Navier-Stokes equations

The Navier-Stokes equation is an equation in fluid mechanics that states:

$\rho {\frac {D\mathbf {V} }{Dt}}=-\nabla p+\mu \nabla ^{2}\mathbf {V} +\rho \mathbf {g}$

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla p} is the pressure difference (expressed as the partial derivative of pressure in each dimension), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{D \mathbf{V}}{D t}} is the total derivative of velocity, $\mu \,$ is the kinematic viscosity of the fluid, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \,} is the density of the fluid, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{g}} is the gravitational acceleration.

In Incompressible flow

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol \tau = \mu (\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T})}