City-Data Forum Simple Differential Equation That For Some Reason Eludes Me (power, functions, problem)
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01-11-2012, 12:22 PM
 Location: Wilkinsburg 1,661 posts, read 2,149,764 times Reputation: 983

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?

01-11-2012, 03:10 PM
 Location: Not where you ever lived 11,544 posts, read 23,643,955 times Reputation: 6103
If you don't get an answer I can ask my math whiz.

01-11-2012, 06:20 PM
 Location: Westwood, MA 3,074 posts, read 3,843,710 times Reputation: 3727
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)

01-11-2012, 08:22 PM
 Location: Westwood, MA 3,074 posts, read 3,843,710 times Reputation: 3727
u = const * (r/R)

is the right way, I had reversed r and R before

01-11-2012, 09:00 PM
 Location: Wilkinsburg 1,661 posts, read 2,149,764 times Reputation: 983
Quote:
 Originally Posted by jayrandom 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.
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