Wednesday, November 28, 2012

Navier–Stokes equations

General form of the equation

The general form of the Navier–Stokes equations for the conservation of momentum is:
\rho\frac{D\mathbf{v}}{D t} = \nabla\cdot\mathbb{P} + \rho\mathbf{f}
where
  • \rho\ is the fluid density,
  • \frac{D}{D t} is the substantive derivatives (also called the material derivative).
  • \mathbf{v} is the velocity vector,
  • \mathbf{f} is the body force vector, and
  • \mathbb{P} is a tensor that represents the surface forces applied on a fluid particle (the stress tensor).
Unless the fluid is made up of spinning degrees of freedom like vortices, \mathbb{P} is a symmetric tensor. In general, (in three dimensions) \mathbb{P} has the form:
\mathbb{P} = \begin{pmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{pmatrix}
where
  • \sigma\ are normal stresses,
  • \tau\ are tangential stresses (shear stresses).
The above is actually a set of three equations, one per dimension. By themselves, these aren't sufficient to produce a solution. However, adding conservation of mass and appropriate boundary conditions to the system of equations produces a solvable set of equations.
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