[Physics] How to compute div&curl of volume velocity field with Helmholtz decomposition if velocity field is known?

[mikelawr at freenetname.co.uk]

Arend,

Hi. Just a small thought as I'm busy at the moment, but will spend more 
time when I can. The issue with anything going infinte when you let 
volume go to zero is common and is why there will not be a simple maths 
soltion. If you accept (as I do) that there is a smallest paticle of 
Planck radius, from which everything is built, then your integral will 
be bounded. Furthermore, since you use a lot of dimensional analysisin 
your argumens, a novel fundamental law is that volume x shear viscosity 
is a constant. So there will always be a drag within fluid motion at the 
basest level,whether that fluid is the aether or water.  To get that law 
requires using my form of dimensional analysis which is deeper than MLT 
etc, as seen in some of the papers I quoted in past postings.
Cheers
Mike



On 2020-06-16 15:50, Arend Lammertink wrote:
> Hi all,
> 
> I ran into a rather interesting mathematical problem around the limit
> of applicability of the Helmholtz decomposition:
> 
> https://math.stackexchange.com/questions/3721666/how-to-compute-divcurl-of-volume-velocity-field-with-helmholtz-decomposition-if
> 
> https://www.researchgate.net/post/How_to_compute_div_curl_of_volume_velocity_field_with_Helmholtz_decomposition_if_velocity_field_is_known
> 
> If you are interested or could forward this to a mathematician that
> could perhaps look into this and provide an answer, that would be
> great.
> 
> It should be noted that if this can be solved, this has tremendous
> consequences for the analysis of fluid dynamics problems, such as
> weather forecasting, the study of vortex behavior, aircraft design,
> oceanography, etc., etc. This is because currently, the potential
> fields have been defined slightly different and without the use of
> separate E and B components. Because the Helmholtz decomposition
> allows these components to be independently computed after which
> superposition may be applied, the definitions shared in this question
> undoubtedly have important applications within fluid dynamics, even if
> no answer to the question can be found because recourse can be taken
> to discrete mathematics as is done anyway with numerical analysis
> techniques.
> 
> Best regards,
> 
> Arend.
> 
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