Diffusion-Reaction in a Cylindrical Pore

Simulate diffusion into a cylindrical dead-end pore with an infinitely fast reaction on the wall. The pore has a radius $R = 1~\mathrm{\mu m}$ and a length $L = 5~\mathrm{\mu m}$. The diffusion coefficient in both radial and axial directions is $D = 10^{-5}~\mathrm{m^{2}~s^{-1}}$.

Questions:

1. Provide the governing PDE for the two-dimensional diffusion in the pore.

2. What are the proper boundary conditions?

3. Perform the spatial discretization of this (axially symmetric) 2D problem to obtain a matrix-vector equation.

4. Provide a Python implementation.

Note that the implementation should be consistent with the governing equations and boundary conditions.