Each and every engineering student finds himself confronted with the Biot-Savart law at one point or another of their undergraduate studies whether it is related to fluid mechanics or classical electromagnetics.

The Biot-Savart carries along the qualitative idea that knowing the curl of a vector field at one point allows us to infer something about the vector field itself at another point.

As attractive as the idea is, it’s often misleading as it frequently leads to confusion concerning cause and effect.

Moreover, the fact that Navier-Stokes equations may be straightforwardly transformed from velocity to vorticity formulation and the use of potential flow related models to create obstructions to the flow strengthens the Biot-Savart frequently inferred view that vorticity induces velocity.

Well, here lies the fallacy. **In the absence of a gravitational or electromagnetic body forces there is no action at a distance in ordinary fluid flows.**

Casting the equations in one form or another and appealing to the Bio-Savart law as a calculus relation between a vector field and its curl does not mean a vortex at point A can cause a velocity at a remote point B.

In conclusion, although the claim that a mathematical relation as the Biot-Savart allows us to infer both quantitative and qualitative information about the velocity field at a distant point is true, in fluid mechanics it does not represent the physics and such a direct cause and effect relation is somewhat misleading as opposed to its counterpart analogy in classical electromagnetics.

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I think there as a need for a deeper look into the definitions and assumptions. Although I could understand where the misunderstanding comes from, in order to discuss it seriously, one has to rely on the textbooks and/or reviewed literature. I suggest to start with Lamb (1932) Hydrodynamics https://archive.org/details/hydrodynamics00horarich

and a comment:

“In the absence of a gravitational or electromagnetic body forces there is no action at a distance in ordinary fluid flows.” – this is at least in one aspect incorrect statement because you neglect the pressure here. Of course pressure is a property of the fluid which has nothing to do with body forces (gravitational, electromagnetic, etc.) and it’s a scalar field that has to solved/provided with the boundary conditions to solve the full problem (as we know the Navier Stokes is at least 4 equations for the incompressible fluid, not 3 and the unknowns are velocity vector field and the pressure field). So, there is a action at distance through pressure.

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Hi dear Alex.

As much as I think a matter of interpretation should always have its place in science, as you’ve mentioned, pressure is not a force. Keeping in mind that NSE is a momentum equation, induction of velocity to let’s say a static fluid parcel (Lagrangian viewpoint and a static fluid parcel for the sake of simplicity) will be somewhat formally problematic to be presumed as a direct cause of pressure.

Even if just for the matter of a straightforward explanation, it may be explained as such, and in any case, the view is incorrect for the curl of the velocity (I.e vorticity – subject of the blog post).

Nonetheless, keeping in mind the reciprocal relation between NS dependent variables (now returning to an Eulerian viewpoint), I am not certain one may easily claim that velocity is induced somehow by pressure and not the other way around. I think the best way to refer to the relationship between pressure and velocity is as circular reciprocal relation.

Saying all that, this is perhaps what is so amazingly wonderful in NSE description of flow in general and in particular turbulence.

Thank you very much for the kind response,

Best

Tomer

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Hi,

Your post and further comments are rather useful. I believe, application of equations depends on the task definition. One can use surely potential flow eq. with so-called free-vortices accompanied by appropriate BC to model large scale turbulence adequately enough for practical and even cognitive analysis.

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Hi,

Your post and further comments are rather useful. I believe, application of equations depends on the task definition. One can use surely potential flow eq. with so-called free-vortices accompanied by appropriate BC to model large scale turbulence adequately enough for practical and even cognitive analysis.

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I would be allays somewhat careful of explicitly stating the Cause-and-Effect Relations existing in confronting NSE. Here’s a short discussion on the subject: https://cfdisraelblog.wordpress.com/2017/08/19/nse-and-a-taste-of-cause-and-effect-relations/

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What I understand by velocity induced by vorticity (which is not necessary a vortex, can be vorticity in a boundary layer) is due to viscous forces between fluid elements. For example, if there was a boundary layer near a wall, then the flow above the wall within a certain distance will be affected due to the vorticity in the boundary layer, because of friction between fluid elements that will transmit the presence of a wall. And I think scientists saw this interpretation as the perfect analogy…

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I would be allays somewhat careful of explicitly stating the Cause-and-Effect Relations existing in confronting NSE. Here’s a short discussion on the subject: https://cfdisraelblog.wordpress.com/2017/08/19/nse-and-a-taste-of-cause-and-effect-relations/

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