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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.

The two wall boundary conditions areDefine the capillary parameter for a falling filmRetain $S=O(1)$ in the long-wave approximation. Thus surface tension is retained in the leading normal stress balance even though geometric slope corrections are small. For $h_x=O(1)$ in the rescaled coordinates, the dimensional surface slope is $\epsilon h_x$.

The three free surface boundary conditions, accurate through $O(\epsilon )$, areThe first is the kinematic boundary condition; the second is the zero tangential traction condition. For the third, the dimensional liquid-side pressure is, to the required order,Division by $\rho g{h}_0\sin \alpha$ gives $p=-3S{h}_{xx}/2+\epsilon w_z$ in dimensionless variables. Subtracting $\bar{p}(h)=(1-h)\cot \alpha$ gives the stated perturbation condition. The additional slope-times-shear contribution to normal stress vanishes at this order because the tangential traction is zero. The omitted geometric and tangential-stress corrections are $O({\epsilon }^2)$ under this ordering.