I have the following two PDEs, which describe steady-state coupled heat transport between a externally heated axi-symmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
$$ \frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}=0 \tag1 $$
$$ \frac{\partial t}{\partial z}+\alpha(t-T)=0 \tag2 $$ Eq. (1) is defined in the domain $r\in[r_1,r_2]$ where $r_1$ and $r_2$ describe the inner and outer radii of the cylinder and $z\in[0,L]$ where $L$ is the length of the cylinder. The boundary conditions for Eq. (1) are: $$ \frac{\partial T(r,0)}{\partial z}=\frac{\partial T(r,L)}{\partial z}=0 \tag3 $$ $$ \frac{\partial T(r_2,z)}{\partial r}=\gamma \tag4 $$ $$ \frac{\partial T(r_1,z)}{\partial r}=\beta(T(r_1,z)-t) \tag5 $$ For Eq. (2) it is known that $t(z=0)=t_{in}$
$\alpha, \beta, \gamma, t_{in}$ are known constants. It seems the solid and fluid temperatures are coupled through the B.C. at $r=r_1$ (solid-fluid interface, Robin condition).
Any suggestion on how to approach this problem analytically is appreciated.