Navier-Stokes solutions: Beltrami flow

casparov · Mar 24, 2024

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,

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

and because w and v are parallel,

$(\mathbf {v} \cdot \nabla ){\boldsymbol {\omega }}=({\boldsymbol {\omega }}\cdot \nabla )\mathbf {v} =0$

, yields a linear DE

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

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 ?

Navier-Stokes solutions: Beltrami flow

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