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## GF, Double-Sum Form

The GF stated below contains a double summation with eigenfunctions norms and , eigenvalues and kernel function , as follows:

 (4)

where are Bessel functions of order . The norms associated with the angular direction are given by and . Norm is given in Table 1 and the conditions defining are given in Table 2. The kernel functions are discussed below.
Table 1. Inverse norm for solid cylinder cases.

 Boundary at Case for for R01 same as R02 R03 same as

Table 2. Eigenconditions which define .
 Case eigencondition R01 R02 R03

Kernel functions

The kernel function must satisfy

 (5)

where is the eigenvalue associated with eigenfunction . The kernel function, suppressing the subscript, may be written as
 (6)

where the subscripts and indicate sides and sides of the cylinder, respectively. Parameters and depend on the boundary conditions on side and are given by
 (7) (8)

Here is the Biot Number for side , and is the conductivity of the cylinder.

The expression for in Eq. (7) is symmetric if and are interchanged and covers several combinations of boundary conditions provided . The special case of is discussed below.

Kernel function with

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

 (9)

as well as the boundary conditions at and . Since there are nine combinations of boundary conditions (type 1, 2, or 3 at and ), there are nine kernel functions which are given in Table 3. Special case Z22 is discussed below.

Table 3. Kernel Function for .
 Case for . Z11 Z12 Z13 Z21 Z22 (( Z23 Z31 Z32 Z33

Special temperature solution needed with this pseudo GF.

Modified GF for Case R02Z22

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:

 (10)

and where is the volume of the cylinder. Physically, the term is a uniformly distributed heat sink to remove the heat introduced by the point source at . 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
 (11)

Modified Green's function 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 ; and, since the spatial average temperature of the body computed from 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).

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