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Laplace equation in an annulus with Neumann BCs

  1. Oct 4, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve Laplace's equation inside a circular annulus (ring) (a < r < b) subject to the BCs:

    [tex]\frac{\partial{u}}{\partial{r}}\left({a,}\theta\right)&={f}\left(\theta\right)[/tex]

    [tex]\frac{\partial{u}}{\partial{r}}\left({b,}\theta\right)&={g}\left(\theta\right)[/tex]

    2. Relevant equations

    [tex]\nabla^{2}{u}&=0[/tex]

    I also believe because of the physics of a ring, we have additional BCs:

    [tex]{u}\left({r,-}\pi\right)&={u}\left({r,}\pi\right)[/tex]

    [tex]\frac{\partial{u}}{\partial{\theta}}\left({r,-}\pi\right)&=\frac{\partial{u}}{\partial{\theta}}\left({r,}\pi\right)[/tex]

    3. The attempt at a solution

    I have used separation of variables to arrive at a general solution:

    [tex]{u}\left({r,}\theta\right)&={A}_0+{B}_0\ln{r}+\sum^{\infty}_{n=1}{r}^{n}\left({A}_{n}\cos{n}\theta+{B}_{n}\sin{n}\theta\right)+{r}^{-n}\left({C}_{n}\cos{n}\theta+{D}_{n}\sin{n}\theta\right)[/tex]

    This solution takes into account the periodic boundary conditions specified above under "Relevant Equations", but I am struggling to correctly apply the Neumann BCs specified by the problem in order to arrive at equations for the arbitrary constants.

    I try to take the partial derivative of u(r,theta) with respect to r and arrive at:

    [tex]\frac{\partial{u}}{\partial{r}}&=\frac{{B}_0}{r}+\sum^{\infty}_{n=1}{n}{r}^{n-1}\left({A}_{n}\cos{n}\theta+{B}_{n}\sin{n}\theta\right)-{nr}^{-n-1}\left({C}_{n}\cos{n}\theta+{D}_{n}\sin{n}\theta\right)[/tex]

    I don't know how I can apply Fourier series (assuming that is what I'm supposed to do) to solve for each of the coefficients (A,B,C,D).
     
  2. jcsd
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