$2.$ Irrotational Vortex on the Ground $(15$ pts $)$

Here, we have an irrotational vortex on the ground by letting a uniform flow $\backslash (\backslash$left$($U$\_\{0\}\backslash$right$)\backslash )$ normal to the vortex on the $\backslash ($ y $\backslash )-$axis $($see Fig. $($i$)).$

A$.$ Find the complex potential $\backslash ($ F$($z$)=\backslash$phi$+$i $\backslash$psi $\backslash )$ in $\backslash ($ x$-$y $\backslash )$ coordinate $[$Hint: Fig. ii$)\backslash$& Appendix $1](2$ pts$)$

B$.$ Calculate stream function $\backslash (\backslash$psi $\backslash )$ and velocity potential $\backslash (\backslash$phi $\backslash )$ in $\backslash ($ x$-$y $\backslash )$ coordinate $[$Hint: Appendix $2](2$ pts$)$

C$.$ Calculate velocity field $\backslash ($ u $\backslash$& v $\backslash )$ from the velocity potential $(2$ pts $)$

D$.$ Find two stagnation points and the value of stream function $\backslash (\backslash$psi$\_\{\backslash$text $\{$stag $\}\}\backslash )$ on the stagnation points $(2$ pts $)$

E$.$ Find the shape function of the solid body $(1$ pts $)$

F$.$ If you find a proper solid body, its shape would be a half dom with two stagnation points on the ground, as shown in Fig. i$).$ However, depending on the strength of the irrotational vortex $\backslash (\backslash$Gamma $\backslash ),$ the number of stagnation point on the ground would be changed from $0$ or $1$ or $2.$ For these three cases, drive the range of $\backslash (\backslash$Gamma $\backslash )$ to have $0$ or $1$ or $2$ stagnation points on the ground, and draw the shape of solid body $(4$pts$).$

G$.$ We may consider this irrotational vortex as a very simplified airfoil $($see figure below$).$ According to the Kutta$-$Joukowski theorem, lift force per unit span $\backslash (\backslash$mathrm$\{$L$\}^\{\backslash$prime$\}=\backslash$rho U$\_\{0\}\backslash$Gamma $\backslash )$ in the uniform flow $\backslash ($ U$\_\{0\}\backslash ).$ Then, do you think the life force of the airfoil near the ground is higher or lower than that in just uniform flow? $[$Hint: WIG ship$](2$ pts$)$