$2.$ An two$-$dimensional eddy of circulation $\backslash (\backslash$Gamma $\backslash )$ is convected with the flow past a circular cylinder, centered at the origin, of radius $\backslash ($ a $\backslash ).$ The free stream velocity $\backslash ($ V$\_\{\backslash$infty$\}\backslash )$ is in the $\backslash ($ x $\backslash )$ direction. Consider the instant when the vortex is located at the position $\backslash (-$a$+$i a $\backslash ).$

a$)$ Determine the complex potential of this flow

b$)$ Decompose the complex potential to reveal the image flows inside the cylinder and determine the total circulation around the cylinder

c$)$ Plot the streamlines of the flow using the Ideal Flow Machine. Take $\backslash (\backslash$Gamma$=5\backslash$mathrm$\{$~m$\}^\{2\}/\backslash$mathrm$\{$s$\},$ a$=\backslash )1\mathrm{.}5$ m and $\backslash ($ V$\_\{\backslash$infty$\}=1\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

d$)$ Using Matlab, determine and plot the pressure coefficient distribution on the cylinder surface as a function of polar angle. Include a listing of your Matlab code with your solution.

e$)$ Derive expressions for the lift and drag forces on the cylinder normalized on $\backslash (1/2\backslash$rho V$\_\{\backslash$infty$\}^\{2\}$ d $\backslash ),$ where $\backslash ($ d $\backslash )$ is the cylinder diameter, in terms of the integral of the pressure

coefficient. These normalized forces are called the lift and drag coefficients. Extend your code to compute these coefficients. Explain your answers given your result for the circulation on the cylinder. Include a listing of your Matlab code with your solution.