• The maximum Reynolds number value corresponding to fully developed laminar flow, for which a stable motion will persist, is 2300.
• In a reasonably careful experiment, laminar flow can be observed up to even Re = 10,000.
• But the value below which the flow will always remain laminar, i.e. the critical value of Re is 2300.