Q$2$: $($i$)$ Consider a $2$D pressure driven steady flow between stationary parallel plates separated

by distance b as shown in the Figure. Coordinate y is measured from the bottom plate. The

velocity field is given by

$(10$ marks$)$

u$=$U$(($y$)/($b$))[1-(($y$)/($b$))]$

$($a$)$ Obtain an expression for the circulation about a rectangular closed contour $1-2-3-4$

$($shown by dashed lines$)$ of height h and length L from basic definition. Consider dashed

line $1$ at y$=0.$

$($b$)$ Show that same expression is obtained from the area integral of the Stokes theorem.

Equation to be used for reference: Stokes theorem. Where vec$($ds$)$ is an element vector tangent

to the closed curve. A is the area of the enclosed curve.

$\backslash$int$\_$A $($vec$($grad$)\backslash$times vec$($V$))$dA$=$o$\backslash$int$\_$c vec$($V$)*$vec$($ds$)$

$($ii$)$ Consider a flow field represented by the stream function $\backslash$psi $($x$,$y$)=-($A$)/(2\backslash$pi $($x$^(2)+$y$^(2)))$ Where A

is a constant. Prove is this a possible $2$D incompressible flow? Further, justify is the flow

irrotational.

$($iii$)$ Differentiate between the following:

Lagrangian and Eulerian description of motion.

Timeline and Streakline

$($iv$)$ Can a variable density flow be incompressible. Justify your answer?

$(2$ marks$)$