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
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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,
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, Cf is defined by
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| (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.
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