Navier-Stokes equations

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    \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.


(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}}{D t}  \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}}{D t}  \right) = -\nabla p + \mathbf{f} .