# Integration of Partial Differential Equations

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September 11, 2017, 14:18
Integration of Partial Differential Equations
#1
Member

Matt Ridzon
Join Date: Jun 2014
Posts: 91
Rep Power: 10
Hello! I know this forum is dedicated to CFD and my question below isn't exactly related to such. But I know this forum is full of fluid analysts, so I thought perhaps someone on here might be able to help me out. And so hopefully the moderators won't delete my post.

In the attached JPG, I boxed in an area where I'm working through the integration of the 2D cylindrical axisymmetric Navier-Stokes equations.

z = axis along the centerline of the pipe
p = pressure, which is a function of z
r = radius of a pipe (varies from centerline to R)
vz = velocity, which is a function of r
mu = constant value (i.e., viscosity)
R = constant value (i.e., outer radius)

Near the top of the boxed area of the attachment, I show the start of the integration. I have to integrate twice with respect to r, to get the answer listed at the bottom of the page. I can't seem to get the right answer and I believe the last term might be causing my trouble (i.e., the term noted as needing integration by parts).

Basically, I need to know how to solve this partial differential equation to its analytical solution, but I'm not sure what I'm doing wrong.

M Ridzon
Attached Images
 Integration Help.jpg (193.6 KB, 17 views)

 September 11, 2017, 14:34 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,257 Rep Power: 67 If you are studying the Hagen-Poiseuille solution, have a look to the book of White, Sec.4.10

September 11, 2017, 14:53
#3
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Matt Ridzon
Join Date: Jun 2014
Posts: 91
Rep Power: 10
Quote:
 Originally Posted by FMDenaro If you are studying the Hagen-Poiseuille solution, have a look to the book of White, Sec.4.10
Yes, this is exactly what I'm working on. Thank you for your reply, but I'm not familiar with what you refer to as the "book of White." Can you share the book's title or upload an excerpt from the book? I have found one or two other books that go through this derivation, but the step-by-step details have been left out, thus leaving me lost as to what tiny detail I'm overlooking in my work. Perhaps your "book of White" has the step-by-step derivation.

Thanks,
M Ridzon

 September 11, 2017, 14:55 #4 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,257 Rep Power: 67

September 11, 2017, 15:26
#5
Member

Matt Ridzon
Join Date: Jun 2014
Posts: 91
Rep Power: 10
Quote:
Thank you again, but that excerpt leaves out the details below Eqn 4.136 to reveal how the derivation is done. And that's where I'm getting stuck. My answer strangely appears more like flow between two plates (i.e., Eqn 4.134 in your excerpt), which is not what I'm after. Would you have any additional insight that might help me along?

Thanks again!
M Ridzon