Hollow Cylinder, steady 1-D.

R11 Hollow cylinder, a $\leq$ r $\leq$ b, with G = 0 (Dirichlet) at r=a and at r=b.

2$$\pi$$G_R11(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \ln (b/r^{\prime })\ln (r/... ...\ln (r^{\prime }/a)/\ln (b/a) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \ln (b/r^{\prime })\ln (r/a)/\ln (b/a) & \text... ... \ln (b/r)\ln (r^{\prime }/a)/\ln (b/a) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \ln (b/r^{\prime })\ln (r/a... ...\ln (r^{\prime }/a)/\ln (b/a) & \text{for }r>r^{\prime } \end{array} }\right.$$

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

2$$\pi$$G_R12(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \ln (r/a) & \text{for }r<r... ...me } \\ \ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \ln (r/a) & \text{for }r<r^{\prime } \\ \ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \ln (r/a) & \text{for }r<r^... ...me } \\ \ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array} }\right.$$

R13 Hollow cylinder, 0 $\leq$ r $\leq$ b, with G = 0 (Dirichlet) at r = 0 and k$\partial$G/$\partial$r + h₂G = 0 (convection) at r = b. Note B₂ = h₂b/k.

2$$\pi$$G_R13(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \ln (r/a)\left[ 1+B_{2}\ln... ...n (b/r)]/[1+B_{2}\ln (b/a)] & \text{for } r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \ln (r/a)\left[ 1+B_{2}\ln (b/r^{\prime })\rig... ...[1+B_{2}\ln (b/r)]/[1+B_{2}\ln (b/a)] & \text{for } r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \ln (r/a)\left[ 1+B_{2}\ln ... ...n (b/r)]/[1+B_{2}\ln (b/a)] & \text{for } r>r^{\prime } \end{array} }\right.$$

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

2$$\pi$$G_R21(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} ln(b/r^{\prime }) & \text{for }r<r^{\prime } \\ ln(b/r) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} ln(b/r^{\prime }) & \text{for }r<r^{\prime } \\ ln(b/r) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} ln(b/r^{\prime }) & \text{for }r<r^{\prime } \\ ln(b/r) & \text{for }r>r^{\prime } \end{array} }\right.$$

R22 Hollow cylinder, a $\leq$ r $\leq$ b, with $\partial$G/$\partial$r = 0 (Neumann) at both boundaries. Note that this geometry requires a pseudo GF, denoted H. The temperature solution found from a pseudo GF requires that the total volumetric heat flow is equal to the boundary heat flow, and the spatial average temperature in the body must be supplied as a known condition.

2$$\pi$$H_R22(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{c} \lbrack (r^{2}+(r^{\prime }... ...rime }}{a})]/(b^{2}-a^{2}) \\ \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{c} \lbrack (r^{2}+(r^{\prime })^{2})/2-b^{2}\ln (\... ...ac{% r^{\prime }}{a})]/(b^{2}-a^{2}) \\ \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{c} \lbrack (r^{2}+(r^{\prime })... ...rime }}{a})]/(b^{2}-a^{2}) \\ \text{for }r>r^{\prime } \end{array} }\right.$$

R23 Hollow cylinder, a $\leq$ r $\leq$ b, with $\partial$G/$\partial$r = 0 (Neumann) at r = a and k$\partial$G/$\partial$r + h₂G = 0 (convection) at r = b. Note B₂ = h₂b/k.

2$$\pi$$G_R23(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} 1/B_{2}+\ln (b/r^{\prime }... ...rime } \\ 1/B_{2}+\ln (b/r) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} 1/B_{2}+\ln (b/r^{\prime }) & \text{for }r<r^{\prime } \\ 1/B_{2}+\ln (b/r) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} 1/B_{2}+\ln (b/r^{\prime })... ...rime } \\ 1/B_{2}+\ln (b/r) & \text{for }r>r^{\prime } \end{array} }\right.$$

R31 Hollow cylinder, a $\leq$ r $\leq$ b, with - k$\partial$G/$\partial$r + h₁G = 0 (convection) at r = a and G = 0 (Dirichlet) at r = b. Note B₁ = h₁a/k

2$$\pi$$G_R31(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \ln (b/r^{\prime })[1+B_{1... ...me }/a)]/[1+B_{1}\ln (b/a)] & \text{for }% r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \ln (b/r^{\prime })[1+B_{1}\ln (r/a)]/[1+B_{1}... ...n (r^{\prime }/a)]/[1+B_{1}\ln (b/a)] & \text{for }% r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \ln (b/r^{\prime })[1+B_{1}... ...me }/a)]/[1+B_{1}\ln (b/a)] & \text{for }% r>r^{\prime } \end{array} }\right.$$

R32 Hollow cylinder, a $\leq$ r $\leq$ b, with - k$\partial$G/$\partial$r + h₂G = 0 (convection) at r = a and $\partial$G/$\partial$r = 0 (Neumann) at r = b. Note B₁ = h₁a/k.

2$$\pi$$G_R32(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} 1/B_{1}+\ln (r/a) & \text{... ... 1/B_{1}+\ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} 1/B_{1}+\ln (r/a) & \text{for }r<r^{\prime } \\ 1/B_{1}+\ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} 1/B_{1}+\ln (r/a) & \text{f... ... 1/B_{1}+\ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array} }\right.$$

R33 Hollow cylinder, a $\leq$ r $\leq$ 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. Note B₁ = h₁a/k and B₂ = h₂b/k.

2$$\pi$$G_R33(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{c} \lbrack B_{1}B_{2}ln(b/r^{\... ..._{1}+B_{2}+B_{1}B_{2}ln(b/a)];\;\text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{c} \lbrack B_{1}B_{2}ln(b/r^{\prime })ln(r/a)+B_{1... ... \lbrack B_{1}+B_{2}+B_{1}B_{2}ln(b/a)];\;\text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{c} \lbrack B_{1}B_{2}ln(b/r^{\p... ..._{1}+B_{2}+B_{1}B_{2}ln(b/a)];\;\text{for }r>r^{\prime } \end{array} }\right.$$