$3.$ Given the two$-$dimensional overflow structure $($open channel flow$)$ below, calculate the

a$.$ flow per unit width of the structure in meters squared per second and

b$.$ the magnitude of the horizontal force in kilonewtons per meter of the fluid on the structure.

Formulas

s$.$g$.=(\backslash$rho $\_($sub$))/(\backslash$rho $\_($H$\_(2)$o$)),\backslash$gamma $\_($water $)=62\mathrm{.}4($lb$\_($f$))/($ft$^(3)),\backslash$gamma $\_($water $)=9810(($N$))/($m$^(3)),\backslash$gamma $=\backslash$rho g$,$g$=9\mathrm{.}81(($m$))/($s$^(2))=32\mathrm{.}2($ft$)/($s$^(2))$

$\backslash$tau $=\backslash$mu $($dv$)/($dy$),\backslash$mu $=\backslash$rho

u $,$p$\_($atm $)=14\mathrm{.}7\backslash$psi $=101\mathrm{.}33$kPa,R$=8\mathrm{.}314($LPa$)/($ Kmol $)$

u $\_(()()$water $)=1\mathrm{.}003\backslash$times $10^(-6)($m$^(2))/(($s$))$ at $10\backslash$deg C$=1\mathrm{.}059\backslash$times $10^(-5)($ft$)/(($s$))$s at $70\backslash$deg F

F$=\backslash$gamma h$\_($c$)$A$,$y$\_($p$)=$y$\_($c$)+($I$\_($c$))/($y$\_($c$)$A$),$Re$=(\backslash$rho vL$)/(\backslash$mu $)=($vL$)/()$

u $,$Q$=$va$,$PV$=$mRT

$($P$\_(1))/(\backslash$gamma $)+$z$\_(1)+($v$\_(1)^(2))/(2$g$)=($P$\_(2))/(\backslash$gamma $)+$z$\_(2)+($v$\_(2)^(2))/(2$g$)=$H$,7\mathrm{.}48$ gallons $=1$ft$^(3),1$lb$\_($f$)=32\mathrm{.}2($lb$\_($m$)$ft$)/($s$^(2))$

$($P$\_(1))/(\backslash$gamma $)+$z$\_(1)+($v$\_(1)^(2))/(2$g$)+$h$\_($u$)=($P$\_(2))/(\backslash$gamma $)+$z$\_(2)+($v$\_(2)^(2))/(2$g$)+$h$\_($e$)+$h$\_($l$),$P$=(\backslash$gamma Qh$\_($u$))/(550)($horsepower$)=(\backslash$gamma Qh$\_($u$))/(1000)($kilowatts$)$

$\backslash$sum vec$($F$\_($ext $))=\backslash$rho Q$($vec$($V$\_(2))-$vec$($V$\_(1))),\backslash$sum F$\_($x$)=\backslash$rho Q$($V$\_($x$,2)-$V$\_($x$,1)),\backslash$sum F$\_($y$)=\backslash$rho Q$($V$\_($y$,2)-$V$\_($y$,1))$

$($a$)$ nectangre

A$=\backslash$pi $($R$^(2))/(2),$I$\_(\backslash$times $,$C$)=0\mathrm{.}109757$R$^(4)$

A$=\backslash$pi a$($b$)/(2),$I$\_(\backslash$times $,$C$)=0\mathrm{.}109757$ab$^(3)$

$($d$)$ Triangle

$($e$)$ Semicircle

$($f$)$ Semiellipse

Figure $3-32$ of Cengel