= Benney equation
{c}
{title2=$h_t+\partial_xq[h]=0$}
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 is
$$
q[h]=\frac23h^3+\varepsilon\left[-\frac23h^3h_x\cot\alpha+S h^3h_{xxx}+\frac8{15}\mathrm{Re}\,h^6h_x\right].
$$
Here $\varepsilon$ 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.
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