Assigned: Tues., Oct. $1,2024$

Due: Tues., Oct. $8,2024$

The velocity distribution for laminar flow in a long circular tube of radius $R$ is

given by

$u=ui=u_{\text{m}\text{a}\text{x}}[1-(\frac{r}{R}{)}^2]i$

The velocity profile is sketched in Figure $1.$ For this profile, obtain expressions

for the volume flow rate, ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}u*ndA,$ and the momentum flux, ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(u)(u*n)dA,$

through a section normal to the pipe axis. Obtain an expression for the kinetic

energy flux, ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{1}{2}|u{|}^2)(u*n)dA,$ through a section normal to the pipe axis.

Figure $1$: Geometry for Question $1.$