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10.5 Unsteady State Bernoulli in Accelerated Coordinates

  • Page ID
    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 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.


    This page titled 10.5 Unsteady State Bernoulli in Accelerated Coordinates is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


    This page titled 10.5 Unsteady State Bernoulli in Accelerated Coordinates is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Genick Bar-Meir via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.