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Solid Cylinder, Steady 3D, R0JZKL$\Phi00$

Consider the steady temperature in the cylinder caused either by heating at the boundaries or by internal energy generation. The temperature satisfies

$\displaystyle \frac{\partial ^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T...
...\frac{\partial ^{2}T}{\partial \phi^{2}}
+\frac{\partial ^{2}T}{\partial z^{2}}$ $\textstyle =$ $\displaystyle -\frac{g(r,\phi,z)}{k}$ (1)
    $\displaystyle 0<z<L;\;0<r<a;\;0<\phi<2\pi$  
$\displaystyle k_{i}\frac{\partial T}{\partial n_{i}}+h_{i}T$ $\textstyle =$ $\displaystyle f_{i}\mbox{ \ for
boundaries }i=1,\; 2,\mbox{ or }3$  

Here $n_{i}$ is the outward normal on each surface of the cylinder (at $r=a$, $z=0$, and $z=L$). The boundary condition represents one of three types at each surface: type 1 for $k_{i}=0$, $h_{i}=1$, and $f_{i}$ a specified temperature; type 2 for $k_{i}=k,$ $h_{i}=0$, and $f_{i}$ a specified heat flux; and, type 3 for $%
k_{i}=k$ and $f_{i}=h_{i}T_{\infty }$ for convection to surroundings at temperature $T_{\infty }$. Heat transfer coefficient $h_{i}$ must be uniform on the i$^{th}$ boundary.

The temperature can be stated in the form of integrals with the method of Green's functions. If the Green's function $G$ is known, then the temperature that satisfies equation (1) is given by:

\begin{displaymath}
T(r,\phi,z)=\int_{z^{\prime }=0}^{L}\int_{\phi^{\prime }=0}^...
...ght.)
\,r^{\prime }d\phi^{\prime } dr^{\prime } dz^{\prime }
\end{displaymath}

(for volume energy generation)

\begin{displaymath}
+\sum_{j=1}^{3}\int_{s_{j}}\frac{f_{j}}{k}
G(r,\phi,z\,\left...
...,\phi_{j}^{\prime },z_{j}^{\prime }\right.)
ds_{j}^{\prime }
\end{displaymath}

(for boundary conditions of type 2 and 3)
\begin{displaymath}
-\sum_{i=1}^{3}\int_{s_{i}}f_{i}\frac{\partial
G(r,\phi,z\,...
...^{\prime }\right.)
}{\partial n_{i}^{\prime }}ds_{i}^{\prime }
\end{displaymath} (2)

(for boundary conditions of type 1 only).


The same Green's function appears in each integral but is evaluated at locations appropriate for each integral. Here position $(r_{i}^{\prime },\phi_{i}^{\prime },z_{i}^{\prime })$ is located on surface $s_{i}$. Surface differential $ds_i^{\prime}$ is associated with appropriate surfaces of the cylinder: on surface $r=a$, $ds_i= a d\phi dz'$; and, at $z'=0$ or $z'=L$, $ds_i^{\prime}= r'd\phi^{\prime} dr'$. The two summations represent all possible combinations of boundary conditions, but with only one type of boundary condition on each of three surfaces of the cylinder. Mixed-type boundary conditions are not treated.



Subsections
next up previous
Next: GF for Solid Cylinder, Up: Radial Cylendrical Coordinates, Steady Previous: Radial Cylendrical Coordinates, Steady
Frank Pribyl 2005-06-07