Navier-Stokes equations

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

<math>\rho \frac{D \mathbf{V}}{D t} = -\nabla p + \mu \nabla^2 \mathbf{V} + \rho \mathbf{g}</math>

where <math>\nabla p</math> is the pressure difference (expressed as the partial derivative of pressure in each dimension), <math>\frac{D \mathbf{V}}{D t}</math> is the total derivative of velocity, <math>\mu \,</math> is the kinematic viscosity of the fluid, <math>\rho \,</math> is the density of the fluid, and <math>\mathbf{g}</math> is the gravitational acceleration.

In Incompressible flow

<math>\boldsymbol \tau = \mu (\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T})</math>