10.5 Unsteady State Bernoulli in Accelerated Coordinates
- Page ID
- 786
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.