Complex Potential Flow and Blasius Formula

1. Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \]
2. Show that \(r=a\) is a streamline.
3. Determine the asymptotic velocity when \(z\to \infty \). Sketch the flow when \(\gamma =0\).
4. Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C \left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \neq 0\).

The fourth quadrant \(x\ge 0\), \(y\le 0\) is occupied by an impermeable solid, around which air flows. Treating the flow as incompressible and irrotational use conformal mapping to map to a simpler problem and give one non-trivial solution for the complex potential \(f(z)\) in the original coordinates. Find the equation in polar coordinates for the streamline passing through \(z=-1\). What happens to the fluid velocity as one approaches the corner?

1. Verify that the real and imaginery parts of a complex analytic function satisfy Laplace’s equation.
2. Give an interpretation of the complex function \[ w(z)=z+\frac {a^2}{z} \] as a flow past some object. That is, determine the streamlines for this function. Where are the stagnation points?

Discuss the flow pattern associated with the complex potential \[ \Omega (z)=Q_0z+\frac {\overline {Q_0}a^2}{z}+\frac {i\gamma }{2\pi }\log z \] with \(a>0\), \(\gamma \) real and \(Q_0=U_0+iV_0\). Sketch the flow when \(\gamma =0\). Given that the complex force on a cylindrical obstacle with boundary \(C\) is \[ \overline {F}=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz, \] where \(\rho \) is the density of fluid, determine the lift when \(\gamma \ne 0\).

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 \(U_x=U\), \(U_y=0\).

1. 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.
2. 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?
3. 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 \(\gamma \).

Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \] Show that \(r=a\) is a streamline. Determine the asymptotic velocity when \(z\to \infty \). Calculate the stagnation points. Sketch the flow when \(\gamma =0\). Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \ne 0\).

Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \] Show that \(r=a\) is a streamline. Determine the asymptotic velocity when \(z\to \infty \). Calculate the stagnation points. Sketch the flow when \(\gamma =0\). Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \ne 0\).

Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \] Show that \(r=a\) is a streamline. Determine the asymptotic velocity when \(z\to \infty \). Calculate the stagnation points. Sketch the flow when \(\gamma =0\). Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \ne 0\).

Using conformal mapping find the electrostatic potential \(\phi (x,y)\) in the region between the circumferences \[ \gamma =\{z:\ |z|=1\}\qquad \text {and}\qquad \Gamma =\{z:\ |z-1|=2.5\} \] satisfying the boundary conditions \[ \phi \equiv 0\text { on }\gamma ,\qquad \phi \equiv 1\text { on }\Gamma . \]