# Equilibrium solution using polar coordinates

Find the equilibrium temperature on a half-disk of radius 1 when the temperature is held to 1 degree on the curved edge, while the straight edge is insulated.

For this problem I think the solution uses Laplace equation in polar coordinates using method of separation of variables. However, I'm not sure how to derive the boundary conditions.

Let $u(r,\theta)$ be the solution in polar coordinates, $0<r\le1$, $0\le\theta\le\pi/2$.
• The temperature is held to 1º on the curved edge: $u(1,\theta)=1$, $0\le\theta\le\pi/2$.
• The straight edge is insulated. This is a little more complicated. In cartesian coordinates it would be $u_y(x,0)=0$. Since $u_y=u_r\sin\theta+u_\theta\cos\theta$, this translates into $u_\theta(r,0)=u_\theta(r,\pi)=0$, $0<r\le1$.
• Could you explain why the insulated condition means $u_y(x,0) = 0$? Feb 5, 2015 at 11:22
• $u_y(x,0)$ represents the heat flux through the line $y=0$ (Newton's law: heat flux is proportional to the gradient of the temperature). Insulation means that there is no heat flux through that line, so $u_y=0$. Feb 5, 2015 at 11:47
• Thank you. Would $0 \leq \theta \leq \bf{\pi}$ since the domain is a half disk? Feb 5, 2015 at 11:49
• Yes. ${}{}{}{}{}{}$ Feb 5, 2015 at 11:51
• I'm a little confused about your $u_y$. I get $u_y = u_r / \sin \theta + u_\theta / (r\cos\theta)$. Feb 5, 2015 at 16:43