Average fluid velocity in pipe from pressure loss and radii

[physea]

I found this equation that supposedly shows the average velocity of a fluid in pipe in respect to the pressure loss, the radius R of the pipe and any smaller radius r.

However I have no idea how they came up with this, is it Darcy equation for laminar flow where f=64/Re?
Can anyone enlighten please?

 

[Chestermiller]

The reason you don't recognize it is that it is incorrect. The left hand side should be the axial velocity as a function of r, not the average axial velocity. Please cite a reference for this equation.

 

[boneh3ad]

I don't know what source you are reading, but that equation is wrong. That is local velocity as a function of $r$ in a pipe of radius $R$. If you want average velocity, you need to integrate that over the cross section and divide by area.

You can derive it from the cylindrical Navier-Stokes equations. I'd actually suggest you do that as a useful exercise.

 

[physea]

OK, even if the left hand side is V(r), I still cannot find that equation anywhere.

Hyperphysics gives

which is very different!
Any hint?

 

[boneh3ad]

Well your two equations use different variables. What is $v_m$ expressed in terms of pressure?

 

[dRic2]

Actually $v(r) = 2v_m \left[ 1-\frac {r^2} {R^2} \right]$. And $v_m = \frac {R^2ΔP} {8uL}$ (which is the average velocity across the pipe section). Put them together. If you want to know how to find those formulas

You can derive it from the cylindrical Navier-Stokes equations. I'd actually suggest you do that as a useful exercise.

 

[Chestermiller]

OK, even if the left hand side is V(r), I still cannot find that equation anywhere.

Hyperphysics gives View attachment 219748 which is very different!
Any hint?

Get yourself a copy of Transport Phenomena by Bird, Stewart, and Lightfoot

 

[boneh3ad]

First of all, $v_m$ is ambiguous notation, and, in fact, both @physea and @dRic2 have correct equations depending on whether $v_m$ is meant to be maximum velocity or mean velocity.

Second, you need further assumptions to replace the gradient with $\Delta p$.

Third, you don't need a textbook to derive Poiseuille flow.

 

[Chestermiller]

http://inspirehep.net/record/44832/

The subsequent paper by Einstein generalizing to a gas of massive particles can be found here:

http://www.lorentz.leidenuniv.nl/hi...ein_1925_publication/Pages/paper_1925_04.html

I guess this latter paper you get with much better quality from the Einstein Paper Project.

Are you sure you meant to post this in the present thread?

 

[vanhees71]

Uups, that was of course for another thread, where the original papers about Bose gases were looked for. I deleted it and posted it there:

https://www.physicsforums.com/threads/bose-einstein-condensate-original-paper.938990/