Infinite body with a circular hole, steady 1-D.

R10 Infinite body surrounding a circular hole, a $\leq$ r < $\infty$, with G = 0 (Dirichlet) at r = a.

2$$\pi$$G_R10(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \ln (r/a) & \text{for }r<r... ...me } \\ \ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \ln (r/a) & \text{for }r<r^{\prime } \\ \ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \ln (r/a) & \text{for }r<r^... ...me } \\ \ln (r^{\prime }/a) & \text{for }r>r^{\prime } \end{array} }\right.$$

R20 Infinite body surrounding a circular hole, a $\leq$ r < $\infty$, with $\partial$G/$\partial$r = 0 (Neumann) at r = a. 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_R20(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} -\ln (r^{\prime }/a) & \te... ...r<r^{\prime } \\ -\ln (r/a) & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} -\ln (r^{\prime }/a) & \text{for }r<r^{\prime } \\ -\ln (r/a) & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} -\ln (r^{\prime }/a) & \tex... ...r<r^{\prime } \\ -\ln (r/a) & \text{for }r>r^{\prime } \end{array} }\right.$$

R30 Infinite body surrounding a circular hole, a $\leq$ r < $\infty$, with - k$\partial$G/$\partial$r + h₁G = 0 (convection) at r = a. Note: B₁ = h₁a/k.

2$$\pi$$G_R30(r $$\left\vert\vphantom{ \,r^{\prime }}\right.$$ r^$\prime$$$\left.\vphantom{ \,r^{\prime }}\right.$$) = $$\left\{\vphantom{ \begin{array}{cc} \ln (r/a)+1/B_{1} & \text{... ... \ln (r^{\prime }/a)+1/B_{1} & \text{for }r>r^{\prime } \end{array} }\right.$$$$\begin{array}{cc} \ln (r/a)+1/B_{1} & \text{for }r<r^{\prime } \\ \ln (r^{\prime }/a)+1/B_{1} & \text{for }r>r^{\prime } \end{array}$$ $$\left.\vphantom{ \begin{array}{cc} \ln (r/a)+1/B_{1} & \text{f... ... \ln (r^{\prime }/a)+1/B_{1} & \text{for }r>r^{\prime } \end{array} }\right.$$