Rectangular Coordinates, Strip and Semi-strip, steady.

by K. D. Cole and D. H. Y. Yen


The GF for the 2D strip satisfies:

$${\frac{\partial ^{2}G}{\partial x^{2}}} {\frac{\partial ^{2}G}{\partial y^{2}}} \delta \prime \delta \prime$$ (18)

 

Figure: Geometry of (a) infinite and (b) semi-infinite strip.

For the infinite strip, shown in Fig. 1a, the domain is (- $\infty$ < x < $\infty$,  0 < y < W) and the boundary conditions are

$${\frac{\partial G}{\partial n_{i}}}$$ at y = 0 and y = W

$${\frac{\partial G}{\partial x}} \rightarrow \pm \infty$$ (19)

For the semi-infinite strip, shown in Fig. 1b, the domain is (0 < x < $\infty$,  0 < y < W)and the homogeneous boundary conditions are

$${\frac{\partial G}{\partial n_{i}}}$$ at x = 0,  y = 0, and y = W

$${\frac{\partial G}{\partial x}} \rightarrow \infty$$ (20)

The GF of infinite and semi-infinite strip are described by number XI0YKL which represents 36 different GF for I = 0, 1, 2, or 3 and K, L = 1, 2, or 3. Note that the GF for I = 0 are for the infinite strip geometry only.

The GF for the strip has a single-summation form with eigenfunction Y_n, eigenvalue $\gamma_{n}^{}$, norm N_y, and kernel function P_n, as follows:

$$\left\vert\vphantom{ \,x^{\prime },y^{\prime }}\right. \prime \prime \left.\vphantom{ \,x^{\prime },y^{\prime }}\right. {\frac{1}{W}} \prime \sum_{n=1}^{\infty } {\frac{Y_{n}(y^{\prime })Y_{n}(y)}{N_{y}(\gamma_{n})}} \prime$$ (21)

The first term with kernel function P₀ is needed only when Y22 is part of the GF number (i.e., when zero is an eigenvalue). There are nine different eigenfunctions associated with the nine possible boundary condition combinations YKL (K, L = 1, 2, or 3). Table 1 contains the eigenfunctions and norms, and Table 2 contains the associated eigenconditions (and eigenvalues for simple cases).
Kernel functions P_n for n $\neq$ 0 are given by

$$\prime {\frac{1}{2\gamma _{n}S^{+}}} \left[\vphantom{ S^{+}\exp(-\gamma _{n}\left\vert x-x^{\prime }\right\vert )+S^{-}\exp (-\gamma_{n}\left\vert x+x^{\prime }\right\vert )}\right. \gamma_{n}^{} \left\vert\vphantom{ x-x^{\prime} }\right. \prime \left.\vphantom{ x-x^{\prime} }\right\vert \gamma_{n}^{} \left\vert\vphantom{ x+x^{\prime }}\right. \prime \left.\vphantom{ x+x^{\prime }}\right\vert \left.\vphantom{ S^{+}\exp(-\gamma _{n}\left\vert x-x^{\prime }\right\vert )+S^{-}\exp (-\gamma_{n}\left\vert x+x^{\prime }\right\vert )}\right]$$ (22)

where the values for S⁺ and S^- are given in Table 3. Kernel functions P_n for n $\neq$ 0 have four different forms, one for each of the boundary condition combinations XI0 for I = 0, 1, 2, and 3.

Kernel functions P₀ satisfy

$${\frac{d^{2}P_{0}}{dx^{2}}} \delta \prime$$ (23)

as well as appropriate homogeneous boundary conditions. Functions P₀, listed in Table 4, must be included in the GF whenever zero is an eigenvalue (XI0Y22 for I = 0, 1, 2, and 3).

Table 1. Eigenfunctions and inverse norm ^{a, b}

Geometry	Yn(y)	Ny-1
Y11	sin($\gamma_{n}^{}$y)	2/W
Y12	sin($\gamma_{n}^{}$y)	2/W
Y13	sin($\gamma_{n}^{}$y)	2$\phi_{2n}^{}$/W
Y21	cos($\gamma_{n}^{}$y)	2/W
Y22	cos($\gamma_{n}^{}$y);$\gamma_{n}^{}$ $\neq$ 0	2/W for $\gamma_{n}^{}$ $\neq$ 0
	1;$\gamma_{n}^{}$ = 0	1/W for $\gamma_{n}^{}$ = 0
Y23	cos($\gamma_{n}^{}$y)	2$\phi_{2n}^{}$/W
Y31	$\gamma_{n}^{}$Wcos($\gamma_{n}^{}$y) + (h1W/k)sin($\gamma_{n}^{}$y)	2$\phi_{1n}^{}$/W
Y32	$\gamma_{n}^{}$Wcos($\gamma_{n}^{}$y) + (h1W/k)sin($\gamma_{n}^{}$y)	2$\phi_{1n}^{}$/W
Y33	$\gamma_{n}^{}$Wcos($\gamma_{n}^{}$y) + (h1W/k)sin($\gamma_{n}^{}$y)	2$\Phi_{n}^{}$/W

^a Index n = 1, 2,... for all cases except Y22 with n = 0, 1, 2,...
^b$\phi_{in}^{}$ = [($\gamma_{n}^{}$W)² + (h_iW/k)²] ÷ [($\gamma_{n}^{}$W)² + (h_iW/k)² + h_iW/k]
$\Phi_{n}^{}$ = $\phi_{2n}^{}$ ÷ [($\gamma_{n}^{}$W)² + (h_iW/k)² + (h₁W/k)$\phi_{2n}^{}$]


Table 2. Eigencondition and eigenvalues for Y_n(y)^a

Geometry	Eigencondition	Eigenvalues
Y11	sin($\gamma_{n}^{}$W) = 0	n$\pi$/W
Y12	cos($\gamma_{n}^{}$W) = 0	(2n - 1)$\pi$/2W
Y13	$\gamma_{n}^{}$Wcot($\gamma_{n}^{}$W) = - h2W/k
Y21	cos($\gamma_{n}^{}$W) = 0	(2n - 1)$\pi$/2W
Y22	sin($\gamma_{n}^{}$W) = 0	n$\pi$/W, n = 0, 1, 2,...
Y23	$\gamma_{n}^{}$Wtan($\gamma_{n}^{}$W) = h2W/k
Y31	$\gamma_{n}^{}$Wcot($\gamma_{n}^{}$W) = - h1W/k
Y32	$\gamma_{n}^{}$Wtan($\gamma_{n}^{}$W) = hW/k
Y33	tan($\gamma_{n}^{}$W) = [$\gamma_{n}^{}$(h1 + h2)/k]/[$\gamma_{n}^{2}$ - h1h2k-2]

^a Index n = 1, 2,... for all cases except Y22 with n = 0, 1, 2,...

Table 3. Coefficients for P_n for n $\neq$ 0, strip and semi-strip.

Geometry	S+	S-
X00	1	0
X10	1	-1
X20	1	1
X30	$\gamma_{n}^{}$W + hW/k	$\gamma_{n}^{}$W - hW/k

Table 4. Kernel function P₀ for strip and semi-strip.

Geometry	P0(x, x$\prime$)
X00	- ${\frac{1}{2}}$$\left\vert\vphantom{ x-x^{\prime} }\right.$x - x$\prime$$\left.\vphantom{ x-x^{\prime} }\right\vert$
X10	- ${\frac{1}{2}}$$\left\vert\vphantom{ x-x^{\prime} }\right.$x - x$\prime$$\left.\vphantom{ x-x^{\prime} }\right\vert$ + ${\frac{1}{2}}$$\left\vert\vphantom{ x+x^{\prime} }\right.$x + x$\prime$$\left.\vphantom{ x+x^{\prime} }\right\vert$
X20	- ${\frac{1}{2}}$$\left\vert\vphantom{ x-x^{\prime} }\right.$x - x$\prime$$\left.\vphantom{ x-x^{\prime} }\right\vert$ - ${\frac{1}{2}}$$\left\vert\vphantom{ x+x^{\prime} }\right.$x + x$\prime$$\left.\vphantom{ x+x^{\prime} }\right\vert$
X30	- ${\frac{1}{2}}$$\left\vert\vphantom{ x-x^{\prime} }\right.$x - x$\prime$$\left.\vphantom{ x-x^{\prime} }\right\vert$ + ${\frac{1}{2}}$$\left\vert\vphantom{ x+x^{\prime} }\right.$x + x$\prime$$\left.\vphantom{ x+x^{\prime} }\right\vert$ + k/h