Fractal foundation of Fluid Mechanics

[jehovajah]

The time varying transmission of force through a viscous medium surrounding a rotating body will be proportional to the inverse square of the radial distance. In fact it will be better described as the logarithm of the radial distance  and the arc motion of the interacting bodies, as per Cotes formula.

ix = ln( cosx + isinx)

Where cosx is some function of the radial distance related to the radian measure x or more generally r(x)( cosx+isinx)

This is a bit restrictive but the point is that the inverse square law is an approximation of a more general formulation.

The time rate of transmission of force was ignored previously because the assumption was that space was empty. Thus force had to be transmitted mysteriously and instantly. However we know s
Space is in fact a viscous plasma with variations in viscosity, so the time rate of transmission has to now be included. It is a consequence of spherical geometry that the tangential surface varies as the square of the radius. However this implies an inversion or the force transmission because the spreading strain decreases not increases with surface area. The inverse square law is a rough guide to tangential forces, and by the Reynolds number discussion to centripetal and centrifugal radial forces.

We know from empirical experiments with rotation solids stressed from the central rotation that the transmission forms a Rayleigh like wave and takes time to spiral outwards and then reflect, by relaxation back into the centre. We also know that eventually a region will attain a block rotation if no other external forces are contingent.
The gravitational effect is a state of disequilibrium  while the viscous fluid assymptopically arrives at steady state,

[jehovajah]

For most viscous mediums there is a rate of force transmission that overcomes the relaxation limit. Particularly in water nd gases this is a very low rate. The result is the separation of the rotating regions into " bubbles". These bubbles develop yjeir own centres of rotation as a result of viscous forces achieving equilibrium. However, this very much depends on the viscosity of the medium and the surrounding media which becomes interstitial on the break up of a viscous region.


Vortex shedding is a commonly seen evnt in flowing water, indicating that rotating regions are dynamic in many ways, and that may not allow bubble formation.

However if bubble formation occurs these interlinked regions will usually have anti spin so that the contact tangent is uniformly directional. This leads to normal forces that do not repel the bubbles. . These structures may form chains similar to gear chains, eventually collapsing as the rotational energy dissipates. Bubbles in the sink ay look static, but the surfaces are highly active, and bubbles coalesce into larger bubbles under these conditions..

The gear chain effect is often associated with antennae and electric current, but in astronomical space or in the atmosphere they may be seen as lightning. As a rule they are short lived at our scale, requiring a conductor to catalyse them, but at huge scales the dynamic rotational energy dominates and these chains can last for aeons!.

The viscosity of space is such a counter intuitive notion that we invent different names for the phenomena. The electro static nomenclature has perhaps done the most to obscure the bubbles of counter rotating space. But recent research findings have supported Ivor Catts long held research findings that there is no static electric charge in a transmission line or it's equivalent a capacitor.  Plasma physics recognises this " double layer" effect in plasma and every planet in the heliosphere has such a surrounding bubble similar to the Van Allen radiation belts.

These are highly energetic active regions as you would expe t from counter rotating space.

[jehovajah]

I generated this sculpture years ago, but then I only suspected it had something to do with gravity. Now I have shifted paradigms to a fluid mechanical one I am more certain it is a model of electro Thermo magneto complexes as viscous space.

[jehovajah]

http://claesjohnsonmathscience.wordpress.com/2012/02/22/why-it-is-possible-to-fly-yvfu3xg7d7wt-18/
This important breakthrough has many implications , including in the diffraction propagation of light .
Google "jehovajah vortices " for more details.

[jehovajah]

http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
This geometry may be useful to describe general pressure systems in fluids. However, the choice of dot products may need to be made flexible rather than rigid to accommodate different views of a system and discrete behaviours.

[jehovajah]

<a href="http://www.youtube.com/v/RGevKSlT0yY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/RGevKSlT0yY&rel=1&fs=1&hd=1</a>

[jehovajah]

fast Fourier Transform by Zedzero on Vimeo.

http://vimeo.com/zedzero

Understanding that the Fourier transform is based on the general concept of " periodicity" . Now this is a "mash up " concept. The underlying Spaciometry is the circle in the plane. Fourier was interested in describing how " heat" flowed in a conductor and he chose a metal ring!  The temperature at a fixed point was increased and the pattern of temperatures in the ring was modelled at various time intervals.

The parameters are the quantities usually independent , that are measured by some Metron.  Position was therefore measured by polar coordinates r,ø which can be transformed into orthogonal coordinates rcodø,rsinø which might be denoted as trig line segments x,y. Temperature was measured by mercury expansion. Thus this " temperature" is in fact a heat pressure, comparable to atmospheric pressure also measured in height of mercury!

So for each position a mercury height could be plotted. The fourth parameter is again an independent measure, usually the duration of the experiment measured by a metronome.

By recording these data sets in tabular form Fourier presented the scientific world with a technique of statistically managing sets of data. We have to review the work of Gauss and Boltzmann to find similar statistical management. But it was reputedly an idea of Laplace that Fourier developed extensively. The idea is based on the representation of a polynomial form by simpler polynomials that sum to it.

The purpose of representing a polynomial by a sum of polynomials was and is to be able to handle discrete information by a general polynomial that contains it. What this means is best illustrated by the function or polynomial y= x².

Typically we calculate a few discrete values, thrn plot them and join the plots with a smooth , but estimated curve. In one sense we have no way of knowing if that curve is right or rather " true" without calculating the intermediary points. This calculation is called interpolation. Thus it became clear that if one only used straight line equations one could approximate the curve by summing line segments from many lines,

The process clearly is based on difference algorithms and consequently in the limit melds with the calculus methods of Descartes , Leibniz and Newton . Thus the curve can be considered as the envelope "sum" of the tangents at each point.

The question that arose is what was the best fit for any number of discrete points ? Could you recover the " true" polynomial by this approach?  In general the newer is no. But under certain constraints you could converge onto the true curve.
Thus it was shown that you could model any complex curved line by a sum of appropriate line segments, and eventually by a minimal choice of polynomials of a lower degree. Within the given range the convergence could be as close as desired, but outside the range the summed polynomials were seen Togo " crazy" . Extrapolation was definitely not possible!

A Fourier transform was based on Fouriers use of the trig functions of Euler. It was known that these functions could disappear if the amplitude was at the right phase in the rotational cycle. This meant and Fourier demonstrated that over the unit circle any complex form could be modelled by a summation of sine and cosines at different frequencies of rotation. The frequency at which the angle was considered to turn determines the phase of the amplitude .,these would thus cancel or add accordingly creating complex patterns of highs lows and blanks.this was called interference either constructive or destructive. Eventually it came to be called interference by superposition.

Now it was clear that in a ring if this dynamic rotation was occurring, then time would be trapped in the ring, and the angular rotation of these sine and coine waves would also be trapped in the ring by polar coordinates. Thus any amplitude would show up as interesting patterns in the ring. 

Nobody knew or could think how heat was transmitted or conducted, but it seemed it flowed from hot to cold. Fourier wondered if it rotated or oscillated through the medium. If so the ring would show this by a travelling amplitude of oscillation which he could measure by temperature.

Actually his ideas for heat were misconceived, but his analytical method revealed how any complex pattern, conformed to a ring could be precisely modelled by sine and cosine summations.

His heat idea was misconceived because he eliminated the rotation concept, replacing it by the sine or cosine concept. Typically with the thought of his day and even today, the imaginary component of the exponential forms were always thrown away.

Heat flows from a higher temperature to a lower temperature Newton observed. Fourier never explained why, but he did show how a temperature form was represented ar different times in the process.

The reason heat flows is explained by Newtons laws of motion. The heat indeed increases the quantity of motioning a region. However, unlike a blow, the heat Los changes the bulk properties of the material. Thus the increase motion is not uni directional, the expansion is not an elastic deformation, except at the edges and the viscosity of the material leads to a longer relaxation time, a Thermo relaxation called thermal contraction. It takes time! Thus a pressure wave does travel in the ring, but with all these constraints varying it moves so slowly governed by other factors like latent heat and specific heat rather thn bulk material properties of compression.

That is not to say these transmissions do not happen. They do, but typically in a heat conduction experiment the parameters measured are not the acoustic ones!

The Fourier transform is sometimes confusingly presented as converting time into phase. This video shows precisely what is happening.a spatial forms modelled by several parameters only one of which is time.once the spatial form os modelled as an interpolation of some fixed positions and amplitudes at those positions. Then a time argument is devised to add to the spatial form so as to move it along.

Of course spatial form cn alo change with time. In this case the additional time argument is combined ith a time dependent form of the spatial disposition. The form however it is disposed in time cannot exceed its boundaries or the fixed point values by which it is defined. If these values do vary with time a massive compute is entailed and that is why the fast Fourier transform algorithm has been so important to the study of Dynmics.

The wave concept is a misnomer, in my opinion. The rotation or swivelling function is what was meant by undulation, and the progression of a deformation in a medium as a consequence of resident forces in the medium , ie inertial forces , is what deformation propagation is.

Newton in proposing light was a ballistic corpuscle was certain that it's collision with the retina produced the sensation of colour .he did an experiment to confirm this! However later he discovered colour could be described by angle of di
Refraction. This did not alter his opinion that sensation required collision with the retina. There were other details he chose yo overlook because he believed his model was the most straightforward explanation. Without a clear understanding of superposition of deformations in space, Huygens could only suggest that wave amplitudes joined to make a wave front, as circles do in a Moire pattern. By the time of Fresnel, Arago and others had a speed for light , Fourier had a method for describing complex forms that demonstrated superposition, and Young had shown interference patterns in ligh. Newtons ballistic concept had to go. But what replaced it was an arcane question bout form!

It is irrelevant whether a wave is shoed like a corpuscle or like a sea wave! What is and always has been the issue is the existence of a viscous medium for the propagation of light, whether as wave or corpuscle. That medium was called the luminiferous aether and it was thought to exist, until some disputed it. Today it is again believed to exist but it is called quantum energy etc.

Light is a strain that travels in this medium by Newyonial principles of motion and transmits energy throughout this viscous space. My best guess is that the form of this strain is a complex trochoidal rotation in the medium, modelled by Fourier transforms in 3d. The motion is generated by the restoration of equilibrium to this luminiferous aether or quantum energy .

<a href="http://www.youtube.com/v/r4Pc-rGBRJA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/r4Pc-rGBRJA&rel=1&fs=1&hd=1</a>

[jehovajah]

<a href="http://www.youtube.com/v/w-XMUQBlcEY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/w-XMUQBlcEY&rel=1&fs=1&hd=1</a>

[jehovajah]

I hope this does not crash anyone's old vomputer.

<a href="https://www.youtube.com/v/Tfi8BLca07M&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/Tfi8BLca07M&rel=1&fs=1&hd=1</a>
The trochoidal nature of rotation is revealed by these computations. These behaviours are complex enough to explain magnetic behaviours which in turn define electric potential as radial due to contra spinning vortices that pass through each other.

The more I study vortices and fluid mechanics the more I understand that rigid dynamics is misleading . Fluids can abn do pass through each other with minimal mixing.

Also streak lines are different to streamlines as streamlines have a structure while streamlines are instantaneous tangent curves that ignore other forces on a material point that precisely give a streak line its observed dynamic structure.

[jehovajah]

Yesterday morning I awoke from a disturbed sleep. I had been researching the history of magnetism on YouTube. I had come upon the SLAC channel and found a lecture on the history of magnetism, but also a series of lucid expositions of various other related topics with the " common" touch. As I meditated on magnetism and other issues these lectures slipped easily into the background of my thoughts.

So it was that I awoke with a unified thought. That space is a fundamental fluid plasma which is in dynamic fractal regional and trochoidal relative motion.

This fundamental plasma consists in trochoidal motions at all scales and at all intensities of motion. In my perception I visualised a complex fractal pattern of plumes with associated vortex rings. This was my fundamental Newtonian motive structure as a fundamental primitive environmental plenum in which and of which I am and from which by formal analogies and analogous ratios and proportioning I must synthesise a model of behaviours observed in and of space at all scales.

My issue is epistemological. What can I know of this fundamental plenum which by its nature is in constant flux?

The Pythagorean solution is to define invariants..

The first and most primitive invariant is the inherent Logos response. In this response I interact with spaceas a conscious entity separating itself from all else, while defining itself by all else!. This I formalised into 2 formal and contra sets. The set FS and the contra set notFS. . These sets are formal and dynamic and as the set FS is dynamically defined so is the contra set as everything external to the set FS.

I describe this as a formal conjugation of Shunya, Shunya being the Sanskrit for Everything possible and impossible, probable and improbable, statistical and non statistical, distinguishable and indistinguishable, orderable and non orderable, dynamic and non dynamic. Essentially: Everything!

As a dynamic entity in a dynamic field my first formal invariant is to sing and dance.

The song I sing is a sequenced lyrical response to a regularly changing event. The dance I perform is a metronomic mimicry of the changing event. If I share that event with others it becomes the basis of all communication responses within the logos response. It generates and shapes a common language which distinguishes by sound and by movement a regularly changing event and becomes the progenitor among other spatial progenitors of conceptual order and sequence, both in dynamic relativity and invariant relativity.

As a " concrete" conception, through the syllabric development process it gives rise to a symbolic and speech act representation that is both alphabetic and numeric. At some stage in the history of the development of human writing these 2 aspect of the logos response become separated and distinct. At the same time later generations became confused about the origin of this bicameral logos response.

Sticking with the Pythagoreans I see a development, formal in nature, that captured analogous thinking in and by this bicameral brain, and this developing bicameral mind. It was called Logos and Analogos philosophy.

Later still it was developed and combined into Arithmos philosophy.

The rich vein of formally defined invariants in the Logos response constitutes a forml spatial philosophy which I defined as Spaciometry. Strictly speaking Spaciometry does not formally constitute until the Arithmos philosophy reaches a sufficiently complex form of expression but the intuitive logos response distinctions of sequence and order and distinguished form capture powerful invariant concept defined as " ideals" of pragmatic response requirements, language development requirements and social communication requirements.

You will find more detailed discussions on these topics in the Fractl foundations of mathematics thread.

Thus , returning to the fundamental conjugation of Shunya I can construct only formal models of the set FS. In so doing I may perceive the set notFS as a murky non modelable contra or inverse " reality". The more universal I claim my model to be, the more nonsensical any alternative cotra model appears to be.

It is therefore wise and less arrogant to consider any model as only an approximate or partial or limited model of the set FS. In fact, because the setFS is dynmic, any model defined on it must either be dynamically adjusted or regularly reviewed, redacted or discarded in the light of New data of a sort I will refer to as empirical.

My fundamental forml primitive is therefore a limited model of a complex constructed or structured primitive designed to be the ground of more formal constructions of metrical systems that is constructed measures based on the invariants of the philosophy of the Arithmoi.

[jehovajah]

Newtons philosophy of quantity marked a distinct break with a qualitative description of magnitudes and behaviours in space.

However his philosophy was firmly based on the Pythagorean principles in the philosophy of the Arithmoi. Using even the simplest expression of this as in Euclids Stoikeia, an introductory course to the philosophy of the Pythagoreans, produced a remarkable set of analogous measures. These measures though well understood were not employed in the service of a quantitative physical theory of the cosmos , that is a more primitive astrological theory of matter, before Newton.

Evenso Newton acknowledges that it was not for want of attempting to do so, for he garnered many of his principles of quantity from the ancient masters of philosophy of natural phenomena. It was chiefly his combination of the principles of Mrchanics with the principles of astrological measures and calculations that founded his philosophy of quantity.

The fact is that the purest ideals of Astrological measures are derived from the philosophy of the Arithmoi. Thehilosophy of the Arithmoi essentially means we can create an analogous mosaic form of any form in our cosmos. Thus every form has an analogous twin which we can explore subjectively by counting and comparing and analysing and synthesising internal relationships and patterns.  It is the invariance of these patterns that we use to found formal systems of measure and logic and comparisons.

At the same time these invariant relationships do not reveal to us the reality of any form, but rather our ability to perceive patterns in a spatial format, as an analogy. Thus they reveal how we may model forms and behaviours analogously without ever knowing the absolute truth of them and their behaviours!

Accepting this limitation was once the humility of philosophers, but later arrogant thinkers declared these analogous forms to be reality , and mans mind as the master of it all!

It will be suitably humbling to read Newtons explanations of his fundamental primitives of quantitative measures, both to learn how he applied the philosophy of the Arithmoi to this end and how we may do so with a greater empirical data set of fundamental behaviours. I am sure we will find that as Ampère and others did that Newton's principles are adequate to the task , but that we must expect to have to demonstrate conformity with the invariants in the philosophy of the Arithmoi.

In addition we now have Grassmanns tools and methods, which while not being different to Newtons own are in fact publicly explained! This was something Newton was loathe to do, and set the general enquirer at a considerable disadvantage as to how he arrived at many of his formulaic expressions for calculating quantities and measures.

Finally newtons first approximation was to the motion of bodies. Ours must be to the motion of fluids and for that we require a whole new conception of environmental structure and primitive motions. I believe that density and pressure will be the fundamental concepts of vis and vis viva, that is newtons fluid motive?

[jehovajah]

This is a very good series on fluid mechanics giving a sound overview. However the historical development sees the application of solid deformation mechanics to fluid space. The concepts of classical hydraulics were the basis built by engineers that led to computational fluid dynamics.

<a href="https://www.youtube.com/v/fa0zHI6nLUo&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/fa0zHI6nLUo&rel=1&fs=1&hd=1</a>

For the first time I have heard an explanation of Newtons  differential equation for Lubricity! It has been worked over through the years to establish a definition for viscosity and a definition for the rate of shear strain.

I am grateful for prof Soms explanation because it shows a fundamental mistake and how it has been redacted to make it " work".

Shear stress is dimensionally [L]/[ T]² the velocity gradient has dimensions [L]/[T][L]

This means that the viscosity is not a dimensionl constant.

Because of this mismatch the early equations tend to be littered with empirically derived constants and indices. The formulae thus have the feel of being squeezed into conformity with empirical data rather than applying philosphical ideas to pragmatic measure construction.

The issue can be resolved by using a stress that depends on time and space distribution . Then this shear strain concept will be defined by a partial differential relationship.

The analysis of the interaction between layers in the 1 dimensional case then can be seen to be misleading as the interaction is not one dimensional but 2 dimensional. The one dimensional velocity gradient is a diagramatic nonsense, as it requires 2 dimensions to represent it.

Further the analysis does not account for lubricity between the laminar flows. Thus one layer acting on another has to slip past without resistance or friction, otherwise we can expect oscillations to develop between the laminates, that is rotation that Makes the model inns curate ans t.he laminar flow should show interleaved streak lines consisting of rotating fluid regions, and laminar streaklines. Laminar flows should be filamental.

The analyis is in practice used for boundary layer calculations. In that case it would be more appropriate to use a terminal velocity analysis. In fact as a first order approximation to a terminal velocity solution this standard introduction is good empirically. The compressive resistance of a fluid is ignored in this analysis and so the dynamics cannot hope to be acceptably captured.

The analogy is comparable precisely to Hookes law of spring force and Newtons law of force. However it is best to avoid the measure force as a symbol of causation and focus on pressure as the fundamental symbol of causation of dynamics

[jehovajah]

While researching the flux moment of a rectangle, newtons fundamental definition of a flux in 2 directions at once, and from which he derives his rules of Fluxions and fluents without recourse to limits, I came across Newtons own words regarding the resistivity of fluids.

It is to be noted that the various laws are mathematical models in an attempt to understand what may best describe the observed behaviour. Newton in fact is so bold as to say they are non physical but purely mathematical!

The result of this experimentation lead him unsuccessfully to build a model of fluid behaviours. While his model came close it was nevertheless untrustworthy and misleading. In abandoning it he remarked that future generations might build a better model, not as is commonly taught that vortices d o not exist in space as DesCartes conjectured!

However DesCartes in Newtons School was hardly respected due to his arrogant writings. This should not obscure our view of his general genius in these matters of  philosophy and geometry. In fact reading his own words it is questionable if his arrogance was any greater than any other self publicist who must survive on patronage! The Puritan streak in the British character would have been highly offended I am sure, but landed gentry or Aristocracy could afford to adopt such effacing attitudes where feudal peasantry could not! In France all were peasants to the king!

[jehovajah]

So now I make progress by regressing.

It so happens that I have been in discussion with Barau A Tour on the Magnetic Universe orim about the fundamentals of Mechnics, and defending Newtons approach from misinformation. However I know my in ledge of Newtons Astrological Principles and the system of Worlds is very Sketchy. In particular it is speculative with regard to several early notions I held by my own translation of parts of a Latin version.

I m no Latin scholar and so only translate bits and bobs as I have interest. There are however some very good Latin translators who have translated the Principles into English. I avoided those until I had formed an independent view. Substantially my view is that most translators and scientists misunderstand Newtons concepts. Bearing that in mind I can now read a translation skeptic ally without credulity obscuring the translators bias.

So I returned to find the section on motive. This had bern a key liberating idea I drew from my translation activities. In essence I took motive to be the " cause" of acceleration on the basis of my translation of one sentence!

Running with this idea and a general paradigm shift on my part from solid mechanics to fluid echanics, I developed a notion of Newyonian fluid motive as cause of fluid motion and accelerations , that is pressures , pressure gradients etc within fluid element boundaries.. In short Netonian fluid motive becomes an alternative " energy" concept.

In a discussion with a " believer" who regarded the ontological argument as unshakeable logically for demonstrating the existence of a " prin" that is first mover who cannot or which cannot itself move, etc I was introduced to aspects of Aristoles research into motion and motive. On the face of it I was able to adduce that Newyon concocted his motive notions under Aristotelian influence. I have yet o thoroughly research hat hpothesis but it seems very plausible.

However it came at last to googles notice that when I googled Newton on motive, that several,scholarly works were available for inspection. In them I found a detailed discussion on Newtons Force concepts in which it became apparent hat motive was part of a triumvirate of concepts of the notion vis and central acting vis at that. Within this context it did not play a causative role at all, but rather a summative role for intel parts of a body . In fact absolute vis was the central cause and accelerative vis embodied effect, with motive vis being the distributed internal action capable of being summed variously.

This was a triumvirate of conceptual action from and around a centre that was purely mathematical nd not physical.

The mathmatical design was so Newton could apply geometrical modelling. Thus absolute vis emanated from the body centre as the cause of all propensities to move within its influence. This then was the ground by which the parts various and any of a body moved in relayed bodily motion acting on each other and other nearby external parts, either in concert or in opposition. Thus these motive parts were admissible of summation relative to each other and the central cause and the external others.

[jehovajah]

Continuing with this brief drive through newtons vis concepts we come to the accelerative vis. This was an effect of the other 2 in which from the centre and round about it it may be observed that a bounded body as a sum of its parts is in acceleration in a distinguished direction. It chiefly transmits cause on impact with other bodies but as a measure it distinguishes preferred is placements in equal times for the whole body omplexity.

We can see then that Newtons scope, though  founded on central emanations of various sorts was to measure the behavioural interactions of particulate bodies fom,the outset.

Being thus founded on central emanations it freed his system from any particular scale or quantity, and allowed that any body thought of as a whole was reduce yo the summation of its parts should that transpire to be the case.

Motive played the cruise ole of being the summation principle of the parts.

Absolute vis , underpinning everything, no longer needed to be discussed in his economy. Motive and accelerative vis were the bedrock of his measure system.

It is important to grasp that the accelerative vis is the product of the motive vis within a body. Should all the vis act in one way the accelerative vis is a sum of these which assuming uniform parts quickly allows a multiplication form to stand in general.the multiplication fom is crucial to all geometrical models, nd relies on books 2, 5-7 of Euclids Stoikeia if a Pythagorean expianation is required. The alternative explanation comes through the Aristotelian scholars of the Islamic schools. Newton likely studied both, but that is not certain because evidence suggests he relied on a vast information processing network to verify his postulates, which clearly became accepted as propositions of " truth".