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Equilibrium heat equation in 2D cylindrical coordinates

  1. Jan 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Plate in the shape of the circular halo (inner radius a, outer radius b>a), the inner edge is being kept at a constant temperature [tex]T_0[/tex], and the outer at the temperature given by the function [tex]f(\phi)=T_0\cos(2\phi)[/tex]. Find the equilibrium distribution of the heat everywhere inside the plate.

    2. Relevant equations
    Laplace equation:

    [tex]\nabla^2 u(r,\phi)=0[/tex]

    Boundary conditions:
    [tex]u(a,\phi)=T_0[/tex]
    [tex]u(b,\phi)=T_0\cos(2\phi)[/tex]

    3. The attempt at a solution

    By separation of variables I get:

    [tex]r^2\frac{R''}{R}+r\frac{R'}{R}=-\frac{\Phi''}{\Phi}=\lambda^2[/tex]

    For [tex]\lambda\neq 0[/tex] I have 2 DE, one is Euler's DE, and the other the equation of harmonic oscillator. Their solution is:

    [tex]R(r)=Ar^\lambda+Br^{-\lambda}[/tex]

    [tex]\Phi(\phi)=C\cos(\lambda\phi)+D\sin(\lambda\phi)[/tex]

    With the condition of consistency of the azimuthal part:

    [tex]\Phi(\phi)=\Phi(\phi+2\pi)[/tex]

    I get that [tex]\lambda=m\in\mathbb{Z}[/tex], the [tex]\lambda=0[/tex] gives different solutions.

    So my first solution is:

    [tex]u_1(r,\phi)=\sum_{m=1}^\infty r^m[A_m\cos(m\phi)+B_m\sin(m\phi)]+\sum_{m=1}^\infty r^{-m}[C_m\cos(m\phi)+D_m\sin(m\phi)][/tex]

    For [tex]\lambda=0[/tex] I have these solutions:

    [tex]R(r)=C_0\ln(r)+D_0[/tex]

    [tex]\Phi(\phi)=A_0\phi+B_0[/tex]

    With the condition of consistency I get:

    [tex]A_0\phi+B_0=A_0\phi+A_02\pi+B_0\Rightarrow A_0=0[/tex]

    So the second solution is: (I have put the constants together)

    [tex]u_2(r\phi)=C\ln(r)+D[/tex]

    The general solution is the superposition of the two solutions:

    [tex]u(r,\phi)=C\ln(r)+D+\sum_{m=1}^\infty r^m[A_m\cos(m\phi)+B_m\sin(m\phi)]+\sum_{m=1}^\infty r^{-m}[C_m\cos(m\phi)+D_m\sin(m\phi)][/tex]

    So here is where my problem starts. I know that I'm supposed to use formulas for Fourier summation but in the case of a string that was the more straightforward, for example:

    [tex]f(x,0)=\sum_{m=0}^\infty A_m\sin\left(\frac{m\pi}{L}x\right)\Big/ \cdot \sin\left(\frac{n\pi}{L}x\right),\ \int_0^L dx[/tex]

    And the integral on the right would give me Kronecker delta which would 'kill' the sum so I would get the [tex]A_n[/tex] from that. But here I don't have this simple way. How should I get the [tex]A_m,\ B_m,\ C_m,\ D_m[/tex]?
     
  2. jcsd
  3. Jan 26, 2010 #2
    Use the same procedure for [tex] r=a [/tex] and [tex] r=b [/tex], multiplying your solution by [tex] sin(n\phi) [/tex] and [tex] cos(n\phi) [/tex]. You will get equations for the unknown coefficients.
     
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