Navier-Stokes equations

Continuity

$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{v})=0$

or, using the substantive derivative:

$\frac{D\rho }{Dt}+\rho (\mathrm{\nabla }\cdot \mathbf{v})=0.$

For an incompressible fluid, $\rho$ is constant, hence the velocity field must be divergence-free:

$\mathrm{\nabla }\cdot \mathbf{v}=0.$

Momentum

(Also known as the Navier-Stokes equation.)

$\rho (\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot \mathrm{\nabla }\mathbf{v})=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot \mathbb{T}+\mathbf{f},$

or, using the substantive derivative:

$\rho (\frac{D\mathbf{v}}{Dt})=-\mathrm{\nabla }p+\mathrm{\nabla }\cdot \mathbb{T}+\mathbf{f},$

where $\mathbf{f}$ is a volumetric force (e.g. $\rho g$ for gravity), and $\mathbb{T}$ is the stress tensor.

Stress

The vector quantity $\mathrm{\nabla }\cdot \mathbb{T}$ is the shear stress. For a Newtonian incompressible fluid,

$\mathrm{\nabla }\mathbb{T}=\mu {\mathrm{\nabla }}^2\mathbf{v},$

with $\mu$ being the (dynamic) viscosity.

For an inviscid fluid, the momentum equation becomes Euler's equation for ideal fluids:

$\rho (\frac{D\mathbf{v}}{Dt})=-\mathrm{\nabla }p+\mathbf{f}.$