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I want, for obscur reasons which would lead us too far to explain, to split my flow into two component, one steady and another one non-steady

[tex]v = v_0 + v'[/tex]

I'm looking for a simple equation governing the evolution of this non steady components. The complete momentum equation gives

[tex]\rho\frac{\partial (v_0 + v')}{\partial t} = -\rho\left(\left(v_0 + v'\right)\nabla\right)\left(v_0 + v'\right) - \nabla P + \rho g + \frac{1}{\mu_0}\left(\nabla \times B\right)\times B + \mu\triangle\left(v_0+v'\right) [/tex]

Assuming that v0 is steady and that both v0 and v' are perturbations of the hydrostatic state, we can get rid of the pressure gradient and the gravity force, leading to the following equation :

[tex]\rho\frac{\partial v'}{\partial t} = -\rho\left(\left(v_0 + v'\right)\nabla\right)\left(v_0 + v'\right) + \frac{1}{\mu_0}\left(\nabla \times B\right)\times B + \mu\triangle\left(v_0+v'\right) [/tex]

I would be pleased to get an even easier equation, but I can't find out the hypothesis necessary to get rid of some of the advection terms or the v0 diffusive term.

Are there some time/space scales which I should compare ?

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# Navier Stokes, separation steady/non-steady

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