# Thread: Heat Equation On an Annulus

1. ## Heat Equation On an Annulus

The following partial differential equation describes the radially symmetric temperature distribution in the annulus $\displaystyle 0<a<r<b$, with constant temperature prescribed at $\displaystyle r=a$ and $\displaystyle r=b$:

$\displaystyle u_t = \frac{1}{r} (ru_r)_r$
$\displaystyle a<r<b, t>0$

Boundary Conditions:
$\displaystyle u(a,t) = 0, u(b,t)=1$

Initial Condition:
$\displaystyle u(r,0) = f(r)$

Find the steady-state temperature distribution $\displaystyle v(r) = \lim_{ t \to \infty} u(r,t)$

2. Originally Posted by Creebe
The following partial differential equation describes the radially symmetric temperature distribution in the annulus $\displaystyle 0<a<r<b$, with constant temperature prescribed at $\displaystyle r=a$ and $\displaystyle r=b$:

$\displaystyle u_t = \frac{1}{r} (ru_r)_r$
$\displaystyle a<r<b, t>0$

Boundary Conditions:
$\displaystyle u(a,t) = 0, u(b,t)=1$

Initial Condition:
$\displaystyle u(r,0) = f(r)$

Find the steady-state temperature distribution $\displaystyle v(r) = \lim_{ t \to \infty} u(r,t)$

Maybe it's a bit too late but steady state means [imath]\frac{\partial u}{\partial t}=0[/imath] therefore you have an ordinary DE to solve which is fairly simple:

$\displaystyle \frac{d}{dr}\left(r\frac{du}{dr}\right)=0$

Can you take it from here?

Coomast