Fluid flows through nano-scale channels depend sensitively on the physical and chemical properties of the walls that surround them. The sub-micron dimensions of such channels, however, are impossible to resolve optically, which rules out most methods for flow visualization. Classic calculations by Squire (Q. J. Mech. Appl. Maths, vol. IV, 1951, pp. 321–329) and Landau & Lifshitz (Fluid Mechanics, vol. 6, 1959, Pergamon) showed that the laminar flow driven outside a capillary, by fluid emerging from the end of the capillary, is identical to the flow driven by a point force proportional to the average velocity in the capillary. Secchi et al. (J. Fluid Mech. 826, R3) analyze the dispersion of a solute that is injected along with the fluid, whose concentration decays slowly with distance but with a strong angular dependence that encodes the intra-capillary velocity. Fluorescence micrographs of the concentration profile emerging from the nanocapillary can be related directly to the average fluid velocity within the nanocapillary. Beyond their remarkable capacity for nano-velocimetry, Landau–Squire plumes will likely appear throughout micro- and nano-fluidic systems.