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10.5 Unsteady State Bernoulli in Accelerated Coordinates

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    786
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    Table 10.1 Table of Basic Solutions to Laplaces' Equation.

    Name Stream Function Potential Function Complex Potential
      \(\psi\) \(\phi\) \(F(z)\)
    Uniform Flow in \(x\) \(U_0\,y\) \(U_0\,x\) \(U_0\,z\)
    Uniform Flow in \(y\) \(U_0\,x\) \(-U_0\,y\) \(U_0\,z\)
    Uniform Flow in an Angle \(U_{0y}\,y - U_{0y}\,x\) \(U_{0y}\,x+U_{0x}\,y\) \(\left(U_{0x}-i\,U_{0y}\right)\,z\)
    Source \(\dfrac{Q}{2\,\pi}\,\theta\) \(\dfrac{Q}{2\,\pi}\,\ln\,r\) \(\dfrac{Q}{2\,\pi}\,\ln\,z\)
    Sink \(-\dfrac{Q}{2\,\pi}\,\theta\) \(-\dfrac{Q}{2\,\pi}\,\ln\,r\) \(-\dfrac{Q}{2\,\pi}\,\ln\,z\)
    Vortex \(-\dfrac{\Gamma}{2\,\pi}\,\ln\,r\) \(\dfrac{\Gamma}{2\,\pi}\,\theta\) \(-\dfrac{i\,\Gamma}{2\,\pi}\,\ln\,z\)
    Doublet

    \(- \dfrac{Q_0}{2\,\pi} \, \dfrac{1}{2} \, \ln \left(
          \dfrac{\dfrac{r^2+{r_0}^2}{2\,r\,r_0\, \cos \theta} + 1}
             {\dfrac{r^2+{r_0}^2}{2\,r\,r_0\, \cos \theta} - 1}\right)\)

    \(\dfrac{Q_0}{2\,\pi} \left( \tan^{-1} \dfrac{y}{x-r_0} - \tan^{-1} \dfrac{y}{x+r_0} \right)\) \(-\dfrac{i\,\Gamma}{2\,\pi}\,\ln\,z\)
    Dipole \(-\dfrac{\Gamma}{2\,\pi}\,\ln\,r\) \(\dfrac{\Gamma}{2\,\pi}\,\theta\) \(-\dfrac{i\,\Gamma}{2\,\pi}\,\ln\,z\)
    \(90^\circ\) Sector Flow \(U\,r^2\,\sin\,2\theta\) \(U\,r^2\,\cos\,2\theta\) \(U\,z^2\)
    \(\pi/n\) Sector Flow \(U\,r^n\,\sin\,n\theta\) \(U\,r^n\,\cos\,n\theta\) \(U\,z^n\)

      Table 10.2 Table of 3D Solutions to Laplaces' Equation.

    Name Stream Function Potential Function
      \(\psi\) \(\phi\)
    Uniform Flow in \(z\) direction \(U_0\,r \,\cos\theta\) \(U_0\,x\)
    Source \(-\dfrac{Q\,\cos\theta}{4\,\pi}\) \(U_0\,x\)

    Contributors

    • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.