# 10.5 Unsteady State Bernoulli in Accelerated Coordinates

- Page ID
- 786

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( |
\(\dfrac{Q_0}{2\,\pi} \left( \tan^{-1} \dfrac{y}{x-r_0} - \tan^{-1} \dfrac{y}{x+r_0} \right)\) | \(-\dfrac{i\,\Gamma}{2\,\pi}\,\ln\,z\) |

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 and Attributions

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.