U.S. CitiesCity-Data Forum Index
Go Back   City-Data Forum > General Forums > Science and Technology
 [Register]
Please register to participate in our discussions with 1.5 million other members - it's free and quick! Some forums can only be seen by registered members. After you create your account, you'll be able to customize options and access all our 15,000 new posts/day with fewer ads.
Jump to a detailed profile or search
site with Google Custom Search

Search Forums  (Advanced)
Business Search - 14 Million verified businesses
Search for:  near: 
 
Old 01-11-2012, 11:22 AM
 
Location: Wilkinsburg
1,661 posts, read 1,263,539 times
Reputation: 956
Default Simple Differential Equation That For Some Reason Eludes Me

So I've been working on a problem at work that deals with the flow of fluids through annular passages. The following differential equation arose from a Navier-Stokes derivation:



U_theta represents fluid velocity in the azimuthal direction, r is the radius, and R distance to the outside of the channel.

I know I solved a lot of ODEs like this in college, but right now I can't figure out how to solve it. I tried doing a numerical solution, as I typically do when I can't figure out an analytical solution, but unfortunately I couldn't get it to converge.

Any math nuts out there that can offer advice?
Reply With Quote Quick reply to this message

 
Old 01-11-2012, 02:10 PM
 
Location: Not where you ever lived
10,785 posts, read 14,570,048 times
Reputation: 5344
If you don't get an answer I can ask my math whiz.
Reply With Quote Quick reply to this message
 
Old 01-11-2012, 05:20 PM
 
Location: Westwood, MA
1,498 posts, read 1,734,833 times
Reputation: 1299
Default Let me wolframalpha that for you

Give a man a fish, he'll eat for a day. Teach a man to fish, he'll eat for a lifetime. Teach a man about wolframalpha and he'll totally forget even the most basic of ODEs ;-)

DSolve[r*u''[r] + u'[r]- u[r]/r==0, u[r], r] - Wolfram|Alpha

Matching the boundary conditions you'll find C[2] = -i*C[1] and C[1] = const/R giving the final, super interesting solution

u = const * (R/r)

If you're interested, that's a Cauchy-Euler ODE, which has a fairly straightforward solution (Cauchy

The trick is to see that each higher derivative has one lower power of r, so a polynomial would be a good trial solution. Substituting u = x^m (after simplifying) gives

m^2 - 1 = 0

Which has solutions m = +/- 1 or u = C[1]*r + C[2]/r

(clearly wolfram's solver uses a different method and comes out with a different combination)
Reply With Quote Quick reply to this message
 
Old 01-11-2012, 07:22 PM
 
Location: Westwood, MA
1,498 posts, read 1,734,833 times
Reputation: 1299
u = const * (r/R)

is the right way, I had reversed r and R before
Reply With Quote Quick reply to this message
 
Old 01-11-2012, 08:00 PM
 
Location: Wilkinsburg
1,661 posts, read 1,263,539 times
Reputation: 956
Quote:
Originally Posted by jayrandom View Post
Give a man a fish, he'll eat for a day. Teach a man to fish, he'll eat for a lifetime. Teach a man about wolframalpha and he'll totally forget even the most basic of ODEs ;-)

DSolve[r*u''[r] + u'[r]- u[r]/r==0, u[r], r] - Wolfram|Alpha

Matching the boundary conditions you'll find C[2] = -i*C[1] and C[1] = const/R giving the final, super interesting solution

u = const * (R/r)

If you're interested, that's a Cauchy-Euler ODE, which has a fairly straightforward solution (Cauchy

The trick is to see that each higher derivative has one lower power of r, so a polynomial would be a good trial solution. Substituting u = x^m (after simplifying) gives

m^2 - 1 = 0

Which has solutions m = +/- 1 or u = C[1]*r + C[2]/r

(clearly wolfram's solver uses a different method and comes out with a different combination)
Ah, thanks so much! Yeah, I recognized the form, but couldn't for the life of me think how to approach it. I psyched myself out, and started thinking that if anyone did reply they were going to tell me that the solution involved Bessel functions, at which point I was going to find someone else to do the work.
Reply With Quote Quick reply to this message
Please register to post and access all features of our very popular forum. It is free and quick. Over $68,000 in prizes has already been given out to active posters on our forum. Additional giveaways are planned.

Detailed information about all U.S. cities, counties, and zip codes on our site: City-data.com.


Reply
Please update this thread with any new information or opinions. This open thread is still read by thousands of people, so we encourage all additional points of view.

Quick Reply
Message:

Over $84,000 in prizes was already given out to active posters on our forum and additional giveaways are planned!

Go Back   City-Data Forum > General Forums > Science and Technology
Similar Threads

All times are GMT -6. The time now is 12:07 PM.

2005-2014, Advameg, Inc.

City-Data.com - Archive 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 - Top