There is a two$-$dimensional flow tube $($Figure b$),$ with the flow field symmetric along the $x-$

axis. The flow field can be expressed as:

vec$(V)=(u,v)=(U+bx)hat(i)-$byhat$(j)$

where $U$ is the horizontal velocity at $x=0,$ and $b$ is a constant.

Answer the following questions $($Challenge problem$)$:

$($a$)$ Find the linear strain rates ${}_x^{},{}_y^{},$ angular strain rate ${}^{},$ vorticity $,$ acceleration vec$(a),$ and the

streamline equation.

$($b$)$ Given that at time $t=0,$ a fluid particle is at position $(x_0,y_0),$ find its position $(x,y)$ after

time $t.$

$($c$)$ The pressure field is given by $P(x,y)=P_0-(\frac{}{2})[2$Ubx$+b^2(x^2+y^2)],$ where $P_0$ is the

pressure at $x=0,$ and $$ is the fluid density. Find the rate of pressure change as the fluid flows

through the point $(x,y)=(0,1).$

$($d$)$ On the x $-$axis, two particles A and B are initially separated by a distance $$ at $t=0.$ After time $t$

$,$ the distance between them becomes $+.$ Find $\frac{}{}.$ Using the result of $\frac{}{}$ and

L$\text{'}$H$$pital$\text{'}$s rule, find the linear strain rate ${}_x^{},$ and compare it with the answer from part $($a$).$

$($e$)$ Similarly, for two particles A and B initially separated by a vertical distance $$ at $t=0,$ the

distance between them becomes $+$ at time $t.$ Find $\frac{}{}.$ Using the result of $\frac{}{}$ and

L$\text{'}$H$$pital$\text{'}$s rule, find the linear strain rate ${}_y^{},$ and compare it with the answer from part $($a$).$

$($f$)$ Continuing from part $($e$),$ if at time $t=0,$ particles A and B are located at $x=x_0,$ and their

distance is $.$ After time $t,$ their distance becomes $\frac{}{2}.$ Find the displacement of the two particles

in the $x-$direction during this time interval.

$($g$)$ Draw a deformation diagram of a square fluid element ABCD $($Figure b$)$ as it moves in the flow

field, showing its transformed shape $A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}{D}^{\text{'}}.$Picture b

Picture c