Editing Navier–Stokes equations (section)

==Non-inertial frame of reference==

The rotating frame of reference introduces some interesting pseudo-forces into the equations through the [[material derivative]] term. Consider a stationary inertial frame of reference <math display="inline">K
</math>&nbsp;, and a non-inertial frame of reference <math display="inline">K'
</math>, which is translating with velocity <math display="inline">\mathbf{U}(t)
</math> and rotating with angular velocity <math display="inline">\Omega(t)
</math> with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes

{{Equation box 1
|indent=:
|title='''Navier–Stokes momentum equation in non-inertial frame''' 
|equation=<math> \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = - \nabla p  + \nabla \cdot \left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\} + \nabla[\zeta (\nabla\cdot\mathbf{u})] + \rho\mathbf{f} - \rho \left[2\mathbf\Omega\times\mathbf u + \mathbf\Omega\times(\mathbf\Omega\times\mathbf x)+ \frac{\mathrm{d} \mathbf U}{\mathrm{d} t} + \frac{\mathrm{d} \mathbf \Omega}{\mathrm{d} t}\times\mathbf x\right].</math>
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Here <math display="inline">\mathbf{x}
</math> and <math display="inline">\mathbf{u}
</math> are measured in the non-inertial frame. The first term in the parenthesis represents [[Coriolis acceleration]], the second term is due to [[centrifugal force|centrifugal acceleration]], the third is due to the linear acceleration of <math display="inline">K'
</math> with respect to <math display="inline">K
</math> and the fourth term is due to the angular acceleration of <math display="inline">K'
</math> with respect to <math display="inline">K
</math>.