Equations

Material Derivative

$\frac{Df}{Dt} = \frac{\partial f}{\partial t} + \vec{v} \cdot \nabla f$

Conservation of Mass

$\frac{D \rho}{Dt} + \nabla \cdot \rho \vec{v} = 0$

Navier-Stokes Equations

$\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v} = -\frac{\nabla p}{\rho} + \nu \nabla^{2} \vec{v} + g \nabla z$

Euler Equations

$\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v} = -\frac{\nabla p}{\rho} + g \nabla z$

Kelvin's Theorem

$\frac{d \Gamma}{dt} = 0$

Bernoulli's Equation

$\rho \frac{\partial \phi}{\partial t} + \rho\!\left(\frac{| \vec{v} |^{2}}{2}\right) + p + \rho gz = constant$

Incompressible Flow

$\nabla \cdot \vec{v} = 0$

Irrotational Flow

$\nabla \times \vec{v} = 0$

Laplace's Equation

$\nabla^{2} \phi = 0$

Point Source/Sink

$\phi = \frac{m}{2\pi}\ln(r)$

$\psi = \frac{m}{2\pi}\theta$

$\vec{v} = \frac{m}{2\pi r}\hat{r}$

Point Vortex

$\phi = \frac{\Gamma}{2\pi}\theta$

$\psi = -\frac{\Gamma}{2\pi}\ln(r)$

$\vec{v} = \frac{\Gamma}{2\pi r}\hat{\theta}$

Dipole

$\phi = -\frac{\mu}{2\pi} \frac{(x - x_o)\cos(\alpha)\ +\ (y - y_o)\sin(\alpha)}{(x - x_o)^{2}\ +\ (y - y_o)^{2}}$

$\psi = \frac{\mu}{2\pi} \frac{(y - y_o)\cos(\alpha)\ -\ (x - x_o)\sin(\alpha)}{(x - x_o)^{2}\ +\ (y - y_o)^{2}}$

$v_x = -\frac{\mu}{2\pi}\frac{\left((x - x_o)^{2}\ +\ (y - y_o)^{2}\right)\!\cos(\alpha)\ -\ 2(x - x_o)\left((x - x_o)\cos(\alpha)\ +\ (y - y_o)\sin(\alpha)\right)}{\left((x - x_o)^{2}\ +\ (y - y_o)^{2}\right)^{2}}$

$v_y = -\frac{\mu}{2\pi}\frac{\left((x - x_o)^{2}\ +\ (y - y_o)^{2}\right)\!\sin(\alpha)\ -\ 2(y - y_o)\left((x - x_o)\cos(\alpha)\ +\ (y - y_o)\sin(\alpha)\right)}{\left((x - x_o)^{2}\ +\ (y - y_o)^{2}\right)^{2}}$

Corner Flow

$\phi = \beta r^{\alpha}\cos(\alpha(\theta - \theta_o))$

$\psi = \beta r^{\alpha}\sin(\alpha(\theta - \theta_o))$

$v_{r} = \frac{\partial \phi}{\partial r} = \beta\alpha r^{\alpha - 1}\cos(\alpha(\theta - \theta_o))$

$v_{\theta} = \frac{1}{r}\frac{\partial \phi}{\partial\theta} = -\beta\alpha r^{\alpha - 1}\sin(\alpha(\theta - \theta_o))$