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Steady State Temperature

  1. Apr 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the steady state temperature ##U(r, \theta)## in one-eighth of a circular ring shown below:
    OGZxSVU.png



    2. Relevant equations



    3. 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##?
     
  2. jcsd
  3. Apr 2, 2013 #2

    haruspex

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    Before worrying about boundary conditions, can you develop the general solution?
     
  4. Apr 2, 2013 #3
    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.
     
  5. Apr 2, 2013 #4

    HallsofIvy

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    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.
     
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