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GF, Double-Sum Form

The GF stated below contains a double summation with eigenfunctions $J_n$ norms $N_{r}$ and $N_{\phi}$, eigenvalues $\beta_{nm}$ and kernel function $P_{nm}$, as follows:

$\displaystyle G(r,\phi,z\,\left\vert \,r^{\prime },\phi^{\prime},z^{\prime }\ri...
...0m}r^{\prime}) }
{N_{r}(\beta_{0m})} \frac{1}{N_{\phi}(0)}
P_{0m}(z,z^{\prime})$     (4)
$\displaystyle + \sum_{n=1}^{\infty }\sum^{\infty}_{m=1}
\frac{ J_n(\beta_{nm}r)...
\frac{\cos[n(\phi - \phi^{\prime})]}{N_{\phi}(n)}

where $J_n$ are Bessel functions of order $n$. The norms associated with the angular direction $\phi$ are given by $N_{\phi}(0) =2\pi$ and $N_n(n \neq 0 ) =\pi$. Norm $N_r$ is given in Table 1 and the conditions defining $\beta_{nm}$ are given in Table 2. The kernel functions are discussed below.
Table 1. Inverse norm for solid cylinder cases.

  Boundary at $(N_{r})^{-1}$ $(N_{r})^{-1}$
Case $r=a$ for $n\neq0$ for $n=0$
R01 $G=0$ $\frac{2}{a^2J_n^{\prime} (\beta_{nm} a) }$ same as $n\neq0$
R02 $\frac{\partial G}{\partial r}=0$ $\frac{2}{a^2 J_n^2 (\beta_{nm} a) } \frac{a^2\beta_{nm}^2}
{(a^2\beta_{nm}^2-n^2)}$ $\frac{2}{a^2} $
R03 $k\frac{\partial G}{\partial r}+h G=0$ $\frac{2}{a^2 J_n^2 (\beta_{nm} a) } \frac{a^2\beta_{nm}^2}{(B_2^2
+ a^2\beta_{nm}^2-n^2)}$ same as $n\neq0$

Table 2. Eigenconditions which define $\beta_{nm}$.
Case eigencondition
R01 $J_n(\beta_{nm}a)=0$
R02 $J_n^{\prime}(\beta_{nm}a)=0$
R03 $\beta_{nm} a J_n^{\prime}(\beta_{nm}a)+B_iJ_n(\beta_{nm}a)=0$

Kernel functions

The kernel function $P_{nm}
(z,z^{\prime})$ must satisfy

\frac{d\mbox{\thinspace }^{2}P_{nm}}{dz^{2}}-\beta _{nm}^{2}P_{nm}=
-\delta (z-z^{\prime })
\end{displaymath} (5)

where $\beta_{nm}$ is the eigenvalue associated with eigenfunction $J_n$. The kernel function, suppressing the $nm$ subscript, may be written as
$\displaystyle P(z,z^{\prime})$ $\textstyle =$ $\displaystyle \frac{S_2^- (S_1^- e^{-\beta (2L_i-\left\vert z-z^{\prime} \right...
...ta(2L_i-z-z^{\prime} )})} {2 \beta (S_1^+ S_2^+ - S_1^- S_2^- e^{-2\beta L_i})}$ (6)
  $\textstyle +$ $\displaystyle \frac{
S_2^+ (S_1^+ e^{-\beta(\left\vert z-z^{\prime} \right\vert...
...e^{-\beta(z+z^{\prime})})}{2 \beta (S_1^+
S_2^+ - S_1^- S_2^- e^{-2\beta L_i})}$  

where the subscripts $1$ and $2$ indicate sides $z=0$ and $z=L$ sides of the cylinder, respectively. Parameters $S^{+}_{M}$ and $S^{-}_{M}$ depend on the boundary conditions on side $M$ and are given by
$\displaystyle S_M^{+}$ $\textstyle =$ $\displaystyle \left\{ \begin{array}{ll}
1 & \mbox{if side $M$\ is type 1 or type 2} \\
\beta L+B_M & \mbox{if side $M$\ is type 3} \\
\end{array}\right.$ (7)
$\displaystyle S_M^{-}$ $\textstyle =$ $\displaystyle \left\{ \begin{array}{ll}
-1 & \mbox{if side $M$\ is type 1} \\
... type 2} \\
\beta L-B_M & \mbox{if side $M$\ is type 3} \\
\end{array}\right.$ (8)

Here $B_M = L h_M/k$ is the Biot Number for side $M$, and $k$ is the conductivity of the cylinder.

The expression for $P$ in Eq. (7) is symmetric if $z$ and $z^{\prime}$ are interchanged and covers several combinations of boundary conditions provided $\beta \neq 0$. The special case of $\beta = 0 $ is discussed below.

Kernel function with $\beta = 0 $

If the $r=a$ face of the cylinder is of type 2, then the zero eigenvalue exists. In this case the kernel function must satisfy

\frac{\partial^2 P_{00}}{dz^2}=-\delta(x-x^{\prime})
\end{displaymath} (9)

as well as the boundary conditions at $z=0$ and $z=L$. Since there are nine combinations of boundary conditions (type 1, 2, or 3 at $z=0$ and $z=L$), there are nine kernel functions $P_{00}$ which are given in Table 3. Special case Z22 is discussed below.

Table 3. Kernel Function for $\beta = 0 $.
Case $P_{00} (x, x^{\prime })$ for $x>x^{\prime }$. $\ \left( \mbox{ Use } P_{00} (x^{\prime }, x)\mbox{
for }x<x^{\prime }.\right) $
Z11 $x^{\prime }(1-x/L)$
Z12 $x^{\prime }$
Z13 $x^{\prime }[1-B_2(x/L)/(1+B_2)]$
Z21 $L-x$
Z22$^a$ (( $x^{\prime})^2+x^{2})/(2L)-x+L/3$
Z23 $L(1+1/B_2-x/L)$
Z31 $(B_1x^{\prime }-B_1x^{\prime }x/L+L-x)/(1+B_1)$
Z32 $L(1/B_1+x^{\prime }/L)$
Z33 $\left( B_1B_2x^{\prime }+B_1x^{\prime }-B_1B_2x^{\prime
}x/L-B_2x+B_2L+L\right) $
  $\div (B_1B_2+B_1+B_2)$

$^a$ Special temperature solution needed with this pseudo GF.

Modified GF for Case R02$\Phi00$Z22

A very special condition occurs if the cylinder's entire boundary has type 2 (Neumann) conditions. In this case the ordinary Green's function, defined above, does not exist. However the GF method can be used if we define a modified GF, following Barton (Elements of Green's Functions and Propagation, Oxford University Press, 1989), as follows:

$\displaystyle \nabla^2 G_m = - \frac{1}{r}\delta(r - r') \delta(\phi - \phi^{\prime})
\delta(z - z')+ \frac{1}{V}$     (10)
$\displaystyle \mbox{where } \; \nabla^2 G_m =
\left( \frac{\partial^2}{\partial...
...frac{\partial^2}{\partial \phi^2}
+ \frac{\partial^2}{\partial z^2} \right) G_m$      

and where $V= \pi a^2 L$ is the volume of the cylinder. Physically, the term $1/V$ is a uniformly distributed heat sink to remove the heat introduced by the point source at $(r',\phi^{\prime},z')$. Since no heat passes through homogeneous boundaries of type 2 (insulated), without this term the differential equation cannot be satisfied. The modified GF actually has exactly the same form as the ordinary GF, except that the kernel function for case Z22 satisfies
\frac{\partial^2 P_{00}}{dz^2}=-\delta(x-x^{\prime}) + \frac{1}{L}
\end{displaymath} (11)

Modified Green's function $G_m$ may be used to find temperature with the Green's function solution equation, Eq. (2), with the following additional constraints: the sum of the heat passing through the boundaries must be equal to the total amount of heat introduced by volume energy generation $g$; and, since the spatial average temperature of the body computed from $G_m$ is zero, the average temperature in the body must be supplied as part of the input data to the problem. Further discussion and numerical examples of steady heat conduction in the cylinder may be found in a paper by Cole (K.D. Cole, "Fast converging series for heat conduction in the circular cylinder," J. of Engineering Mathematics, vol. 49, pp. 217-232, 2004).

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Next: Solid Cylinder, Steady 2D, Up: Solid Cylinder, Steady 3D, Previous: GF for Solid Cylinder,
Frank Pribyl 2005-06-07