Deriving Half-Max Velocity in Pipe Flow

[jonnygeordie]

Could someone help me with this. In pipe flow show that the average velocity is half of the maximum velocity. I have it written in my notes that this is true, but not how to derive it.

 

[Hao]

You can get the answer if you solve Navier-Stokes equation in cylindrical coordinates with the following assumptions:

1) Flow is incompressible.
2) Pipe pressure decreases linearly.
3) Flow rate is finite.
4) Flow rate is static, hence, the fluid only flows in one particular direction (you can guess which one based on the symmetry of a cylindrical pipe).
5) Flow rate at the sides of the pipe = 0.

 

[Mene]

Sure, I'd be happy to help with this!

The average velocity in pipe flow is defined as the total volume flow rate divided by the cross-sectional area of the pipe. Mathematically, we can express this as:

V_avg = Q/A

Where V_avg is the average velocity, Q is the volume flow rate, and A is the cross-sectional area of the pipe.

Now, let's consider the maximum velocity in pipe flow. This occurs at the center of the pipe, where the flow is most streamlined. We can express this as:

V_max = 2V_avg

Where V_max is the maximum velocity and 2 represents the maximum velocity at the center of the pipe being twice the average velocity.

Next, we can use the continuity equation to relate the volume flow rate to the maximum velocity. The continuity equation states that the volume flow rate is constant throughout the pipe, and can be expressed as:

Q = AV_max

Where A is the cross-sectional area of the pipe.

Substituting this into our equation for average velocity, we get:

V_avg = Q/A = (AV_max)/A = V_max

Therefore, we can see that the average velocity is equal to the maximum velocity, which we previously expressed as 2 times the average velocity. This means that the average velocity is half of the maximum velocity, as stated in your notes.

I hope this helps in understanding how to derive the relationship between average and maximum velocities in pipe flow. Let me know if you have any further questions!