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Engineering LibreTexts

10.5 Unsteady State Bernoulli in Accelerated Coordinates

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.