# 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 $0, with constant temperature prescribed at $r=a$ and $r=b$:

$u_t = \frac{1}{r} (ru_r)_r$
$a0$

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

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

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

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

$u_t = \frac{1}{r} (ru_r)_r$
$a0$

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

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

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

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:

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

Can you take it from here?

Coomast