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Next: Plate, transient 1-D. Up: Rectangular coordinates. Transient 1-D. Previous: Infinite body, rectangular coordinate

Semi-infinite body, transient 1-D.

X10 Semi-infinite body, 0 < x < $ \infty$, with G = 0 (Dirichlet) at x = 0.

GX10(x, t | x$\scriptstyle \prime$,$\displaystyle \tau$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha... ...ment_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha ... ...ent_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$    

X20 Semi-infinite body, with $ \partial$G/$ \partial$x = 0 (Neumann) at x = 0.

GX20(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha... ...ment_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha ... ...ent_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$    

X30 Semi-infinite body, with - k$ \partial$G/$ \partial$x + h2G = 0 (convection) at x = 0.
GX30(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha... ...ment_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha ... ...ent_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{h}{k}}$exp$\displaystyle \left[\vphantom{ \alpha (t-\tau )\frac{h^{2}}{k^{2}}+(x+x^{\prime })% \frac{h}{k}}\right.$$\displaystyle \alpha$(t - $\displaystyle \tau$)$\displaystyle {\frac{h^{2}}{k^{2}}}$ + (x + x$\scriptstyle \prime$)$\displaystyle {\frac{h}{k}}$ $\displaystyle \left.\vphantom{ \alpha (t-\tau )\frac{h^{2}}{k^{2}}+(x+x^{\prime })% \frac{h}{k}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+% \frac{h}{k}[\alpha (t-\tau )]^{1/2}}\right.$$\displaystyle {\frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}}$ + $\displaystyle {\frac{h}{k}}$[$\displaystyle \alpha$(t - $\displaystyle \tau$)]1/2$\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+% \frac{h}{k}[\alpha (t-\tau )]^{1/2}}\right]$  

X40 Semi-infinite body, with - k$ \partial$G/$ \partial$x + ($ \rho$cb)1$ \partial$G/$ \partial$t = 0 (thin film) at x = 0.
GX40(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha... ...ment_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha ... ...ent_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    + $\displaystyle {\frac{1}{bP}}$exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}% }\right]$  
    x rerf$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+% \frac{1}{P}\frac{[\alpha (t-\tau )]^{1/2}}{b}}\right.$$\displaystyle {\frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}}$ + $\displaystyle {\frac{1}{P}}$$\displaystyle {\frac{[\alpha (t-\tau )]^{1/2}}{b}}$ $\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+% \frac{1}{P}\frac{[\alpha (t-\tau )]^{1/2}}{b}}\right]$  
where P = $\displaystyle {\frac{\left( \rho c\right) _{1}}{\rho c}}$ and rerf (z) = exp(z2)erfc(z)  

X50 Semi-infinite body, with - k$ \partial$G/$ \partial$x + hG + ($ \rho$cb)1$ \partial$G/$ \partial$t = 0 (thin film) at x = 0.
GX50(x, t$\displaystyle \left\vert\vphantom{ x^{\prime },\tau }\right.$x$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ x^{\prime },\tau }\right.$) = [4$\displaystyle \pi$$\displaystyle \alpha$(t - $\displaystyle \tau$)]-1/2  
    x $\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha... ...ment_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x-x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(x-x^{\prime })^{2}}{4\alpha ... ...ent_mark>\left[ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    + $\displaystyle {\frac{1}{2bAP}}$exp$\displaystyle \left[\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(x+x^{\prime })^{2}}{4\alpha (t-\tau )}% }\right]$  
    x $\displaystyle \left(\vphantom{ (1+A)rerf\left[ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1+A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}\right] }\right.$(1 + A)rerf$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1+A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}}\right.$$\displaystyle {\frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}}$ + (1 + A)$\displaystyle {\frac{[\alpha (t-\tau )]^{1/2}}{2bP}}$ $\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1+A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}}\right]$ $\displaystyle \left.\vphantom{ (1+A)rerf\left[ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1+A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}\right] }\right.$  
    $\displaystyle \left.\vphantom{ -(1-A)rerf\left[ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1-A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}\right] }\right.$ - (1 - A)rerf$\displaystyle \left[\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1-A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}}\right.$$\displaystyle {\frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}}$ + (1 - A)$\displaystyle {\frac{[\alpha (t-\tau )]^{1/2}}{2bP}}$ $\displaystyle \left.\vphantom{ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1-A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}}\right]$ $\displaystyle \left.\vphantom{ -(1-A)rerf\left[ \frac{x+x^{\prime }}{[4\alpha (t-\tau )]^{1/2}}+(1-A)\frac{[\alpha (t-\tau )]^{1/2}}{2bP}\right] }\right\}$  
where P = $\displaystyle {\frac{\left( \rho c\right) _{1}}{\rho c}}$ , B = hb/k, A = (1 - 4BP)1/2 for 4BP < 1  
  rerf (z) = exp(z2)erfc(z)  

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Next: Plate, transient 1-D. Up: Rectangular coordinates. Transient 1-D. Previous: Infinite body, rectangular coordinate
Kevin D. Cole
2002-12-31