Problem

$($I need help writing the code and graphing the plots in python because I cannot get my python working$)$

The superposition of the two$-$dimensional potentials of parallel flow, a source of positive strength $$ and

a source of negative strength $(sink)$ represents the flow around an oval contour $($Rankine ovoid, see

sketch below$)$

The velocity potential in Cartesian coordinates is:

$=-U_{}x+\frac{}{2}ln\sqrt[\phantom{2}]{(x-a{)}^2+y^2}-\frac{}{2}ln\sqrt[\phantom{2}]{(x+a{)}^2+y^2}$

$($a$)$ Derive the velocity field $v_{?}=gra{d}_{}$ in Cartesian coordinates

$($b$)$ Find the locations of forward and aft stagnation point, i$.$e$.$ compute $\frac{L}{2}.$ It is safe to assume that the

stagnation points lie on the $x-$axis. Adapt the streamline visualization program which we used in

the recitation session to the new flow field and check your stagnation point calculation.

$(4$ of $40$ points$)$

$($c$)$ Submit a plot of the streamlines in which you marked the location of your stagnation point results

from part $($b$).$ Use the following flow parameters for the plot: $U_{}=1\mathrm{.}4\frac{m}{s},=6\mathrm{.}0\frac{m^2}{s},$ and

$a=1\mathrm{.}2m.$

$(3$ of $40$ points$)$

$($d$)$ Extract the coordinates of the upper part of the dividing streamline $($at least very close to it$)$ and

compute the pressure coefficient $C_P$ along the streamline. Submit a plot of the pressure coefficient

$C_P$ over the $x-$coordinates of the dividing streamline.

$(6$ of $40$ points$)$

$($e$)$ What result would you obtain for the resultant pressure force on the body and why?

$(2$ of $40$ points$)$