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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot \mathbf{v} =0. }

Momentum

(Also known as the Navier-Stokes equation.)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \left(\frac{D \mathbf{v}}{D t} \right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{f} } is a volumetric force (e.g. $\rho g$ for gravity), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{T} } is the stress tensor.

Stress

The vector quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot\mathbb{T} } is the shear stress. For a Newtonian incompressible fluid,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \mathbb{T} = \mu \nabla^2 \mathbf{v}, }

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} being the (dynamic) viscosity.

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

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