(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A circular disc of radius a is heated in such a way that its perimeter [itex]r=a[/itex] has a steady temperature distribution [itex]A+B \cos ^2 \phi [/itex] where r and [itex]\phi[/itex] are plane polar coordiantes and A and B are constants. Find the temperature [itex]T(\rho, \phi)[/itex] everywhere in the region [itex]\rho < a[/itex]

2. The attempt at a solution

I have been able to come to a few conclusions. First i assumed we should use the diffusion equation

[itex]\nabla ^2 u = \frac{1}{\alpha ^2} \frac{\partial u}{\partial t}[/itex]

Since we are talking about a steady state problem

[itex] \frac{\partial u}{\partial t} = 0[/itex]

and thus we get the Laplace

[itex]\nabla^2 u = 0 [/itex]

Laplaces equation in 2d polar coordinates is

[itex]\nabla ^2 u = \frac{1}{r}\frac{\partial}{\partial r}\left ( r \frac{\partial u}{\partial r} \right ) + \frac{1}{r^2}\frac{\partial ^2 u}{\partial \phi ^2} [/itex]

and we assume a solution of the form

[itex]u = R(r) \Phi(\phi) [/itex]

and thus

[itex] \frac{1}{R}\frac{1}{r}\frac{\partial}{\partial r}\left ( r \frac{\partial R}{\partial r} \right ) + \frac{1}{\Phi}\frac{1}{r^2}\frac{\partial ^2 \Phi}{\partial \phi ^2} = 0 [/itex]

[itex] \frac{r^2}{R}\frac{1}{r}\frac{\partial}{\partial r}\left ( r \frac{\partial R}{\partial r} \right ) + \frac{1}{\Phi}\frac{\partial ^2 \Phi}{\partial \phi ^2} = 0 [/itex]

and thus

[itex] \frac{\partial^2 \Phi}{\partial \phi^2} + n^2 \Phi = 0 \rightarrow \Phi = c_1 cos(n\phi) + c_2 sin(n\phi) [/itex]

and

[itex]r^2\frac{\partial^2 R}{\partial r^2} - n^2 R = 0 \rightarrow R = e^n + e^{-n} [/itex]

and the solutions for u become

[itex]u = \{ (e^n + e^{-n})cos(n\phi) \}, \{ (e^n + e^{-n})sin(n\phi) \} [/itex]

This is how far i have gotten. I am not sure if it is completley correct. If it is any hints on where to go from here would be great. Thanks a bunch!

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# Homework Help: Steady-State Temperature Distribution of a circular disk

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