Equilibrium heat equation in 2D cylindrical coordinates

[dingo_d]

Homework Statement

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 $$T_0$$, and the outer at the temperature given by the function $$f(\phi)=T_0\cos(2\phi)$$. Find the equilibrium distribution of the heat everywhere inside the plate.

Homework Equations

Laplace equation:

$$\nabla^2 u(r,\phi)=0$$

Boundary conditions:
$$u(a,\phi)=T_0$$
$$u(b,\phi)=T_0\cos(2\phi)$$

The Attempt at a Solution

By separation of variables I get:

$$r^2\frac{R''}{R}+r\frac{R'}{R}=-\frac{\Phi''}{\Phi}=\lambda^2$$

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

$$R(r)=Ar^\lambda+Br^{-\lambda}$$

$$\Phi(\phi)=C\cos(\lambda\phi)+D\sin(\lambda\phi)$$

With the condition of consistency of the azimuthal part:

$$\Phi(\phi)=\Phi(\phi+2\pi)$$

I get that $$\lambda=m\in\mathbb{Z}$$, the $$\lambda=0$$ gives different solutions.

So my first solution is:

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

For $$\lambda=0$$ I have these solutions:

$$R(r)=C_0\ln(r)+D_0$$

$$\Phi(\phi)=A_0\phi+B_0$$

With the condition of consistency I get:

$$A_0\phi+B_0=A_0\phi+A_02\pi+B_0\Rightarrow A_0=0$$

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

$$u_2(r\phi)=C\ln(r)+D$$

The general solution is the superposition of the two solutions:

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

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:

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

And the integral on the right would give me Kronecker delta which would 'kill' the sum so I would get the $$A_n$$ from that. But here I don't have this simple way. How should I get the $$A_m,\ B_m,\ C_m,\ D_m$$?

 

[Maxim Zh]

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:

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

And the integral on the right would give me Kronecker delta which would 'kill' the sum so I would get the $$A_n$$ from that.

Use the same procedure for $$r=a$$ and $$r=b$$, multiplying your solution by $$sin(n\phi)$$ and $$cos(n\phi)$$. You will get equations for the unknown coefficients.