Math 6420 • PDE

Consider the flow around a thin airfoil of length \(2a\) that is inclined at an angle
\(\alpha\) to the \(x\)-axis. Assume that there is uniform flow at \(\infty\) with velocity
\(U_x=U\), \(U_y=0\).
\begin{enumerate}
\item[(a)] Show that the conformal map
\[
  z=\frac{ae^{-i\alpha}}{2}\left(\omega+\frac{1}{\omega}\right)
\]
maps the unit circle in the \(\omega\)-plane to the airfoil in the \(z\)-plane.
\item[(b)] Consider the complex potential
\[
  \Gamma(\omega)=\frac{aU}{2}\left(e^{-i\alpha}\omega+\frac{e^{i\alpha}}{\omega}\right)
  -\frac{i\gamma}{2\pi}\log\omega.
\]
Calculate the corresponding potential \(\Omega(z)=\Gamma(\omega(z))\) where \(\omega(z)\) is the
inverse of the conformal map in part (a). What does the term in \(\gamma\) represent?
\item[(c)] Calculate \(\Omega'(z)\) and show that it satisfies the correct asymptotic flow.
Determine what happens at the end points of the airfoil, and use this to calculate \(\gamma\).
\end{enumerate}