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Question

Math 6420 • PDE

Spring 2004 • Question 8

Question

Math 6420 • PDE Spring 2004 • Question 8

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 \(\gamma \ne 0\).

Raw LaTeX

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 \(\gamma\ne0\).

Topic Tags

Complex Potential Flow and Blasius Formula Linear Programming and Optimization