Navier-Stokes and why DP/Dz is constant

[yosimba2000]

So in trying to solve for Poiseuille's Law using the continuty eq and Navier-Stokes, the differential EQ becomes DP/dz = some function of R only. The professor says that because DP/dz is ONLY a function of Z, and the left side is ONLY a function of R, the only way they can be equal is if both are constants.

What's the intuition or reasoning behind this?

 

[Chestermiller]

If the flow is fully developed, then the fluid velocity does not depend on z. Also, the flow velocity is purely axial. Are you comfortable with this?

Chet

 

[yosimba2000]

Oh sorry, I should have labeled. the P means pressure of flow and z is the direction.
I have found a similar explanation on this video which may give you a better understanding of what I'm asking. This is at time 1:50.

Sorry, the explanation still isn't working for me!

 

[Chestermiller]

What I was trying to do is to explain why your professor is correct. For fully developed flow in a horizontal pipe, the velocity vector is given by:
$$\vec{V}=v(r)\vec{i}_z$$
where $\vec{i}_z$ is the unit vector in the axial direction. This relationship satisfies the continuity equation exactly and, if you substitute it into the z component of the momentum equation in cylindrical coordinates, you end up with your professor's equation. The right hand side of this equation is independent of z. That means that dp/dz is independent of z. Right now, I don't see how to prove that dp/dz is not a function of r, but I'll think about it. I know that it is.

Chet

 

[Chestermiller]

After thinking about it some more, I realized that dp/dz not being a function of r comes from the components of the NS equation in the other two directions. For these other two directions, the pressure variations are hydrostatic.

Chet

 

[yosimba2000]

I just reread my question and it's more vague than I thought it was, sorry about that!
I do agree with all the equations in the video and what you've posted, but here's my real question:

I'm curious as to why, at time 1:50 in the video, a function of Z only being equal to a function of R only means that both of them have to be constants for this to be true. It doesn't haveWhat is the logic behind this?

 

[Chestermiller]

I just reread my question and it's more vague than I thought it was, sorry about that!
I do agree with all the equations in the video and what you've posted, but here's my real question:

I'm curious as to why, at time 1:50 in the video, a function of Z only being equal to a function of R only means that both of them have to be constants for this to be true. It doesn't haveWhat is the logic behind this?

If f(r) = g(z), give me a specific example of a case in which f and g are not constants.

Chet

 

[pasmith]

So in trying to solve for Poiseuille's Law using the continuty eq and Navier-Stokes, the differential EQ becomes DP/dz = some function of R only. The professor says that because DP/dz is ONLY a function of Z, and the left side is ONLY a function of R, the only way they can be equal is if both are constants.

What's the intuition or reasoning behind this?

Start with: $$\frac{\partial p}{\partial z} = f(r).$$ Using the given information, $$\frac{\partial f}{\partial r} = \frac{\partial}{\partial r}\left(\frac{\partial p}{\partial z}\right) = 0$$ and $$\frac{\partial}{\partial z}\left(\frac{\partial p}{\partial z}\right) =\frac{\partial f}{\partial z} = 0.$$ Hence $f(r) = \frac{\partial p}{\partial z}$ is constant.