CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

Integration of Partial Differential Equations

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   September 11, 2017, 14:18
Default Integration of Partial Differential Equations
  #1
Member
 
Matt Ridzon
Join Date: Jun 2014
Posts: 91
Rep Power: 10
m_ridzon is on a distinguished road
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.

Thanks in advance,
M Ridzon
Attached Images
File Type: jpg Integration Help.jpg (193.6 KB, 17 views)
m_ridzon is offline   Reply With Quote

Old   September 11, 2017, 14:34
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,257
Rep Power: 67
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
If you are studying the Hagen-Poiseuille solution, have a look to the book of White, Sec.4.10
FMDenaro is offline   Reply With Quote

Old   September 11, 2017, 14:53
Default
  #3
Member
 
Matt Ridzon
Join Date: Jun 2014
Posts: 91
Rep Power: 10
m_ridzon is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
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
m_ridzon is offline   Reply With Quote

Old   September 11, 2017, 14:55
Default
  #4
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,257
Rep Power: 67
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
https://docs.google.com/file/d/0B9Jt...Zjg/edit?pli=1
FMDenaro is offline   Reply With Quote

Old   September 11, 2017, 15:26
Default
  #5
Member
 
Matt Ridzon
Join Date: Jun 2014
Posts: 91
Rep Power: 10
m_ridzon is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
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
m_ridzon is offline   Reply With Quote

Reply

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
grid generation with elliptic partial differential equations. gorkemocalan Main CFD Forum 0 October 1, 2015 15:39
Low Reynolds K Epsilon Launder Sharma Model Functions Doubt... Ruonin Main CFD Forum 17 February 17, 2014 10:52
modelling Differential equations in a udf RikardMNorén Fluent UDF and Scheme Programming 2 October 1, 2013 04:36
numerical integration of a set of ordinary partial differential equation wlt_1985 FLUENT 0 November 5, 2010 17:17
MultiDimensional Partial Differential Equations. Aditya Main CFD Forum 2 October 28, 2005 06:03


All times are GMT -4. The time now is 12:56.