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The Benney equation is a long-wave approximation for a gravity-driven thin film that expresses its volume flux per unit width locally in terms of height and its derivatives. For surface-velocity scaling on a plane inclined by $\alpha$, one normalization isHere $\epsilon$ is the film aspect ratio, $\mathrm{Re}$ the thickness-based Reynolds number, and $S$ the capillary parameter for a falling film. The leading velocity is parabolic. Its first inertial correction uses the leading mass conservation equation to eliminate the time derivative of the height. This local closure requires the correction to remain small; it does not describe arbitrary large-amplitude short-wave dynamics.

A nanostructure has structural features on the nanoscale. The relevant dimension may be the thickness of a thin film, the diameter of a nanowire, or the overall size of a nanoparticle. A material can therefore be macroscopically large while containing nanostructures internally.

The geometry matters because it changes the relative importance of surfaces, the distances over which charge carriers move, and sometimes the allowed quantum states. A thin film and a nanoparticle made from the same chemical material need not have the same properties: their quantum confinement dimensions and exposed surface areas differ. Characterizing a nanostructure requires specifying which dimensions are small, its shape and its chemical chemical composition, rather than reporting a single size alone.^[ref]