You are tasked to prevent erosion in a wide rectangular open channel flow system. The Uniform fiow Gradually varied flowopen channel has a bed slope of $S=0\mathrm{.}0002$ and is approaching a vertical drop at the downstream end. The design flow rate is $q=0\mathrm{.}2\frac{m^2}{s}$ and you can assume a constant Manning's $n$ of $0\mathrm{.}025$ independent of the grain sizes and the location. Figure illustrates the open channel flow system including the uniform flow region as well as the gradually varied free$-$surface profile approaching the free$-$overfall. Complete the following tasks:

a$)$ For the uniform flow region, assess if a sand bed with mean grain size of $2$ mm is stable

b$)$ Determine the mean sediment grain size that should be used in the gradually varied flow region as an extra protection layer to prevent erosion in this part of the channel. You can assume that the sediment material is quartz. $[$Hint: use the most erosion affected area.$)$

c$)$ Estimate the extend of the gradually varied flow region to provide guidance for the length of the extra protection layer. You should use at least $3$ steps for your calculations.

d$)$ Briefly discuss the physical processes that could occur if the flows in your system exceed the critical shear stress.