For flow water down an incline, the Manning equation (in SI units) reads: $$v=\left(\frac{1}{n}\right)R^{2/3}S^{1/2}$$where v is the average flow velocity parallel to the incline, R is the hydraulic radius of the flow channel (R = h for an flat incline, with h representing the thickness of the fluid layer), and S is the slope of the incline (##S = \sin{\alpha}##), with ##\alpha## representing the sine of the angle of incline relative to the horizontal), and n is a roughness parameter for the surface, equal to 0.01 for a smooth surface.
This equation is inconsistent with a dimensional analysis of the system. From dimensional analysis, we obtain the following relationship in terms of two key dimensionless parameters: $$\frac{vd}{\nu}=F\left(\frac{h^3 g\sin{\alpha}}{\nu^2}\right)$$where ##\nu## is the kinematic viscosity of the fluid, and g is the acceleration of gravity (9.8 m/s^2). If the velocity is supposed to vary with the liquid thickness h to a power, this equation must be expressible as $$\frac{vd}{\nu}=k\left(\frac{h^3 g\sin{\alpha}}{\nu^2}\right)^m$$ where k and m are dimensionless constants. For v to vary with h to the 2/3 power, the exponent m must have the value 5/9, such that $$v=kg^{5/9}\nu^{-1/9}h^{2/3}\sin {\alpha}^{5/9}$$
So we see that, according to dimensional analysis, the exponent of the channel slope in the Manning equation should be 5/9 = 0.555, rather than 1/2 = 0.5.
It is also possible to perform a standard turbulent Flow frictional analysis for this system and show that the Fanning friction factor is expressible as $$f=\frac{0.0311}{Re^{0.2}}$$ where Re is the Reynolds number for the flow, given by $$Re=\frac{vh}{\nu}$$ This compares with $$f=\frac{0.0791}{Re^{0.25}}$$ for flow in a pipe.
