A case of heat flow which is virtually one-dimensional arises when the conducting medium is a circular cylinder and the temperature function u depends only on the time t and the distance r from the axis of the cylinder, $\displaystyle u=(r,t)$. For example, imagine a cylindrical pipe filled with a hot fluid and suppose that one wishes to study the loss of heat through the sides of the pipe.

Let c, p, k, K denote the thermal constants of the cylinder. By considering the heat energy contained in a section of pipe of length H and lying between the radiir = a, r = b(Why are the radii different? Shouldn't they be the same?)show that

$\displaystyle \displaystyle 2\pi H\int_a^b\left[c\cdot p\cdot r\frac{\partial u}{\partial t}-\frac{\partial}{\partial r}\left(k\cdot r\frac{\partial u}{\partial r}\right)\right]dr=\text{source term}$

Hence,obtain the equation for the source-free radial heat flow in a cylinder:

$\displaystyle \displaystyle\frac{\partial u}{\partial r}=\frac{K}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)=K\left[u_{rr}+\frac{1}{r}u_r\right]$.

How do I start obtaining the equation?