Chapter 30, Problem 14P

The problem of transient radial heat flow in a circular rod in nondimensional form is

described by

$\frac{{\partial }^{2}u}{\partial {\bar{r}}^2}+\frac{1}{r}\frac{\partial u}{\partial \bar{r}}=\frac{\partial u}{\bar{t}}$

Boundary conditions	$u\left(0,\bar{t}\right)=1$	$\frac{\partial u}{\partial \bar{t}}\left(0,\bar{t}\right)=0$
Initial conditions	$u\left(\bar{x},0\right)=0$	$0\le \bar{x}\le 1$

Solve the nondimensional transient radial heat-conduction equation in a circular rod for the temperature distribution at various times as the rod temperature approaches steady state. Use second-order accurate finite-difference analogues for the derivatives with a Crank-Nicolson formulation. Write a computer program for the solution. Select values of $\Delta \bar{r}$ and $\Delta \bar{t}$ for good accuracy. Plot the temperature u versus radius $\bar{r}$ for various times $\bar{t}$.