Graduate Navier-Stokes solutions: Beltrami flow

Thread starter casparov
Start date Mar 24, 2024
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Navier-stokes Pde Viscosity

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Beltrami flow is recognized as a known solution for 3D Navier-Stokes equations, yet it is often categorized as inviscid in literature. The discussion highlights that Beltrami flow appears to involve kinematic viscosity, raising questions about its classification. The relationship between vorticity and velocity suggests a linear differential equation that accommodates viscosity and external forces. Participants are seeking clarification on whether the linearization process contributes to the perception of Beltrami flows as inviscid. The conversation ultimately seeks to reconcile the apparent contradiction regarding the role of viscosity in Beltrami flow solutions.

Mar 24, 2024

casparov

TL;DR: 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,

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

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 ?