The 1-D Saint-Venant momentum equation can be derived from the Navier–Stokes equations that describe fluid motion. The x -component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the x -direction – can be written as: ∂ ∂ u ∂ ∂ t + u ∂ ∂ u ∂ ∂ x + v ∂ ∂ u ∂ ∂ y + w ∂ ∂ u ∂ ∂ z = − − ∂ ∂ p ∂ ∂ x 1 ρ ρ + ν ν ( ∂ ∂ 2 u ∂ ∂ x 2 + ∂ ∂ 2 u ∂ ∂ y 2 + ∂ ∂ 2 u ∂ ∂ z 2 ) + f x ,