Diffusion in a fixed direction

For a fixed unit vector $v$ and $\nu>0$ , diffusion in direction $v$ is the operator

\nu D_v^2f=\nu(v\cdot\nabla)^2f =\nu\sum_{i,j}v_iv_j\partial_i\partial_jf.

The direction is constant in space; a varying vector field would also contribute derivatives of its coefficients.

Axial diffusion

For the axial direction $v=e_z$ , the term is $\nu\partial_z^2f$ . Under $z=L_zZ$ , it becomes $\nu L_z^{-2}\partial_Z^2f$ . Comparing it with radial diffusion therefore requires the two length scales and the radial operator, including any coordinate-basis terms.