Question $4(40$ points$)$: Consider a flow device

as shown in the figure below. The system is at

steady$-$state and the fluid is incompressible

with density $\backslash$rho $=1050$k$($g$)/($m$^(3)).$ There are

three regions of interest $(1,2,3)$ with their

unique inle$($t$)/($o$)$utlet areas A$\_(1)=0\mathrm{.}10$m$^(2),$A$\_(2)=$

$0\mathrm{.}02$m$^(2),$A$\_(3)=0\mathrm{.}05$m$^(2),$ and velocities V$\_(1)=$

$5$hat$()($m$)/($s$)$ and V$\_(2)=-10$hat$()($m$)/($s$)$V$\_(3)\backslash$theta $=45\backslash$deg $.$

a$.$ Write down the normal vectors in components at the control surfaces at location $1,2$ and

$3($widehat$($n$)\_(1),$widehat$($n$)\_(2),$widehat$($n$)\_(3)).$ Remember that the normal is always pointed outwards!

b$.$ The time$-$rate of change of mass of a system is given by:

$($Dm$\_($sys$))/($Dt$)=($del$)/($delt$)\backslash$int$\_($CV$)\backslash$rho dAA$+\backslash$int$\_($CS$)\backslash$rho V$*$widehat$($n$)$dA

Why is $($Dm$\_($sys$))/($Dt$)=0$ in this case? Why is also $($del$)/($delt$)\backslash$int$\_($CV$)\backslash$rho dAA$=0$ in this situation?

c$.$ Simplify the time$-$rate of change of mass, and write out the expressions for $\backslash$int$\_($CS$)\backslash$rho V$*$hat$($n$)$dA

d$.$ Solve at location $3$ for the magnitude of the velocity V$\_(3),$ and the velocity vector V$\_(3).$ Show

that V$\_(3)=|$V$\_(3)|.$