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In fluid dynamics, I have come across two sets of equations, the Navier-Stokes equation and the Lorenz equations. From what I have read the Navier-Stokes equations always hold. So why do we need the Lorenz equations also? and when would we use them over the Navier-Stokes equations?

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  • $\begingroup$ What the Lorenz equations are used for is described in the wiki article you link. And the reason one might use them over the Navier-Stokes equation is that it's much simpler to solve and study (just like we use Newton's second law even though we know the Schroedinger equation). $\endgroup$ – lemon Aug 11 '16 at 10:10
  • $\begingroup$ What is the similarity between the physical phenomena that Navier-Stokes equation describes and what Lorentz equation describes? $\endgroup$ – Deep Aug 12 '16 at 8:04
  • $\begingroup$ @lemon he didn't link those, a moderator added the links to the question. Click on "edited Aug 11 at 8:51" to see the history. $\endgroup$ – Suzu Hirose Oct 14 '16 at 8:43
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The Navier-Stokes equations are a pretty good description of fluid dynamics, whereas the Lorenz equations are a toy model for the atmosphere.

Like with any toy model, you use the Lorenz equations when you want to model some basic aspect of a real system that you believe is already conveyed by the simplified model. It won't give you realistic results, but might be enough to give you valuable insight into the problem.

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The Lorenz equations are often used to think about Navier-Stokes dynamics from the perspective of "phase space". To do this, parameters such as distance and time from the Navier-Stokes equations must be de-dimensionalized and normalized such that coefficients in the Lorenz equations are dimensionless and only range from [0,1].

In order to do this, LOTS of assumptions about fluid flows need to be made.

To name only a few:

  • Fluid flows generate no internal heat (by way of internal friction)
  • Fluid behaves in a Newtonian way
  • Fluid conduction obeys Fourier's law (roughly meaning that the thermal flow obeys Netwon's second law, is smooth and doesn't have weird disjunctive thermal boundaries across the body)
  • ...other stuff

Once all that magic happens, you get static and variable coefficients like the Reynolds number, where the idealized dynamics of fluid flow can be examined as a set of state-based relations.

Confusingly, and contrary to what you might infer from looking at the Lorenz equations on Wikipedia, the Lorenz equations don't have to exist in only a single time-based form. Note the transformation from Navier-Stokes to Lorenz is a transform from a PDE to an ODE.

I found this link to be very helpful and concise at 'splaining the connection between the two equations: http://www.ub.edu/simba/slides/simba10102018JuanPello.pdf

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