We can model potential flow over a jellybean by using a source $(Q),$ vortex $(),$ and sink $(-Q)$

placed on the $x$ axis as shown below. When combined with uniform flow $(U_{}),$ the superimposed

elementary flows will model flow over a closed shape that resembles a jellybean. For the numerical

parts below, use $U_{}=5\frac{m}{s},=10\frac{m^2}{s},Q=8\frac{m^2}{s},$ and $h=0\mathrm{.}5m.$

a$)$ Write the complex potential, $F(z),$ for this flow, in terms of $z$ and the given parameters $($not

numerical values$).$

b$)$ Write the equation for the complex velocity, $w(z),$ again in terms of the given parameters.

c$)$ Determine the location of the stagnation points in this flow, using the given numerical values.

You should find three points, with two at the leading$/$trailing edge of the closed shape.

d$)$ Provide a computer$-$generated plot of $20$ streamlines over the range $-1x1,-1y1.$

Space these equally in terms of stream function value in the range $-4,5,$ and overlay in

a different color $(1)$ the dividing streamline that separates the interior and exterior flows,

and $(2)$ the stagnation points. Hint: you can use the leading$/$trailing$-$edge stagnation point

locations to determine the value of $$ on the dividing streamline.

e$)$ Plot $20$ iso$-$potential lines for this flow, spaced equally in the velocity potential $($use the range

$[-12,6],$ and verify that they are perpendicular to the streamlines.