Here we discus the diffusion of particles in a curved tube. This kind of transport phenomenon is observed in biological cells and porous media. To solve such a problem, we discuss the three dimensional diffusion equation with a confining wall forming a thinner tube. We find that the curvature appears in a effective diffusion coefficient for such a quasi-one-dimensional system. As an application to higher dimensional case, we discuss the diffusion in a curved surface with thickness. In this case the diffusion coefficient changes to the tensor form depending on the mean and Gaussian curvatures. Then the diffusion flow can be interpreted as usual flow plus anomalous flow. The anomalous flow shows not only the diffusion but also the concentration depending on mean and Gaussian curvatures, and also it includes the flow proportional to the gradient of Gaussian curvature.
"Diffusion Under Geometrical Constraint." J. Geom. Symmetry Phys. 34 35 - 49, 2014. https://doi.org/10.7546/jgsp-34-2014-35-49