10.5 Unsteady State Bernoulli in Accelerated Coordinates
- Page ID
- 786
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
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.