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I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines on the manifold"? If so, would it be fair to say that the Navier-Stokes equations are partial differential equations which "embed" this manifold in $R^3$?

This would be somewhat analogous to the way that the Poincare Disk Model transforms straight lines in the hyperbolic plane into arcs of circles that are orthogonal to the boundary circle. Except in this case, the "straight lines" are transformed into fluid particle trajectories in $R^3$.

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up vote 12 down vote accepted

For the Navier Stokes equation, I am rather doubtful, because the presence of the viscosity term makes the equation not time-symmetric. And the geodesic flow is necessarily time-symmetric.

If you remove the viscosity term, you are down to the Euler equation, for which there is a well-known characterisation due to V I Arnold.

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Maybe Lagrangian coordinates is what you are after.

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