Hollow Sphere, steady 1-D.

RS11 Hollow sphere, a $\leq$ r $\leq$ b, with G = 0 (Dirichlet) at r = a and at r = b.

4$$\pi$$G_RS11(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} (b-r^{\prime })(1-a/r)/[r^... ...-r)(1-a/r^{\prime })/[r(b-a)] & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} (b-r^{\prime })(1-a/r)/[r^{\prime }(b-a)] & \t... ... & \\ (b-r)(1-a/r^{\prime })/[r(b-a)] & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} (b-r^{\prime })(1-a/r)/[r^{... ...-r)(1-a/r^{\prime })/[r(b-a)] & \text{for }r>r^{\prime } \end{array} }\right.$$

RS12 Hollow sphere, a $\leq$ r $\leq$ b, with G = 0 (Dirichlet) at r = a and $\partial$G/$\partial$r = 0 (Neumann) at r = b.

4$$\pi$$G_RS12(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} 1/a-1/r & \text{for }r<r^{\prime } \\ 1/a-1/r^{\prime } & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} 1/a-1/r & \text{for }r<r^{\prime } \\ 1/a-1/r^{\prime } & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} 1/a-1/r & \text{for }r<r^{\prime } \\ 1/a-1/r^{\prime } & \text{for }r>r^{\prime } \end{array} }\right.$$

RS13 Hollow sphere, a $\leq$ 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.

4$$\pi$$G_RS13(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{c} \lbrack B_{2}(\frac{a}{r}-\... ...] \\ \div (B_{2}a-B_{2}b-a);\;\text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{c} \lbrack B_{2}(\frac{a}{r}-\frac{ba}{rr^{\prime ... ...prime }}+1] \\ \div (B_{2}a-B_{2}b-a);\;\text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{c} \lbrack B_{2}(\frac{a}{r}-\f... ...] \\ \div (B_{2}a-B_{2}b-a);\;\text{for }r>r^{\prime } \end{array} }\right.$$

RS21 Hollow sphere, a $\leq$ r $\leq$ b, with $\partial$G/$\partial$r = 0 (Neumann) at r = a and G = 0 (Dirichlet) at r = b.

4$$\pi$$G_RS21(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} 1/r^{\prime }-1/b & \text{for }r<r^{\prime } \\ 1/r-1/b & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} 1/r^{\prime }-1/b & \text{for }r<r^{\prime } \\ 1/r-1/b & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} 1/r^{\prime }-1/b & \text{for }r<r^{\prime } \\ 1/r-1/b & \text{for }r>r^{\prime } \end{array} }\right.$$

RS22 Hollow sphere, 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.

4$$\pi$$H_RS22(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \lbrack r^{2}/2+\left( r^{... ...ime }+b^{3}/r]/(b^{3}-a^{3}) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \lbrack r^{2}/2+\left( r^{\prime }\right) ^{2}... ...{3}/r^{\prime }+b^{3}/r]/(b^{3}-a^{3}) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \lbrack r^{2}/2+\left( r^{\... ...ime }+b^{3}/r]/(b^{3}-a^{3}) & \text{for }r>r^{\prime } \end{array} }\right.$$

RS23 Hollow sphere, 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.

4$$\pi$$G_RS23(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} 1/r^{\prime }+(1/B_{2}-1)/... ...rime } \\ 1/r+(1/B_{2}-1)/b & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} 1/r^{\prime }+(1/B_{2}-1)/b & \text{for }r<r^{\prime } \\ 1/r+(1/B_{2}-1)/b & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} 1/r^{\prime }+(1/B_{2}-1)/b... ...rime } \\ 1/r+(1/B_{2}-1)/b & \text{for }r>r^{\prime } \end{array} }\right.$$

RS31 Hollow sphere, a $\leq$ r $\leq$ b, with - k$\partial$G/$\partial$r + h₂G = 0 (convection) at r = a and G = 0 (Dirichlet) at r = b. Note B₁ = h₁a/k.

4$$\pi$$G_RS31(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{c} \lbrack B_{1}\frac{ba}{rr^{... ...] \\ \div (B_{1}b-B_{1}a+b);\;\text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{c} \lbrack B_{1}\frac{ba}{rr^{\prime }}-B_{1}\frac... ...r}+B_{1}+1] \\ \div (B_{1}b-B_{1}a+b);\;\text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{c} \lbrack B_{1}\frac{ba}{rr^{\... ...] \\ \div (B_{1}b-B_{1}a+b);\;\text{for }r>r^{\prime } \end{array} }\right.$$

RS32 Hollow sphere, 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.

4$$\pi$$G_RS32(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} -1/r-(1/B_{1}+1)/a & \text... ... -1/r^{\prime }-(1/B_{1}+1)/a & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} -1/r-(1/B_{1}+1)/a & \text{for }r<r^{\prime } \\ -1/r^{\prime }-(1/B_{1}+1)/a & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} -1/r-(1/B_{1}+1)/a & \text{... ... -1/r^{\prime }-(1/B_{1}+1)/a & \text{for }r>r^{\prime } \end{array} }\right.$$

RS33 Hollow sphere, 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.

4$$\pi$$G_RS33(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{c} \lbrack B_{1}(\frac{a}{r}+1... ..._{2}b+B_{1}a+B_{1}B_{2}(b-a)];\;\text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{c} \lbrack B_{1}(\frac{a}{r}+1)+B_{1}B_{2}(1-\frac... ... \lbrack B_{2}b+B_{1}a+B_{1}B_{2}(b-a)];\;\text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{c} \lbrack B_{1}(\frac{a}{r}+1)... ..._{2}b+B_{1}a+B_{1}B_{2}(b-a)];\;\text{for }r>r^{\prime } \end{array} }\right.$$