Navier-Stokes equations

Continuity

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

or, using the substantive derivative:

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

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

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

Momentum

(Also known as the Navier-Stokes equation.)

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

or, using the substantive derivative:

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

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

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

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

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

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

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