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A Laplace Eq. in Cylindrical coordinates (no origin)

  1. Jun 8, 2017 #1
    Hi,
    I need to solve Laplace equation:##\nabla ^2 \Phi(x,r)=0## in cylindrical domain ##r_1<r<r_2##, ##0<z<+\infty##. The boundary conditions are:
    ##
    \left\{
    \begin{aligned}
    &\Phi(0,r)=V_B \\
    & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} \\
    &\frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_1}=0 \\
    &\lim_{z\rightarrow \infty} \Phi(z,r)=0
    \end{aligned}
    \right.
    ##
    By separation of variables, I can write the solution as:
    ##
    \Phi_(z,r)=e^{-\lambda z}(A J_0(\lambda r)+BY_0(\lambda r))
    ##
    By applying the first boundary condition I get:
    ##
    \sum_{m=1}^{\infty} A_m J_0(\lambda_m r)+B_m Y_0(\lambda_m r)=V_B
    ##
    Now I can compute a first relation between ##A_m## and ##B_m## by exploiting orthogonality between different scaled Bessel functions (##i \ne m##):
    ##
    \left\{
    \begin{aligned}
    & \int_{r_1}^ {r_2} r J_0(\lambda_i r)J_0(\lambda_m r) dr =0 \\
    & \int_{r_1}^ {r_2} r J_0(\lambda_i r)Y_0(\lambda_m r) dr =0 \\
    & \int_{r_1}^ {r_2} r Y_0(\lambda_i r)Y_0(\lambda_m r) dr =0
    \end{aligned}
    \right.
    ##
    by virtue of the second and the third boundary conditions, as it is explained here (http://www.hit.ac.il/staff/benzionS/Differential.Equations/Orthogonality_of_Bessel_functions.htm)
    and it was pointed out also here
    (https://www.physicsforums.com/threads/laplace-eq-in-cylindrical-coordinates.915420/)

    My question is the following:
    Assume the second boundary condition was ## {C^{'}}_{ox} (V_G - \Phi(x,r_2))=C_0 \frac{\partial \Phi(x,r)}{\partial r}\rvert_{r=r_2} ##, ##V_G=constant##. It seems to me that in this case the orthogonality conditions are not valid anymore, because of the presence of the constant term ##V_G##.
    1) Is this true ?
    2) How can I proceed in this case?

    Thank you all

    Edit: I am having some doubts also about the integrals concerning ##Y_0##. Are they the same as the ones that involve ##J_0##?
     
    Last edited: Jun 8, 2017
  2. jcsd
  3. Jun 13, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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