To solve the Heat Conduction Equation on a 2-D disk of radius
, try to separate the equation using
![\begin{displaymath}
T(r, \theta, t) = R(r)\Theta (\theta)T(t).
\end{displaymath}](h_698.gif) |
(1) |
Writing the
and
terms of the Laplacian in Spherical Coordinates gives
![\begin{displaymath}
\nabla^2={d^2R\over dr^2}+ {2\over r} {dR\over dr} + {1\over r^2} {d^2\Theta \over d\theta^2},
\end{displaymath}](h_700.gif) |
(2) |
so the Heat Conduction Equation becomes
![\begin{displaymath}
{R\Theta\over \kappa}{d^2T\over dt^2} =
{d^2R\over dr^2} \T...
...ver dr} \Theta T + {1\over r^2} {d^2\Theta\over d\theta^2} RT.
\end{displaymath}](h_701.gif) |
(3) |
Multiplying through by
gives
![\begin{displaymath}
{r^2\over \kappa T}{d^2T\over dt^2}
= {r^2\over R} {d^2R\ov...
...over R} {dR\over dr}+{d^2\Theta\over d\theta^2}{1\over\Theta}.
\end{displaymath}](h_703.gif) |
(4) |
The
term can be separated.
![\begin{displaymath}
{d^2\Theta \over d\theta^2} {1\over \Theta } = -n(n+1),
\end{displaymath}](h_704.gif) |
(5) |
which has a solution
![\begin{displaymath}
\Theta(\theta ) = A\cos\left[{\sqrt{n(n+1)}\, \theta}\right]+B\sin\left[{\sqrt{n(n+1)}\,\theta}\right].
\end{displaymath}](h_705.gif) |
(6) |
The remaining portion becomes
![\begin{displaymath}
{r^2\over \kappa T}{d^2T\over dt^2}= {r^2\over R} {d^2R\over dr^2} + {2r\over R} {dR\over dr} -n(n+1).
\end{displaymath}](h_706.gif) |
(7) |
Dividing by
gives
![\begin{displaymath}
{1\over \kappa T}{d^2T\over dt^2}
= {1\over R} {d^2R\over d...
...ver rR} {dR\over dr} -{n(n+1)\over r^2} = -{1\over \lambda^2},
\end{displaymath}](h_708.gif) |
(8) |
where a Negative separation constant has been chosen so that the
portion remains finite
![\begin{displaymath}
T(t)=C e^{-\kappa t/\lambda^2}.
\end{displaymath}](h_709.gif) |
(9) |
The radial portion then becomes
![\begin{displaymath}
{1\over R} {d^2R\over dr^2} + {2\over rR} {dR\over dr} -{n(n+1)\over r^2} +{1\over \lambda^2}=0
\end{displaymath}](h_710.gif) |
(10) |
![\begin{displaymath}
r^2{d^2R\over dr^2} + 2r {dR\over dr} +\left[{{r^2\over \lambda^2}-n(n+1)}\right]R=0,
\end{displaymath}](h_711.gif) |
(11) |
which is the Spherical Bessel Differential Equation. If the initial temperature is
and the boundary
condition is
, the solution is
![\begin{displaymath}
T(r,t)=1-2\sum_{n=1}^\infty {J_0(\alpha_n r)\over \alpha_nJ_1(\alpha_n)} e^{{\alpha_n}^2t},
\end{displaymath}](h_714.gif) |
(12) |
where
is the
th Positive zero of the Bessel Function of the First Kind
.
© 1996-9 Eric W. Weisstein
1999-05-25