Hydrostatic equilibrium and Navier-Stokes equations

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SUMMARY

The discussion centers on deriving the condition for hydrostatic equilibrium and the Navier-Stokes equations for self-gravitating fluids, specifically in the context of non-homogeneous density. Participants reference the Hamiltonian structure of fluids and express uncertainty about the applicability of variational principles, particularly when viscosity is involved. The conversation shifts towards exploring the Euler equations as an alternative to include in the variational framework. Key concepts include the relationship between fluid dynamics and gravitational effects.

PREREQUISITES
  • Understanding of Navier-Stokes equations
  • Familiarity with hydrostatic equilibrium concepts
  • Knowledge of variational principles in physics
  • Basic grasp of fluid dynamics and self-gravitating systems
NEXT STEPS
  • Research Hamiltonian mechanics in fluid dynamics
  • Explore variational principles applied to fluid systems
  • Study Euler equations and their implications in fluid dynamics
  • Investigate the effects of viscosity on fluid behavior in gravitational fields
USEFUL FOR

Researchers in fluid dynamics, physicists studying gravitational effects on fluids, and advanced students in applied mathematics or engineering focusing on fluid mechanics.

tom.stoer
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Is it possible to derive the condition for hydrostatic equilibrium or the Navier-Stokes equation for a self-gravitating fluid - e.g. for water on a planet with non-homogeneous density - based on a variational principle?

(the planet itself is assumed to be a fixed hard core not subject to the variation)
 
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My answer is I don't know. But I do know there is work on the Hamiltonian structure of fluids (see link). I also imagine relativists have work on fluids coupled to gravity in a Lagrangian formulation. However, since you want to produce Navier-Stokes which includes viscosity I worry that might prevent the use of a variational principle.
 
Haborix said:
My answer is I don't know.
Hm, are there other replies than just that?

Haborix said:
But I know there is work on the Hamiltonian structure of fluids (see link).
Thanks a lot. I'll check that as soon as possible.

Haborix said:
However, since you want to produce Navier-Stokes which includes viscosity I worry that might prevent the use of a variational principle.
Not necessarily including viscosity. I would be glad to see any variational principle. So let's discuss with the Euler equations instead.
 

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