next up previous
Next: Solid Sphere, transient 1-D. Up: Radial-spherical coordinates. Transient 1-D. Previous: Infinite body, spherical coordinate,

Infinite body with a spherical void, transient 1-D.

RS10 Infinite body with a spherical void, a < r < $ \infty$, with G = 0 (Dirichlet) at r = a.

GRS10(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{8\pi r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha... ...-\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha ... ...\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right\}$    

RS20 Infinite body with a spherical void, a < r < $ \infty$, with $ \partial$G/$ \partial$r = 0 (Neumann) at r = a. In the formula below, B1 = 1 .
GRS20(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{8\pi r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha... ...-\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha ... ...\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{B_{1}}{4\pi r\,r^{\prime }a}}$exp$\displaystyle \left[\vphantom{ B_{1}\frac{r+r^{\prime }-2a }{a^{{}}}+B_{1}^{2}\frac{\alpha (t-\tau )}{a^{2}}}\right.$B1$\displaystyle {\frac{r+r^{\prime }-2a }{a^{{}}}}$ + B12$\displaystyle {\frac{\alpha (t-\tau )}{a^{2}}}$ $\displaystyle \left.\vphantom{ B_{1}\frac{r+r^{\prime }-2a }{a^{{}}}+B_{1}^{2}\frac{\alpha (t-\tau )}{a^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{r+r^{\prime }-2a}{[4\alpha (t-\tau )]^{1/2}} +\frac{B_{1}\left[ \alpha (t-\tau )\right] ^{1/2}}{a^{{}}}}\right.$$\displaystyle {\frac{r+r^{\prime }-2a}{[4\alpha (t-\tau )]^{1/2}}}$ + $\displaystyle {\frac{B_{1}\left[ \alpha (t-\tau )\right] ^{1/2}}{a^{{}}}}$ $\displaystyle \left.\vphantom{ \frac{r+r^{\prime }-2a}{[4\alpha (t-\tau )]^{1/2}} +\frac{B_{1}\left[ \alpha (t-\tau )\right] ^{1/2}}{a^{{}}}}\right]$  

RS30 Infinite body with a spherical void, a < r < $ \infty$, with - k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = a. In the formula below, B1 = 1 + ha/k.
GRS30(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{8\pi r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha... ...-\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha ... ...\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{B_{1}}{4\pi r\,r^{\prime }a}}$exp$\displaystyle \left[\vphantom{ B_{1}\frac{r+r^{\prime }-2a }{a^{{}}}+B_{1}^{2}\frac{\alpha (t-\tau )}{a^{2}}}\right.$B1$\displaystyle {\frac{r+r^{\prime }-2a }{a^{{}}}}$ + B12$\displaystyle {\frac{\alpha (t-\tau )}{a^{2}}}$ $\displaystyle \left.\vphantom{ B_{1}\frac{r+r^{\prime }-2a }{a^{{}}}+B_{1}^{2}\frac{\alpha (t-\tau )}{a^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{r+r^{\prime }-2a}{[4\alpha (t-\tau )]^{1/2}} +\frac{B_{1}\left[ \alpha (t-\tau )\right] ^{1/2}}{a^{{}}}}\right.$$\displaystyle {\frac{r+r^{\prime }-2a}{[4\alpha (t-\tau )]^{1/2}}}$ + $\displaystyle {\frac{B_{1}\left[ \alpha (t-\tau )\right] ^{1/2}}{a^{{}}}}$ $\displaystyle \left.\vphantom{ \frac{r+r^{\prime }-2a}{[4\alpha (t-\tau )]^{1/2}} +\frac{B_{1}\left[ \alpha (t-\tau )\right] ^{1/2}}{a^{{}}}}\right]$  

See case X30 for approximate values at small $ \alpha$(t - $ \tau$)/a2.
next up previous
Next: Solid Sphere, transient 1-D. Up: Radial-spherical coordinates. Transient 1-D. Previous: Infinite body, spherical coordinate,
Kevin D. Cole
2002-12-31