next up previous
Next: Solid cylinder transient 1-D. Up: Cylindrical Coordinates. Transient 1-D. Previous: Infinite body, cylindrical coordinate,

Infinite body with circular hole, transient 1-D.

R10 Infinite region with circular hole, a < r < $ \infty$, with G = 0 (Dirichlet) at r = a.
GR10(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}{2\pi a^{2}}}$$\displaystyle \int_{0}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta ^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta^{2}_{}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta ^{2}\alpha (t-\tau )/a^{2}}\right]$$\displaystyle {\frac{% \beta \,R(r)\,R(r^{\prime })}{\left[ J_{0}(\beta )\,\right] ^{2}+\left[ Y_{0}(\beta )\,\right] ^{2}}}$d$\displaystyle \beta$  
where R(r) = J0($\displaystyle \beta$r/a)Y0($\displaystyle \beta$) - Y0($\displaystyle \beta$r/a)J0($\displaystyle \beta$)  


R20 Infinite region with circular hole, a < r < $ \infty$, with $ \partial$G$ \partial$r = 0 (Neumann) at r = a.

GR10(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}{2\pi a^{2}}}$$\displaystyle \int_{0}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta ^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta^{2}_{}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta ^{2}\alpha (t-\tau )/a^{2}}\right]$$\displaystyle {\frac{% \beta \,R(r)\,R(r^{\prime })}{\left[ J_{1}(\beta )\,\right] ^{2}+\left[ Y_{1}(\beta )\,\right] ^{2}}}$d$\displaystyle \beta$  
where R(r) = J0($\displaystyle \beta$r/a)Y1($\displaystyle \beta$) - Y0($\displaystyle \beta$r/a)J1($\displaystyle \beta$)  

Note that J0(z)Y1(z) - Y0(z)J1(z) = - 2/($ \pi$z).


R30 Infinite region with circular hole, a < r < $ \infty$, with - k$ \partial$G$ \partial$r + hG = 0 (convection) at r = a.

GR10(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}{2\pi a^{2}}}$$\displaystyle \int_{0}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta ^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta^{2}_{}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta ^{2}\alpha (t-\tau )/a^{2}}\right]$$\displaystyle {\frac{% \beta \,R(r)\,R(r^{\prime })}{U_{0}^{2}+W_{0}^{2}}}$d$\displaystyle \beta$  
where R(r) = W0J0($\displaystyle \beta$r/a) - U0Y0($\displaystyle \beta$r/a)  
W0 = - $\displaystyle \beta$Y1($\displaystyle \beta$) - BY0($\displaystyle \beta$)  
U0 = - $\displaystyle \beta$J1($\displaystyle \beta$) - BJ0($\displaystyle \beta$)  
and B = $\displaystyle {\frac{ha}{k}}$  


next up previous
Next: Solid cylinder transient 1-D. Up: Cylindrical Coordinates. Transient 1-D. Previous: Infinite body, cylindrical coordinate,
Kevin D. Cole
2002-12-31