$(15$ pt$)$ For a general two$-$dimensional incompressible flow,

$($a$)$ find the principal axes of the strain rate tensor.

$($b$)$ Using the obtained general relation, determine if it is possible to have eigenvalues as zero. If the answer

is affirmative, describe under what conditions this occurs.

$($c$)$ Following part b$,$ explain the physical situation that corresponds to that condition.

$(10$ pt$)$ Find the principal axis for vec$(V)=(x+y)hat(i)+(x-y)hat(j).$

$(15$ pt$)$ Consider the following $2$ D velocity field:

$V=-$kxhat$(i)+$kyhat$(j)$

where $k$ is a constant.

$($a$)$ Verify whether the flow is incompressible.

$($b$)$ Find the vorticity vector, and comment on its physical interpretation.

$($c$)$ Determine the location$($s$)$ where the vorticity vanishes.

$($d$)$ Sketch the streamlines for this flow and describe the general behavior of fluid elements near the origin.

For the latter, you may briefly discuss about the velocity close to the origin.