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

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

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

$\phi = \frac{\Gamma}{2\pi}\theta$
$\psi = -\frac{\Gamma}{2\pi}ln(r)$
$v_{x} = \frac{\Gamma}{2\pi r}\hat{\theta}$

Dipole

$\phi = \frac{-\mu}{2\pi} \frac{x - x_{0}cos(\alpha) + y - y_{0}sin(\alpha)}{(x - x_{0})^{2} + (y - y_{0})^{2}}$
$\psi = \frac{\mu}{2\pi} \frac{x - x_{0}sin(\alpha) + y - y_{0}cos(\alpha)}{(x - x_{0})^{2} + (y - y_{0})^{2}}$
$v_{x} = \frac{((x - x_{0})^{2} + (y - y_{0})^{2})cos(\alpha) - 2(x - x_{0})((x - x_{0})cos(\alpha) + (y - y_{0})sin(\alpha))}{((x - x_{0})^{2} + (y - y_{0})^{2})^{2}}$
$v_{x} = \frac{((x - x_{0})^{2} + (y - y_{0})^{2})sin(\alpha) - 2(y - y_{0})((x - x_{0})cos(\alpha) + (y - y_{0})sin(\alpha))}{((x - x_{0})^{2} + (y - y_{0})^{2})^{2}}$

Corner Flow

$\phi = r^{\alpha}cos(\alpha \theta)$
$\psi = r^{\alpha}sin(\alpha \theta)$
$u_{r} = \frac{\partial \phi}{\partial r} = \alpha r^{\alpha - 1}cos(\alpha \theta)$
$u_{\theta} = \frac{1}{r}\frac{\partial \phi}{\partial \theta} = -\alpha r^{\alpha - 1}sin(\alpha \theta)$