$($i$)$ Obtain the relationships between the Ch$$zy$\text{'}$s coeflicient $C,$ Manning's roughness coefticient $n$ and the Darcy$-$Weisbach friction factor $f$ for uniform flow conditions taking place in an open channel in terms of the hydraulic geometry of the channel section.

$6$

$($ii$)$ Determine the diameter of a semi$-$circular open channel section which will carry the same discharge as that taking place in a rectangular section open channel of ${7\mathrm{.}38}^{\text{'}}$ width with a normal depth of flow of $4\mathrm{.}10.$ The bed slope and Manning's roughness coefticient have the same values in both the channels.

$[6]$

$2$a$.$ Water flows in a $$ shaped channel shown in the figure. Critical depth occurs at a section in this canal reach at $1\mathrm{.}40m.$ Estimate the normal depth and determine whether the flow is hydraulically smooth or rough $($Take roughness height as $0\mathrm{.}025m).$

$($or$)$

$2$b$.$ For a non uniform flow taking place in a river $($which may be approximated to a trapezoidal section channel$)$ of bottom width $10\mathrm{.}0m$ with side slopes of $1\mathrm{.}5H$:$1\mathrm{.}0V,$ the depth and velocity of flow observed at a section are $3\mathrm{.}0m$ and $2\mathrm{.}36\frac{m}{s}$ respectively. If the longitudinal bed slope observed in a $500m$ reach of the river is $3{10}^{-4},$ find $($i$)$ the velocity of flow taking place in upstream section of the river where the depth of flow is observed to be $3\mathrm{.}42m,$ and $($ii$)$ the discharge taking place in the downstream section. The equivalent Manning's roughness coefficient $n=0\mathrm{.}026$ for the river channel bed and the river banks.

$8$

$3$a$.($i$)$ Derive the governing equation for Gradually varied flow $($GVF$)$ conditions taking place in a prismatic channel of constant bed slope.

$($ii$)$ The normal depth of flow in a prismatic channel having the cross$-$section as shown in the figure below with bottom width of $15m$ is $1\mathrm{.}5m.$ The longitudinal slope of the channel is $1{10}^{-4}.$ A weir$/$hump at the downstream portion of the channel reach raises the flow depth to $3\mathrm{.}0m$ just upstream of the weir$/$hump$.$ Determine the stale of flow, and type of slope just U$/$S of the weirhump.