(a) Taylor dispersion of a solute (blue) in steady, laminar pipe flow (Poiseuille flow, with velocity profile indicated by the arrows). The solute is spread by the velocity shear, while diffusion in the radial direction (red arrows) establishes a uniform radial distribution. The velocity shear continues to spread out the solute further downstream. (b) For comparison, the effect of diffusion alone in a (non-physical) uniform flow (without shear). (c) Taylor dispersion in a purely oscillatory flow: the velocity shear spreads out the solute, but only to an extent limited by the fluid particle paths, and there is no downstream transport. (Online version in colour.)