Axisymmetric Cylindrical Green’s Function for the Steady-State Advection-Diffusion Operator with Uniform Velocity

The advection-diffusion equation has a wide range of applications as it governs energy and mass transport in a moving medium. Analytical solutions of the advection-diffusion equation for classes of boundary conditions amounts to finding exact expressions of the Green’s function of the advection-diffusion operator. In this work, the Green’s function of the steady-state advection-diffusion operator is obtained for an axisymmetric cylindrical problem with uniform velocity along the axis of the cylinder. The solution is exact, as axial diffusion (or conduction) is not neglected. The method of solution utilizes a formalism relating the Green’s function of the diffusion operator to that of the steady-state advection-diffusion operator with uniform velocity. The Green’s function obtained is applied to a Dirichlet boundary value problem. The distribution, (either particle concentration, or temperature), is plotted for various values of Péclet numbers. This work demonstrates how to invert the steady-state advection-diffusion operator with uniform velocity in a non-trivial geometry, namely the cylinder.