A fluid passes through a cylindrical cavity with thick walls. A constant heat flux acts on the outer wall while the fluid is in thermal interaction with the inner wall. $T$ represents the solid temperature and the fluid counterpart is denoted by $T_f$. $T$ is coupled with $T_f$ in the sense that they influence each other (In other words, the fluid cools the solid along its flow length ). While modelling this situation, I have reduced the phenomenon to the following PDE problem:
In the domain , $r\in[a_1,a_2]$ and $z\in[0,L]$ (considering axisymmetry, $\frac{\partial T}{\partial \theta}=0$) so $T=T(r,z)$
$a_1,a_2$ are the inner and outer radii of the cavity respectively while $L$ is the cylinder length. The boundary condition at $r=a_1$ takes into account the coupling between the two mediums. $$ \frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}=0 $$
is subjected to
$$\frac{\partial T}{\partial z} \vert_{z=0} = \frac{\partial T}{\partial z} \vert_{z=L}=0$$
$$\frac{\partial T}{\partial r} \vert_{r=a_2}=\gamma$$
$$ \frac{\partial T}{\partial r} \vert_{r=a_1}=\delta\Bigg[T-\alpha e^{-\alpha z}\bigg(\int_0^z e^{\alpha s} T(r,s) \mathrm{d}s + \frac{A}{\alpha} \bigg)\Bigg] $$
$A,\alpha,\gamma,\delta$ are constants.
In Cartesian coordinates, I could have assumed a solution form involving $\cos(f(z))$ because of the homogeneous Neumann boundaries along the $z$-direction. Can some one suggest such an alternative in the Cylindrical coordinate system. Additionally, how to deal with the integral type boundary conditions at $r=a_1$ ?
Supporting information The boundary condition at $r=a_1$ is of third kind (Robin) $$ \frac{\partial T}{\partial r} \vert_{r=a_1}=\delta(T-T_f) $$ where $T_f$ is evaluated as $$ \frac{\partial T_f}{\partial z} + \alpha (T_f - T(r,z))=0 \Rightarrow T_f=e^{-\alpha z}\int e^{\alpha z} T \mathrm{d}z \\ \Rightarrow T_f=\alpha e^{-\alpha z} \Bigg[\int_0^z e^{\alpha s}T(r,s)\mathrm{d}s+\frac{A}{\alpha}\Bigg] $$ $T_f(z=0)=A$ is known