An infinite straight metal pipe has annular cross-section $a \leq r \leq b$. The temperature of the inner surface of the pipe is equal to $\cos(\theta)$, and the outer surface is thermally insulted.
(i) Write down the differential equation and the boundary conditions that the temperature must satisfy
(ii) Find the steady state temperature in $a \leq r \leq b$ and show that the temperature of the outer surface is $\dfrac{2ab}{a^2+b^2}\cos(\theta)$
(iii) Find the heat flux across unit area of the inner surface, given that the thermal conductivity of the pipe is equal to $\kappa$.
For (i) it must satisfy Laplace's equation in polar coordinates, so
$\dfrac{\partial^2T}{\partial r^2} + \dfrac{1}{r}\dfrac{\partial T}{\partial r} + \dfrac{1}{r^2}\dfrac{\partial^2 T}{\partial\theta^2}=0$
and $T(a, \theta) = cos(\theta)$
How do I use the fact it's thermally insulated at $r=b$? In cartesians I know this would imply that $-\kappa \dfrac{\partial T}{\partial x} = 0$ so $T_x = 0$. Does this need to be solved via the Neumann problem in polar coordinates, so $\dfrac{\partial T}{\partial n}(b,\theta) = 0$
For (ii) I don't know what 'steady state' is (it's not mentioned in my notes at all). Is it the temperature independent of time? If so, how is it possible to determine this while working in polar coordinates?
Thanks for any help.