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fluidistic
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Homework Statement
I'm unable to solve a problem of heat equation in a cylinder in steady state. The problem is a cylinder of radius a and a height L. The boundary condition are ##T(\rho , \theta , 0)=\alpha \sin \theta##, ##T(\rho, \theta , L)=0## and ##\frac{\partial T}{\partial \rho} (a, \theta , z)=0##.
I must calculate the temperature inside the cylinder.
Homework Equations
The temperature ##T(\rho, \theta , z)## satisfies the Laplace equation inside the cylinder. Namely ##\triangle T=0##.
The Attempt at a Solution
I solved the Laplace equation in cylindrical coordinates using separation of variables looking for solutions of the form ##T=R(\rho ) \Theta (\theta) Z(z)##.
This produced 3 ODE's, namely:
[tex]\begin{array} \frac{Z''}{Z}=k^2 \\ \frac{\Theta '' }{\Theta } =-m^2 \\ \rho ^2 R'' + \rho R' + R(k^2 \rho ^2 - m^2) \end{array}[/tex]. Solving the 3 equations, I got the eigenfunctions of the problem, they have the form ##T_{m,k}=J_m (k \rho )[C_m \cos m \theta + D_m \sin m \theta ](A_k e^{kz}+B_k e^{-kz})##.
However the function that satisfies the boundary conditions is a linear combination of these eigenfunctions.
Namely ##T(\rho , \theta , z)=\sum _{m=0}^\infty \sum _{k=0}^\infty J_m (k \rho ) [C_m \cos (m \theta )+ D_m \sin (m \theta )](A_k e^{kz}+B_k e^{-kz})##.
So far so good. Now I apply the 2nd boundary condition. This gives me that ##A_k=-B_ke^{-2kL}## so that T simplifies to ##T(\rho , \theta , z )=T(\rho , \theta , z)=\sum _{m=0}^\infty \sum _{k=0}^\infty J_m (k \rho ) [C_m \cos (m \theta )+ D_m \sin (m \theta )][B_k (e^{-kz}-e^{k(z-2L)})]##.
Now I apply the first boundary condition. This gives me ##\sum _{k=0}^\infty \sum _{m=0}^\infty E_k J_m (k\rho )[C_m \cos (m \theta ) +D_m \sin (m\theta )]=\alpha \sin \theta##. Where ##E_k=B_k(1-e^{-2kL})##, is a constant for a given k.
So I'm looking for the constant ##C_m## and ##B_m##'s. By inspection it's obvious that ##C_m=0 \forall m## and ##D_m =0 \forall m \neq 1##.
So I am left with ##\sum _{k=0}^\infty E_k J_1 (k \rho ) D_1 \sin \theta = \alpha \sin \theta##. This gives me that ##D_1= \frac{\alpha}{\sum _{k=0}^\infty E_k J_1 (k \rho )}##.
Which is impossible since ##D_1## is a constant and MUST NOT depend on rho, while here I reached that it does depend on rho. I've absolutely no idea where I went wrong. Any pointer would be immensily appreciated.