# 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 $$-\dfrac{Q\,\cos\theta}{4\,\pi}$$ $$U_0\,x$$