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Next: Radial-spherical coordinates. Transient 1-D. Up: Cylindrical Coordinates. Transient 1-D. Previous: Solid cylinder transient 1-D.

Hollow cylinder, transient 1-D.

R11 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and r = b.
GR11(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}\,\left[ J_{0}(\beta _{m})\,\right] ^{2}R(r)... ...ft[ J_{0}(\beta _{m})\,\right] ^{2}-\left[ J_{0}(\beta _{m}b/a)\,\right] ^{2}}}$  
where R(r) = J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) - Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  

with eigenvalues given by J0($ \beta_{m}^{}$)Y0($ \beta_{m}^{}$b/a) - Y0($ \beta_{m}^{}$)J0($ \beta_{m}^{}$b/a) = 0.


R12 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and $ \partial$G/$ \partial$r = 0 (Neumann)at r = b.

GR12(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}\left[ J_{0}(\beta _{m})\,\right] ^{2}}{\left[ J_{0}(\beta _{m})\,\right] ^{2}-\left[ J_{1}(\beta _{m}b/a)\,\right] ^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)Y1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) - Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)J1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  

with eigenvalues given by J0($ \beta_{m}^{}$)Y1($ \beta_{m}^{}$b/a) - Y0($ \beta_{m}^{}$)J1($ \beta_{m}^{}$b/a) = 0.


R13 Hollow cylinder a < r < b, with G = 0 (Dirichlet) at r = a and k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = b.

GR13(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}\left[ J_{0}(\beta _{m})\,\right] ^{2}}{% (B^{2}+\beta _{m}^{2})\left[ J_{0}(\beta _{m})\,\right] ^{2}-V_{0}{}^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = S0J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$) - V0Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)  
with V0 = -$\displaystyle \beta_{m}^{}$J1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) + BJ0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
S0 = - $\displaystyle \beta_{m}^{}$Y1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) + BY0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
B = ha/k  

Eigenvalues are given by S0J0($ \beta_{m}^{}$) - V0Y0($ \beta_{m}^{}$) = 0 .


R21 Hollow cylinder a < r < b, with $ \partial$G/$ \partial$r = 0 (Dirichlet) at r = a and G = 0 (Neumann) at r = b.

GR21(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}\left[ J_{1}(\beta _{m})\,\right] ^{2}}{\left[ J_{1}(\beta _{m})\,\right] ^{2}-\left[ J_{0}(\beta _{m}b/a)\,\right] ^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) - Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  

with eigenvalues given by J1($ \beta_{m}^{}$)Y0($ \beta_{m}^{}$b/a) - Y1($ \beta_{m}^{}$)J0($ \beta_{m}^{}$b/a) = 0.


R22 Hollow cylinder a < r < b, with $ \partial$G/$ \partial$r = 0 (Dirichlet) at r = a and G = 0 (Neumann) at r = b.

GR22(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{\pi (b^{2}-a^{2})}}$ + $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}\left[ J_{1}(\beta _{m})\,\right] ^{2}}{\left[ J_{1}(\beta _{m})\,\right] ^{2}-\left[ J_{1}(\beta _{m}b/a)\,\right] ^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)Y1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) - Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)J1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  

with eigenvalues given by J1($ \beta_{m}^{}$)Y1($ \beta_{m}^{}$b/a) - Y1($ \beta_{m}^{}$)J1($ \beta_{m}^{}$b/a) = 0.


R23 Hollow cylinder a < r < b, with $ \partial$G/$ \partial$r = 0 (Neumann) at r = a and k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = b.

GR23(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}\left[ J_{1}(\beta _{m})\,\right] ^{2}}{% (B^{2}+\beta _{m}^{2})\left[ J_{1}(\beta _{m})\,\right] ^{2}-V_{0}{}^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = S0J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$) - V0Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)  
with V0 = -$\displaystyle \beta_{m}^{}$J1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) + BJ0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
S0 = - $\displaystyle \beta_{m}^{}$Y1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) + BY0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
B = ha/k  

Eigenvalues are given by S0J1($ \beta_{m}^{}$) - V0Y1($ \beta_{m}^{}$) = 0 .


R31 Hollow cylinder a < r < b, with - k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = a and G = 0 (Dirichlet) at r = b.

GR31(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}U_{0}^{2}}{U_{0}^{2}-(B^{2}+\beta _{m}^{2})% \left[ J_{0}(\beta _{m}b/a)\,\right] ^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) - Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
with   U0 = - $\displaystyle \beta_{m}^{}$J1($\displaystyle \beta_{m}^{}$) + BJ0($\displaystyle \beta_{m}^{}$)  
W0 = - $\displaystyle \beta_{m}^{}$Y1($\displaystyle \beta_{m}^{}$) + BY0($\displaystyle \beta_{m}^{}$)  
B = ha/k  

Eigenvalues are given by U0Y0($ \beta_{m}^{}$b/a) - W0J0($ \beta_{m}^{}$b/a) = 0.


R32 Hollow cylinder a < r < b, with - k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = a and $ \partial$G/$ \partial$r = 0 (Neumann) at r = b.

GR32(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}U_{0}^{2}}{U_{0}^{2}-(B^{2}+\beta _{m}^{2})% \left[ J_{1}(\beta _{m}b/a)\,\right] ^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)Y1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) - Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)J1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
with  U0 = - $\displaystyle \beta_{m}^{}$J1($\displaystyle \beta_{m}^{}$) + BJ0($\displaystyle \beta_{m}^{}$)  
W0 = - $\displaystyle \beta_{m}^{}$Y1($\displaystyle \beta_{m}^{}$) + BY0($\displaystyle \beta_{m}^{}$)  
B = ha/k  

Eigenvalues are given by U0Y1($ \beta_{m}^{}$b/a) - W0J1($ \beta_{m}^{}$b/a) = 0.


R33 Hollow cylinder a < r < b, with - k$ \partial$G/$ \partial$r + h1G = 0 (convection) at r = a and k$ \partial$G/$ \partial$r + h2G = 0 (convection) at r = b.

GR33(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{\pi }{4a^{2}}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/a2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/a^{2}}\right]$  
    x $\displaystyle {\frac{\beta _{m}^{2}U_{0}^{2}}{(B^{2}+\beta _{m}^{2})U_{0}^{2}-(B^{2}+\beta _{m}^{2})V_{0}^{2}}}$ R(r) R(r$\scriptstyle \prime$)  
where R(r) = S0J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$) - V0Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{a}}$)  
with  S0 = - $\displaystyle \beta_{m}^{}$Y1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) + B2Y0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$)  
with  U0 = - $\displaystyle \beta_{m}^{}$J1($\displaystyle \beta_{m}^{}$) - B1J0($\displaystyle \beta_{m}^{}$)  
V0 = - $\displaystyle \beta_{m}^{}$J1($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b}{a}}$) + B2J0($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{b% }{a}}$)  
W0 = - $\displaystyle \beta_{m}^{}$Y1($\displaystyle \beta_{m}^{}$) + B1Y0($\displaystyle \beta_{m}^{}$)  
B1 = $\displaystyle {\frac{h_{1}a}{k}}$; B2 = $\displaystyle {\frac{h_{2}a}{k}}$  

Eigenvalues are given by S0U0 - V0W0 = 0.
next up previous
Next: Radial-spherical coordinates. Transient 1-D. Up: Cylindrical Coordinates. Transient 1-D. Previous: Solid cylinder transient 1-D.
Kevin D. Cole
2002-12-31