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.)

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. 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 g} for gravity), and $\mathbb {T}$ is the stress tensor.

Another form of the equation, more similar in form to the continuity equation, stresses the fact that the momentum density is conserved. For each of the three Cartesian coordinates 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 \alpha=1,2,3} :

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 \frac{\partial \rho v_\alpha}{\partial t} + \nabla \cdot {\rho v_\alpha \mathbf{v}} = -\frac{\partial p}{\partial x_\alpha} + \sum_\beta \frac{\partial }{\partial x_\beta} \mathbb{T}_{\beta\alpha} + f_\alpha. }

In vector form:

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 \frac{\partial \rho v_\alpha}{\partial t} + \nabla \cdot {\rho \mathbf{v} \mathbf{v}} = -\nabla p + \nabla\cdot\mathbb{T} + \mathbf{f}. }

The term 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{v} \mathbf{v} } is a dyad (direct tensor product).

Stress

The vector quantity $\nabla \cdot \mathbb {T}$ 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:

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 + \mathbf{f} . }