A Square Fluid Element $(30\%)$

For this question, use the same velocity field as in question $1.$ Consider a fluid element that, at

t $=0,$ is a square of side length $0\mathrm{.}1$ with its lower$-$left corner at the origin.

A

C D

B$0\mathrm{.}1$

$0\mathrm{.}1$

x

y

a$)$ Calculate the velocity of each corner of the fluid element at t $=0.$ Use these velocities to

obtain the coordinates of the corners a short time later, at t $=0\mathrm{.}05.$ Draw the new fluid

element shape at this time by connecting the four corners.

b$)$ Comparing your t $=0\mathrm{.}05$ corner coordinates to those at t $=0,$ verify that

$111$

$$xAB

d$$xAB

dt $,221$

$$yAC

d$$yAC

dt $,121$

$2$

d$\backslash$kappa

dt $,$

where $\backslash$kappa is the angle between AB and AC$.$

c$)$ Again comparing the t $=0\mathrm{.}05$ and t $=0$ coordinates, verify that your angular velocity from

the previous problem indeed predicts the rotation of the fluid element. Calculate the rotation

by taking the average of the angle change in the segment AB and the segment AC