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Question

Math 6420 • PDE

Fall 2005 • Question 8

Question

Math 6420 • PDE Fall 2005 • Question 8

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 \(\gamma \ne 0\).

Raw LaTeX

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 \(\gamma\ne0\).

Topic Tags

Boundary Value Problems Complex Potential Flow and Blasius Formula Linear Programming and Optimization Real Integral Evaluation by Complex Methods