Navier-Stokes Equation

$\rho \left(\frac{\partial \vec{v}}{\partial t}+(\vec{v}\cdot \nabla )\vec{v}\right)=-\nabla p+\eta {\nabla }^2\vec{v}+{\vec{f}}_{\text{external}}$

• $\rho$ is the mass density
• $\vec{v}$ is the velocity vector
• $(\vec{v}\cdot \nabla )\vec{v}$ is the advection of $\vec{v}$
• $p$ is the pressure as a function of position
• $\eta$ is the viscosity of the fluid
• ${\vec{f}}_{\text{external}}$ is the external force per unit area

$\rho \frac{\mathrm{D}\vec{v}}{\mathrm{D}t}=\dots$

• A shorter way of expressing the left hand side of the above equation
• $\frac{\mathrm{D}\vec{v}}{\mathrm{D}t}$ is called the Material Derivative