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I'm considering to study some high-dimensional Navier-Stokes equations. One problem is to do write the viscous equation for vorticity, helicity and other conserved quantities. I think it might be better if it is possible to work with differential form and exterior calculus? Is there any reference that I may find somewhere?

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You mean in a form other than $\rho (\partial_t + v \cdot \nabla) v= f+ \dot{\overline \sigma}(\dot \nabla)$? Or formulas for the quantities you mention? – Muphrid Dec 25 '12 at 22:09

1 Answer 1

MS Mohamed et al begin with a "standard vector calculus formulation of the NS equations",

$$ \frac{∂ \bf u}{∂ t} - \mu ∆ {\bf u} + ({\bf u} \cdot \nabla) {\bf u} + \nabla p = 0 $$ $$ \nabla \cdot {\bf u} = 0 $$

and use the rotational form,

$$ \frac{∂ \bf{u}}{∂ t} + \mu \nabla \times \nabla \times {\bf u} - {\bf u}\times(\nabla \times {\bf u}) + \nabla p^d = 0 $$

to eventually derive

$$ \frac{∂ {\bf u}^\flat}{∂ t} + (-1)^{N+1} \mu \star d \star d {\bf u}^\flat + (-1)^{N+2}\star ({\bf u}^\flat \wedge \star d{\bf u}^\flat) + d p^d = 0, $$ $$ \star d \star {\bf u}^\flat = 0 $$

But they also later derive it as

$$ \frac{∂ d {\bf u}^\flat}{∂ t} + (-1)^{N+1} \mu d \star d \star d {\bf u}^\flat + (-1)^{N}d\star({\bf u}^\flat \wedge \star d{\bf u}^\flat) = 0, $$

Judging from the stars, -1s, and flats, these results could probably be simplified by substitution of the codifferential operator $d^\dagger$. Both are also the rotational form; I would rather provide a convective form, but I'm unaware of such a derivation in the literature.

I urge Troy to actually post an answer rather than referring us to his book, but lacking the points I cannot leave a comment to that effect.

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