What is the Steady State Temperature in a Circular Ring?

[Karnage1993]

Homework Statement

Find the steady state temperature ##U(r, \theta)## in one-eighth of a circular ring shown below:

Homework Equations

The Attempt at a Solution

I start by assuming a solution of the form ##u(r,\theta) = R(r)\Theta(\theta)##. I also note that ##u(r,\theta)## satisfies the equation ##u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2}u_{\theta\theta} = 0## where ##a \le r \le b## and ##0 \le \theta \le \frac{\pi}{4}##. I know that ##r## is bounded, but I am not sure if the temperature is periodic, ie, if ##\Theta(\theta + 2\pi) = \Theta(\theta)##. Where I'm stuck is I do not know how to incorporate the other boundary conditions into what I have, ie, what do I do with the pieces where ##u = 0## and ##u = 100##?

 

[haruspex]

Before worrying about boundary conditions, can you develop the general solution?

 

[Karnage1993]

Yes, I believe so. (That is for ##0 \le r \le a, 0 \le \theta \le 2\pi##, for some finite ##a##, right?)

The process to get to it is quite lengthy, but I can do it.

 

[HallsofIvy]

Well, then, once you have the general solution use the fact that [itex]u(r, 0)= u(r, \pi/4)= 0[/itex], for all r between a and b, to solve for two of the constants.