Chézy drag law ($\tau =\rho c_{d}u|u|$)

A quadratic boundary-drag model relates shear stress to mean flow velocity, with dimensionless resistance coefficient $c_d>0$. Uniform steady flow down slope $S$ balances $\rho gAS=\tau \ell$, giving $|u|=\sqrt{gRS/c_d}$. Its sign is chosen so drag opposes the motion.