Navier-Stokes solutions: Beltrami flow

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SUMMARY

The discussion centers on the characteristics of Beltrami flow as a solution to the 3D Navier-Stokes equations. It highlights a discrepancy in the literature that categorizes Beltrami flow as inviscid, despite the presence of kinematic viscosity in the equations. The conversation raises critical questions about the role of viscosity and external forces in modeling these flows, particularly regarding the linearization of the governing equations. Participants seek clarification on why Beltrami flows are classified as inviscid despite these considerations.

PREREQUISITES
  • Understanding of 3D Navier-Stokes equations
  • Familiarity with Beltrami flow concepts
  • Knowledge of kinematic viscosity and its implications
  • Basic grasp of linear differential equations
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  • Research the implications of kinematic viscosity in fluid dynamics
  • Study the derivation and properties of Beltrami flow solutions
  • Explore the role of external forces in fluid modeling
  • Learn about linearization techniques in differential equations
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Researchers in fluid dynamics, mathematicians studying differential equations, and engineers working on Navier-Stokes solutions will benefit from this discussion.

casparov
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Question on viscosity with Beltrami flow
There are some known solutions for 3D Navier-Stokes such as Beltrami flow.
In the literature these Beltrami flow solutions are said to not take into account viscosity, however when I read the information on Beltrami flow, they do seem to involve (kinematic) viscosity:

From incompressible vorticity eq,

{\displaystyle {\frac {\partial {\boldsymbol {\omega }}}{\partial t}}+(\mathbf {v} \cdot \nabla ){\boldsymbol {\omega }}-({\boldsymbol {\omega }}\cdot \nabla )\mathbf {v} =\nu \nabla ^{2}{\boldsymbol {\omega }}+\nabla \times f,}


and because w and v are parallel,
{\displaystyle (\mathbf {v} \cdot \nabla ){\boldsymbol {\omega }}=({\boldsymbol {\omega }}\cdot \nabla )\mathbf {v} =0}
, yields a linear DE

{\displaystyle {\frac {\partial {\boldsymbol {\omega }}}{\partial t}}=\nu \nabla ^{2}{\boldsymbol {\omega }}+\nabla \times f.}

Which is satisfied with Beltrami flow. With nu the kinematic viscosity and f an external force.

Is the problem the linearization ?
Also, might the necessary viscosity also not be modelled through a suitable "external" force expression?

Can someone explain why Beltrami flows are considered inviscid solutions ?
 

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