From: jos.bergervoet@xs4all.nl   
      
   On 20/11/22 10:08 PM, Rock Brentwood wrote:   
   > On Sunday, November 8, 2020 at 10:18:06 AM UTC-6, Jos Bergervoet wrote:   
   >   
   >> The Navier-Stokes equations can be simplified in two ways:   
   >> by putting to zero the compressibility and/or the viscosity,   
   >> which then leaves us with 4 cases..   
   >   
   > I won't repeat what was said in an earlier reply (about surveying the   
   > field before jumping in),   
      
   Yes, but surveying the field was exactly my aim! By posting in s.p.r.   
   I was hoping to find the experts' opinion about the state of affairs..   
   In particular: which of the 4 cases has been, or has not been solved?!   
      
   >   ... but will note a few things. The best way to   
   > address the problem is to remove the constraints and broaden it back out   
   > to the simple and elegant form   
   >   
   > d_t(rho) + del . (rho u) =3D 0   
   > d_t(rho u) + del . (rho u u + P) =3D rho g   
   > with constitutive laws   
   > (d_t + u.del) rho =3D 0 - non-compressibility   
   > P =3D (p - lambda del.V) I - mu (del u + (del u)^+) - the stress model   
   > where I is the identity dyad, and P the stress tensor dyad   
   >   
   > ... and to broaden it to include the *other* transport equations for the   
   > other Noether 4-currents of the kinematic group. The 2 equations above   
   > are the transport equations for mass and momentum. The kinematic group -   
   > the Bargmann group - also has kinetic energy, and *especially* angular   
   > momentum and moment. These transport equations should also be included   
   > and the whole system dealt with in its entirety   
      
   The additional equations will be added as constraints, like angular   
   momentum conservation is a useful constraint in solving for elliptical   
   planet orbits?   
      
   >   ... especially the   
   > equations for angular momentum, because this figures prominently in the   
   > actual fluid dynamics that come out of the Navier-Stokes equation!   
      
   To 'broaden it' as you write, seems like the opposite of what I was   
   looking for (looking at simplified cases) although adding constraints   
   of course does simplify things.. Still, I'm curious about the simple   
   question: which ones of the simplified cases have been solved already?   
      
   > You want to make money on this, and that's your motivation?   
      
   I was planning to solve one of the open millennium problems each year,   
   which would give me a decent income. :-) But OK, if it's not appreciated   
   I'll just have to predict the stock market. QM is well-suited for it:   
     https://phys.org/news/2018-02-stock-quantum-harmonic-oscillator.html   
     https://arxiv.org/abs/1009.4843   
      
   --   
   Jos   
      
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