NSE and a Taste of Cause-and-Effect Relations

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As much as I think a matter of interpretation should always have its place in science,
even if just for the matter of a straightforward explanation it may be expressed as such, the view that the curl of the velocity (i.e. vorticity) inducing a velocity field is clearly misleading and o n the harsh side, it’s incorrect (as explained thoroughly in Is velocity induction by vorticity a fallacy?.)

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According to Newton’s second law, the direct cause of any acceleration, or change in the
velocity vector, must be a net force applied to the fluid. In the absence of a gravitational or electromagnetic body forces there is no action at a distance in ordinary fluid flows. 

In the effectively inviscid flow outside of boundary layers, a pressure gradient is the only source of such forces. One may easily claim that the pressure field causes the accelerations in the velocity field from a distance.

Nevertheless, it is a fallacy to see the causation as going only one way. keeping in mind the reciprocal relation between NS dependent variables (turning to an Eulerian viewpoint). The pressure gradient causes the accelerations, and the accelerations sustain the pressure gradients. The obscure meaning of this is that the cause-and-effect relationship between the flow field and the integrated pressure force on a body is reciprocal and mutual.
The best way to refer to the relationship between pressure and velocity is as circular reciprocal relation.

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


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