A rectangular river channel is $5$ m wide with manning's roughness $n=0\mathrm{.}22\frac{s}{m^{\frac{1}{3}}}$ and a bed slope $S_0=0\mathrm{.}002.$ The largest known flow in the river is $10\frac{m^3}{s}$ when the normal depth is $1$ m $.$ A model investigation is to be undertaken in a rectangular laboratory flume that is $0\mathrm{.}5$ m wide. The viscosity of water is ${10}^{-6}\frac{m^2}{s}.$

$($a$)$ To reproduce the river hydraulic performance in the flume, what similitude needs to be considered? What is the value of the relevant dimensionless number?

$($b$)$ Outline how the model should be constructed to reproduce the largest flow in the river by specifying the slope, roughness, water depth and flow rate in the flume.

$[8/25]$

$($c$)$ What has been neglected so far? What would be the values of acceleration of gravity and kinematic viscosity, if no approximation was taken?

$[12/25]$