Beltrami flows have zero parallel gradients?

In summary, the wiki article for Beltrami flows states that the nonlinear terms are identically zero. The terms can be proven to be equivalent and cancel each other out, resulting in a linear equation. However, the reasoning behind the vorticity vector being parallel with velocity and causing the parallel gradients to go to 0 is unclear. This may be a mistake and the correct equation should be ##\displaystyle (\mathbf {v} \cdot \nabla ){\boldsymbol {\omega }}-({\boldsymbol {\omega }}\cdot \nabla )\mathbf {v} =0##. Otherwise, there should be a physical explanation for ##J_\mathbf{v} \cdot \mathbf
  • #1
TheCanadian
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In the following wiki article for Beltrami flows, it is stated that the nonlinear terms are identically zero. I can easily prove the terms are equivalent and thus cancel one another, to yield the linear equation. But after a bit of algebra, I don't see why the terms are zero themselves? Why would a vorticity vector being parallel with velocity result in the parallel gradients going to 0?
 
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  • #2
I think this is a mistake. It should be ##\displaystyle (\mathbf {v} \cdot \nabla ){\boldsymbol {\omega }}-({\boldsymbol {\omega }}\cdot \nabla )\mathbf {v} =0## which is all we need.

Otherwise there should be a physical meaning behind ##J_\mathbf{v} \cdot \mathbf{v} =\mathbf{0}##.
 

1. What are Beltrami flows?

Beltrami flows are a type of flow in fluid mechanics that have the property of zero parallel gradients. This means that the flow velocity vectors are always perpendicular to the surfaces of constant vorticity, resulting in a simplified mathematical description of the flow.

2. How do Beltrami flows differ from other types of flows?

In other types of flows, such as irrotational or rotational flows, the velocity vectors are not necessarily perpendicular to the vorticity surfaces. This makes the mathematical analysis of these flows more complex, whereas Beltrami flows have a simpler and more elegant description.

3. What is the significance of zero parallel gradients in Beltrami flows?

The property of zero parallel gradients in Beltrami flows allows for a more intuitive understanding and mathematical description of the flow. It also has practical applications in fields such as fluid dynamics and plasma physics, where the use of simpler models can greatly enhance our understanding of complex phenomena.

4. Can Beltrami flows exist in real-world situations?

Yes, Beltrami flows can exist in real-world situations. They have been observed in various natural and engineered systems, such as in the Earth's atmosphere and in laboratory experiments with fluids and plasmas.

5. Are there any limitations or drawbacks to using Beltrami flows?

While Beltrami flows have many advantages in terms of mathematical simplicity and physical understanding, they also have limitations. For example, they cannot accurately capture certain flow phenomena, such as turbulence, and may not be applicable in all situations. It is important for scientists to carefully evaluate the use of Beltrami flows in their research and consider other flow models when necessary.

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