Problem $36($Streamlines$,$ Streaklines, and Pathlines$).$

Given the steady$-$state, non$-$dimensional velocity field:

vec$(V)(x,y)=cscxhat(i)+\frac{1}{2y}$hat$(j)$ over the domain $0a<mfrac><mrow><mi></mi></mrow><mrow><mn>2</mn></mrow></mfrac>$ and $0y<msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup>$

$[2$pts$]($a$)$ Compute the equation of the strcamlines $($i$.$e$.,$ family of curves$)$ for this flow field. Leave your answer in the form of a graph: $g(x,y)=0.$ Warning: Do not attempt to find a functional form: $y=f(x),$ or $x=h(y),$ at this stage in the problem.

$[2$pts$]($b$)$ Determine the cquation for the streamline going through the point $(0,1)$ in the form of a function: $y=f(x).$

$[2$pts$]($c$)$ Compute the pathline for a particle that is moving with the velocity field starting from the initial position $(0,1)$ at time $t=0($left$-$most dot in figure$).$

$2pts($d$)$ At what time $T$ does this particle reach the point $(\frac{}{2},Y)$ at the right side of the flow field $($right$-$most dot in figure$)?$ Notice that you will also need to find $Y.$ The final point is fonnd from the equation for the trajectory: $(x(T),y(T))=(\frac{}{2},Y).$ You should check your results by comparing them to the location of the dots in the figure

For your convenience, a graph of the velocity field is shown in figure $3$ below.

$[2$pts$]($c$)$ Verify mathematically that the pathline solutions satisfy the streamline equations, this confirming that the pathlines and streamlines are the same for steady flow.