# Equations of fluid dynamics and differential geometry [closed]

Where can I look for equations of fluid motion written in terms of nifty things from differential geometry like exterior derivative, Hodge dual, musical isomorphism?

Preferably both with and without assumption of Newton viscosity model.

Since one can get momentum and continuity equation from the full energy equation + Galileo transform, maybe this one will suffice though continuity equation would be nice as a simple example.

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## closed as too broad by David Z♦Aug 21 '13 at 19:51

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This is probably on the edge of unpractical and uselessly general. Anyway, I don't really have an answer, but have you googled something like "Navier-Stokes Riemannian". Thinks like arxiv.org/abs/1205.4888 will show up, which at first sight looks like the things you want. There is also a math.overflow thread, maybe there are some references. – NikolajK Aug 2 '12 at 18:43
There does exist a well-known duality between Einstein's equations in GR and Navier-Stokes equations of fluid dynamics. See arxiv.org/abs/1107.5780 . Perhaps, to the extent that GR might be described using the items you've listed, you might be able to do the same for fluids. – kleingordon Aug 2 '12 at 21:58
@kleingordon: I wouldn't call it "well known", it's a few years old. The duality comes from linking the dynamics of thermal gauge fields, which is like a fluid with viscosity, through AdS/CFT with the 90's membrane paradigm which treats the surface of a black hole as a viscous fluid. My feeling is that it is more useful for GR than for fluids. – Ron Maimon Aug 3 '12 at 5:16
@NickKidman actually I saw equations in this form in some kind of presentation. It was told they come in hand in CFD to solve equations in curvilinear coordinates. genneth's answer indicates it too. It isn't unpractical at least for engineers. – Yrogirg Aug 3 '12 at 8:53
result due to google: qcpages.qc.cuny.edu/~swilson/formsfluidsfinal.pdf – Christoph Aug 3 '12 at 9:49

The standard reference for this is Arnold and Khesin "Topological Methods in Hydrodynamics", which is excellent.

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As far as I know it is some sort of generalization of Hamiltonian approach, but on infinite dimensional manifolds. And I wanted just usual form equations on 2D and 3D manifolds. Moreover, thus it deals only with ideal fluid. But I think I still get this book to see, maybe even today, thanks. – Yrogirg Aug 3 '12 at 5:20
@Yrogirg: This is nothing difficult--- it's just treating the Euler fluid equation as the Euler geodesic motion (spinning-top) for the diffeomorphism manifold, and this is an important conceptual relation. It is the theory of the usual form of the equations on 2d and 3d manifolds. It doesn't really only deal with ideal fluids, although this is where the topological methods are most insightful. Also, I don't understand "even today", it's not that old. – Ron Maimon Aug 3 '12 at 17:54
"even today" I meant I'll buy it today, and indeed I did. – Yrogirg Aug 3 '12 at 18:15

This sort of thing is actually very well developed in the computational setting, under discrete differential geometry. There is an approach called Simplicial Fluids: http://www.geometry.caltech.edu/pubs/ETKSD07.pdf

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unfortunately is says nothing about the stress tensor, it's type for example. – Yrogirg Aug 6 '12 at 4:09

I've made a small document featuring fluid dynamics equations in terms of vector-valued differential forms. The document with information on any further developments can be found on my page.

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Link is currently dead (which is one of the reasons that "link-only" answers are discouraged). – Kyle Kanos Dec 1 '15 at 21:58