Navier-Stokes Equation

#Physics

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

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

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

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