Consider steady, incompressible, two$-$dimensional flow through a converging duct

$($Fig$.2).$ A simple approximate velocity field for this flow is

vec$(V)=(u,v)=(U_o+bx)vec()-$byvec$()$

Where $U_0$ is the horizontal speed at $x=0.$ A fluid particle $(A)$ is located at $x=x_A$ and

$y=y_A$ at time $t=0.$ At some later time $t,$ the fluid particle has moved downstream with

the flow to some new location $x=x_{A^{\text{'}}}$ and $y=y_{A^{\text{'}}}$ as shown in the figure $2.$ Generate an

analytical expression for the $y-$location of the fluid particle at some arbitrary time $t$ in

terms of its initial location $y=y_A$ and constant b$.$ In other words, develop an

expression for $y_{A^{\text{'}}}.($Hint: We know that $v=d\frac{y_{\text{p}\text{a}\text{r}\text{t}\text{i}\text{c}\text{l}\text{e}}}{d}t$ following a fluid particle. Plug in v $,$

separate variables, and integrate.$)$