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Solid cylinder, steady 1-D.

R01 Solid cylinder, 0 $ \leq$ r $ \leq$ b, with G < $ \infty$ (natural boundary) at r = 0, and G = 0 (Dirichlet) at r = b.

2$\displaystyle \pi$GR01(r $\displaystyle \left\vert\vphantom{ \,r^{\prime }}\right.$ r$\scriptstyle \prime$$\displaystyle \left.\vphantom{ \,r^{\prime }}\right.$) = $\displaystyle \left\{\vphantom{
\begin{array}{cc}
ln(b/r^{\prime }) & \text{for }r<r^{\prime } \\
ln(b/r) & \text{for }r>r^{\prime }
\end{array}
}\right.$$\displaystyle \begin{array}{cc}
ln(b/r^{\prime }) & \text{for }r<r^{\prime } \\
ln(b/r) & \text{for }r>r^{\prime }
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{cc}
ln(b/r^{\prime }) & \text{for }r<r^{\prime } \\
ln(b/r) & \text{for }r>r^{\prime }
\end{array}
}\right.$

R02 Solid cylinder, 0 $ \leq$ r $ \leq$ b, with G < $ \infty$ (natural boundary) at r = 0, and $ \partial$G/$ \partial$r = 0 (Neumann) at r = b . Note that this geometry requires a pseudo GF, denoted H. The temperature solution found from a pseudo GF requires that the total volumetric heat flow is equal to the boundary heat flow, and the spatial average temperature in the body must be supplied as a known condition.

2$\displaystyle \pi$HR02(r $\displaystyle \left\vert\vphantom{ \,r^{\prime }}\right.$ r$\scriptstyle \prime$$\displaystyle \left.\vphantom{ \,r^{\prime }}\right.$) = $\displaystyle \left\{\vphantom{
\begin{array}{cc}
\left( r^{2}+(r^{\prime })...
...\right) /(2b^{2})+\ln (b/r) & \text{for }%
r>r^{\prime }
\end{array}
}\right.$$\displaystyle \begin{array}{cc}
\left( r^{2}+(r^{\prime })^{2}\right) /(2b^{2}...
...ime })^{2}\right) /(2b^{2})+\ln (b/r) & \text{for }%
r>r^{\prime }
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{cc}
\left( r^{2}+(r^{\prime })^...
...\right) /(2b^{2})+\ln (b/r) & \text{for }%
r>r^{\prime }
\end{array}
}\right.$

R03 Solid cylinder, 0 $ \leq$ r $ \leq$ b, with G < $ \infty$ (natural boundary) at r = 0 and k$ \partial$G/$ \partial$r + h2G = 0 (convection) at r = b. Note B2 = h2b/k.

2$\displaystyle \pi$GR03(r $\displaystyle \left\vert\vphantom{ \,r^{\prime }}\right.$ r$\scriptstyle \prime$$\displaystyle \left.\vphantom{ \,r^{\prime }}\right.$) = $\displaystyle \left\{\vphantom{
\begin{array}{cc}
\ln (b/r^{\prime })+1/B_{2...
...rime } \\
\ln (b/r)+1/B_{2} & \text{for }r>r^{\prime }
\end{array}
}\right.$$\displaystyle \begin{array}{cc}
\ln (b/r^{\prime })+1/B_{2} & \text{for }r<r^{\prime } \\
\ln (b/r)+1/B_{2} & \text{for }r>r^{\prime }
\end{array}$ $\displaystyle \left.\vphantom{
\begin{array}{cc}
\ln (b/r^{\prime })+1/B_{2}...
...rime } \\
\ln (b/r)+1/B_{2} & \text{for }r>r^{\prime }
\end{array}
}\right.$



Kevin D. Cole
2002-12-31