Fractal foundation of Fluid Mechanics

[jehovajah]

Why would I need a product or a multiplication process in describing a physical system behaviour?

The nature of dynamic systems is that they vary. They are thus naturally expressed in differential likenings..

Newton's Fluxions are based on the binomial product design. Thus his solution to differential likenings will be a binomial product of some description.

Product design is also conflatable onto function theory or vice versa..

Thus if you can design the product for the system you wish to explore you do not need to use the inferential equation approach.

In general products are the way any system likenings capture rotation in a system as well as extension/ intension

[jehovajah]

While I have posted this video in V9 thread it naturally belongs here.

<a href="http://www.youtube.com/v/eCJxhxDMIb0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/eCJxhxDMIb0&rel=1&fs=1&hd=1</a>
http://m.youtube.com/watch?v=eCJxhxDMIb0

The hyperbolic design is reversible in the sense that flow through a small hole or venturri channel should work optimally if designed accordingly.

[jehovajah]

Stress stressors topology and magnetic bonds.
One of the plane facts is that matter is variable in its intensity and that it has continuous and discrete forms .

Further we recognise energy -phase changes that is to say simply : solid liquid gas plasma. States of matter with different vibrational signatures.

It was determined by Newton et al that heat was best described or pictured mentally as a vibration in a corpuscular region corpuscles were thus conceived as flexible plasmas from the nascent biological discoveries of the time.their flexibility was merely a way of accommodating the vibrational modes conceived as heat .
We have since developed the heat model into the Energy ,Odell.nsuch a model really has its foundation in Gilbert's magnetic philosophy which recognised 2 forms of magnetism: crystalline lattice or mineral magnetism and biological magnetism, that is organic lattice type. The one was called the iron or ferro magnetism, the other Electra magnetism or amber magnetism.

These specs were later intensively studied and distinguished into models that were fundamentally mysteriously related. However the various guilds of magnet makers and electric charge makers  jealously hung on tontheirbtrade secrets and further distinguished the common link, that is the fundamental material experience and phases of matter at different energy or vibrational states.

Matter and space were philosophically divided in the most natural way , but space was called empty for religious reasons. It was desired that spirit be distinct from matter, and invisible able to manifest to conscious souls, themselves spiritual. 

Such a model kept the material world that technology was exploiting as a distinct environment that mankind could "play" with like a child, but spirit was the playground of the gods and those ascended humans who struggled to grasp it.

Of course the material philosophers were brought up in that Matrix of ideas ad models, but younger more ambitious material scientists saw no valid reason to limit material science in this way and technology indeed advanced to miraculous inventions gadgets and contraptions which supported that viewpoint. 

Unfortunately the mathematical clan or guild retained enough obscuranting mystics to cover the material  viewpoint with a pseudo mystical or religious analogue of religious theology . Bishop Berkely wrote a powerful attack on mathematical religious views pointing out that material scientist of the mechanical philosophy were in fact no more clear of superstitious nonsense or beliefs and practices than religious theologians and clergy

Accepting that point we may progress toward a common view proposed in detail by Hegel et sl and eastern mystics since way back, that spce is in fact something that is connected to a singular concept that contains all: if you like spiritual and material are two aspects of an entity with many aspects.

The Aether or Ether philosophies attempted to place that view as foundational to the material sciences, but all clans and guilds fell apart in disarray: a modern Babel of jargon and gobble de gook .

We do not have to be so inclined.  A simple and useful combinatorial model underpins the technological achievements of mankind. It is rather humble in its scope and often  symbolically and ritually obscure. Attempts to make it clearer usually do the opposite, academically. But those who actually make things using their whole organism develop inexpressible expertises that often seem to those not involved in the physical actions of making , well they seem like magic, even magycke of the old kind.

Regularly organic and rock crystals cry out to us ! We call them TV's and Radios!! 

So we can take material science or specifically the science of material and develop an expertise that Tribology or tribologists express. This level of expertise is an art form found among the few material scientists with a broad enough range of experiences and understanding. It is eclectic and very much in its infancy .
For example, topology would help to provide a structural overlay akin to corpuscular philosophy of Newtons time. Topology fundamentally is a combinatorial art that gives forms or formats for space in its most general Aetheric model . These formats may then be used to count and measure regions and distinguish behavioural changes in material over time, circumstance and internal and external conditions. 
Topology allows us to place atomic or molecular models into regions of space so we can characterise certain regions by their phase and nearby status and their visual characteristics . The stresscandvstressors in such a space become characteristics of the phase of the element or molecular material as well as contact behaviours between such characterised materials.

Clearly a chemistry and organic/ biochemistry based on these characteristic measurements and observations has proved very fruitful in organising our technological advances. But we have found some who want to isolate these behavioural and technical expertises away from common knowledge to support some particular world view or religious ideology. 

While the knowledge is presented as iconic, eclectic, difficult to understand, even iconoclastic, it is in fact not. The simplicity of the energy/ vibration - phase model of matter is how it has been hidden in plain sight.

It is a simple robust and powerful combinatorial system that was advanced by Navier Stokes and yes even Newton into the fluid dynamic model. Helmholtz and Kelvin added the Vottex model rules (inaccurately: see Claes Johnson work) but all of this was obscured by the Mathrmaticians. Helmholtz following perhaps Euler and Lagrange felt that the differential equation format was the safest way to express natural laws of motion in space-like material especially if they were fluid.

In a sense , given the prevailing attitude that aether was basically solid billiard balls at the femto metre scale( an assumption since demonstrated to be misleading especially by quantum Mechanics) his reliance on the continuous nature of differential equations by design does preserve the fluidity of the foundational corpuscular theory of Newton et al. 

The use of hard billiard ballasts representing corpuscles is a mistake we can trace back to Leibniz et sl . But even then he was not insisting that matter was thus. The mathematical topology was simpler if this was assumed , and produced useful models. It also produced strange aberrational models which rightly were ignored! Now they are mystically clumped together in the quantum mechanical world viee.

The fluid mechanical expertise is very difficult to establish except in the simplest cases, but in fact computationally it has provided us with some excellent models within the boundary conditions appropriate to the study. The advent of more powerful computers and computational schemes has made it possible to model complex data increasingly accurately, by applying a whole mish mash of equations topologically to a spatial region.

The one fundamental breakthrough for physical topology is ascribed to Benoit Mandelbrot. His Fractal Geometry  inspired many computer geeks to model the particular types of inductive or iterative( recursive) equations that hitherto were avoided by Mathematicians as monsters. Ad yet we know Focault and many others were studying them quietly prior to the wars with little academic support . 

We can laugh, but it was mathematicians that were the Luddites when computers were first introduced!! 

Today I watch a lovely flowing scene of actual waves lapping onto a shore with all the sound effect on a flat, crystalline organic matrix presenting the display of raw data as processed by billions of logic gates in some silicon(rock) chip processor with a certain vibration/ energy which has to have the heat removed by invisible gases passing over it as winds! 

The fluid dynamic models expressed as differential Equations may be nonsense. But the expertise that organises itself in this way around the energy phase model really has a powerful set of models with which to develop further models and technological advances .

Why have I not mentioned Maxwell, and Einstein? Because they represent a deliberate mystification of the simple model. Faraday was not happy with how Maxwell fundamentally distorted his ideas. In fact Örsted Faraday, Ampère Volta and others held that a circular force o rather a rotational force was fundamental to our understanding of all models built on observation. Newton, unlike his successors took a circular or rotational force for granted. 

This simple topological element distinguishes models that account for quantum behaviours from those that do not. 

And yes the mathematics becomes confusing and complicated, but only when the circle and the trochoids based on circles are excluded.

Include trochoids in space in your models and you can model magnetic behaviours topologically in a consistent way.

The gravity of this point should not be missed, as Gravity to this day is still touted as a mysterious force,  all forces are fundamentally mysterious characteristics of topologically described space. Their iscavsimple reason why conic curves describe both gravity and magnetic attraction and repulsion. The topology of space that best describes them is rotational or circular. Add to that fractal geometrical topologies and you can model fluid dynamics in appropriate small boundary conditions.
Add phase changes at different energy and thus vibrational states( frequency amplitude and trochoidal topologies) and you arrive at discrete regional distributions with phase changes marking the boundaries of regions fractally.
What we know about bonds is dependent on these fractal regional surface phase change behaviours at differing energy/ vibrational statuses

We require a new label for application of fractal geometry to real physical behaviours and forms : fractal topology .

[jehovajah]

Claes Johnson helped to pioneer the finite element model of computational solutions to partial differential equations . Another way of saying that clearly is the Fractal Element Model.

We can see how complex the tedious algorithms are , but the results are not only intuitive but spectacular for fluid dynamical modelling.

I really feel that we can dispense with the high mathematical presentation and get down to brass tacks. Fractal applications should be able to implement the regional designs to produce the same results .

http://m.youtube.com/watch?v=hQjo4DkSFyA
https://m.youtube.com/channel/UCCPenA6XwM3REzmqoAXMw5A

[Max Sinister]

Just wanted to tell you that I plan to read this thread in the future, but at the moment I can't say anything yet about it.

[jehovajah]

Just wanted to tell you that I plan to read this thread in the future, but at the moment I can't say anything yet about it.

Welcome aboard!
I hope others too will contribute to the thread.
I post occasionally as insights come to mind and to keep the thread alive.

Currently I am gradually exploring the progress in this field to glean insights especially for understanding some fractal sculpture outputs, but any contribution or questions are welcome

[jehovajah]

http://publishing.cdlib.org/ucpressebooks/view?docId=ft4489n8zn&chunk.id=d0e16359&toc.id=&brand=ucpres
This guide to the Natural Philosophy of the Pronciple for Astologers breaks down the key structures of Newtons presentation . It is important to read the Latin presentation before assuming the English translation is accurate.
Now my conception is that our proprioception of space and spatial dynamics is necessarily representational of a fractal mesh of sensory experiences synaesthesically ombined . Thus a superpositional or interference pattern of sensory signals underpins our conception of extension and intensity of magnitude( that is extrnsive and intensive magnitudes, so called) . The consequence of this is that the simple must be conceptually derived from the complex by analysis. We thn construct the complex by synthesis of our analytical products.

So Newton attempts o didactically impart this notion in synthetical steps after he had done the Anlysis privately and to his satisfaction.
It is the classical duty of all Rhetoricists to impart instructive information at the level of their audience!

Thus we find axiom, definition, proposition, Lemma , demonstration all woven together as set out by some of the greatest classicl master taught. We name only a few such as Plato and his version of Pythagorean scholasticism, Aristotle and his critical revision of Pythagorean ideals, Eudoxus,Euclid and Apollonius .
To these we may add some of Newtons contemporaneous teachers and Philosophers especially Wallis, Barrow and Hooke et al.
For me then a concept of pressure precedes any concept of force, just as a concept of the experience of magnitude precedes any concept of motion .

Finally the concept of spatial distribution of experiences of magnitudes now distinguishable by dynamism into at rest or in motion relative to the observer and perceivable as both intensive experience of magnitude nd or extensive experience of magnitude precede the concpt of Tyme or in modern guise Time.

Tyme itself is a conception of record keeping , whereas by the epoch of Galileos pendulum and Huygens formulaic expression of the pendulum Period time had become an experience of the regularity of motions.

We may go back to Timmaeus and other nicest engineers of worders who expressed the records of spatial positions by ratios of lengths of arcs on the circular perimeters of wheels and gears thus animating ' dead and lifeless " records and creating miniature models of their "Universe"
It is eye opening to realise that our brightest minds only a century ago thought the Milky way was the universe and now we think it is only a small local part of it!!

In defining pressure therefore I acknowledge that a space or region in space is always assumed, and itching that region a surface divides space into  two relative to which side of the surface the remaining space is situated.
Thus a surface is hardly defineable if it does not have the extensive magnitude that dissects spatial perceptions into at least 2 relative positions we may call " sides. The record of point like objects such asvstarsvor planets therefore are conceived to be on or in a surface and on one side of that surface. Classicàlly that surface was universally designated as a spherical one with the gods being on the other side of that surface essentially unknowable but conceptually present.
As we have pushed back the radius of that sphere, the diameter though huge has not expunged the notion of sides to surfaces in our universal knowledge .

Today we have an alternative to this central topology and that is a fractal topology. We can now think of fractally distributed regions of space like objects the totality of which is not conceivable and hich is not bounded by a spherical surface and thus has no sides but one corvl spatial distribution within which surfaces and ides are locl nd relative characteristics of a surface.

Pressure is now conceptual usable as acting on th sides of a surface and as regionally differentiable( differing in some way determined by the side of the surface it acts on, and the region the surface dissects, whether open or closed.
 A closed region now underpins notions of orporeslity or corpuscularity and the pressure inherent in such regions are often considered to be energies, body forces, tendencies and tensions .

We may derive from sides to surfaces notions of orientation hich then feed into notions of direction. Thus we now are able to speak of surface relative irections and distinguish between sides of. Surface and notions of a point in a surface!
The motion of a point is therefore derived from the general notion of a surface in space evn though a general concept of a point seems to be obvious in space! This is because in identifying a point in space the surfaces associated with that point always guide the perception of the precise point.

The action of pressure may now because of a surface be described giving pont direction and effect of that pressure

It is when we come to effect of pressure that we enter into a synthesis that is more complex and yet still fundamental. In particular, the dynamics b hich side ha Ben is cussed nd indeed perceived are revealed to be founded in the general dynmic concept of rotation, which itself in generality is now discussed under the heading of Roulettes, but previously was discussed as Tochoids.

It is this complex trochoidal behaviour we call chaos or turbulence and yet we May approximate to it by Fractal dynamical means computationally.

[jehovajah]

http://m.youtube.com/watch?v=WmKgT0a2KiY
Claes Johnson has substantially improved this method for computing fluid flows including Virtex outcomes in the potential flow solution
http://m.youtube.com/watch?v=CMz8KtvWKW4
A fractal generator cannot depict these situations with specificity, Yet it can still provide interesting outcomes that visualise these flows on the negative! ( photographic negative)
The iterated equation therefore has to describe the flow under the fractal generator set up. That is indirectly the or Mandrlbrot format With the bailout restrictions set up a circular/ spheroidal boundary condition that relates to certain flows of fluids in a region. .

[jehovajah]

http://m.youtube.comwatch? V=/QPTjuS5G3dY
Claes explains clearly if disappointedly the way fluid dynamics for aeroofoils( and thus boundary condition flow in hydrofoils and other situations ) was hindered by the lack of understanding . But that itself is an untenable position. Planes flew, boats sailed because experimentalists conformed to nature not theory or computation!
Today our computers can process the complexity of the descriptions we give them, but they can not predict an outcome. We do that as experts in factual reality not Theiretical mythology!

Is myth less useful than fact? Clealy not. Our humanity utilises both to make life more interesting. However to obscure myth by calling it law and fact by denying it is observable by at least a few others is a recipe for shock, disaster and perplexity.

For Claes to expect to be revered for solving a mathematical conundrum they thought they had already solved is the sad part. His work ithbHoffmann is truly groundbreaking but one of those small but vital details, like your little toe, that just keeps getting overlooked!
Johnson and Hoffmann have transformed the finite element method into a powerful tool for all sorts of materials, especially fluids .
Despite the jargon, it is essentially a fractal generator optimised for this purpose .
The equation controls the material motion or behaviour under iteration , the bailout defines the boundary conditions and can be made as complex or conditional as your application allows , and the surface plotter colour cycle reveals not just the extension of the calculation, but also the intensivity of the resultant outcome.

All three have to match a physical example in the limit to model that behavioural outcome , and that still does not guarantee that the computational model is applicable to any other situation!
How nice to be able to say , by fiat, this is a universal law!

Clearly one can not replace fact by such mythology no matter how useful that myth is in its original environment.

[jehovajah]

http://m.youtube.com/watch? V=B2XOrNyirhc
Claes explains the basics . The one dimensional coordinate system actually includes curvilinear coordinates.
In fluid dynamics Eulers reference was the particle moving with the fluid.LaGrange adopted the reference of the particle measured by an observer I dependent of the fluid and the particle. However the particle then becomes a means of describing the fluid flow at a reference point!

Now we can isolate the particle ( Euler data) and the fluif( LaGrange data) to recombine them by superposition to give a more informative flow description.

The Finite Element method does not allow its elements to flow with the fluid, so it records a laGrangian or Laboratory measurement of the flow. So a test particle has to be supposedly introduced to indicate the effect of the laboratory forces on the particle.

If the test particle itself recorded position and velocity by GPS we would develop a far more useful description of fluid flow.

Currently weather satellites compute wind flow patterns in the atmosphere based on best fit for observed motion of an identified structure or phenomna in the data.  These are always checked against ground measurements by Nasas dedicated teams of verifiers to give us confidence in the model data .
Every do often they update the model to make it even better at describing the observed measurements.

Our weather map models are now increasingly fractal in nature, design and execution.

The myth is that we can then predict the weather!! What we can do is simulate the fractal pattern evolving for a few more time step iterations , but then we have to reconnect to factual day to keep relevant!

[jehovajah]

http://m.youtube.com/watch?v=LNLXdlXKvL8

Now we know Cles's view, we see that time steps as update sequencers are fundamental to fractal generators making them ideal for performing all the elements necessary for a computational solution to any integral or differential equation. In addition the color cycle maps can display that information visually, and the surface plotters sculpt out the consequences of such motions on an initial uniform fractal mesh.

The issue I am addressing is the comprehension of the desired or final product of a fractal rendering. Is the art just art or is it a model of surfaces that could be generated by applying our current laws of motion to familiar topological starting points?
For example do plants grow by just reproduction, or is some external varying bailout shaping the outcome as well? Do the shape and colours of leaves for example reflect this combination and a colour distribution based on chemical density of all those elements that have moved a certain distance from the roots?

These are simple suggestions to highlight how the three main factors of a fractal generator may be shaped to model real life outcomes. The question is : would that significantly enhance our understanding of nature and natural laws  or would it mislead us into thinking mathematics is gods language for Nature! ?
 God forbid that we should ever subscribe to that bit of Mathmythics !!

[jehovajah]

http://m.youtube.com/watch?v= NPJImYibx48
Some teachers are more accessi LE than others.
In this case I wanted to get a feel of all the constraints one has to place on a fractal generator to get it to calculate a recognisable sculpture.

The surfaces plotted represent points that satisfy all the constraints for each region but how the points are defined in a region is based on a calculation sequence truncated or equated to a condition . That sequence of calculations is not defined in a fractal generator, but may be defined in a Computational Fractal Element Dynamic calculation as a Euler stream line .

The mass of calculation results are isolateable in many ways, but how representative of a physical situation it is depends on more detailed constraints.

[jehovajah]

It occurrs to me this morning that the Grassmann Twistors now come into the discussion.
These computational fractal elemnt fluid Dynmical calculations have one simple idea. Specify the form( surface topology) so that boundary conditions can be defined .
Whatever else happens happens or is defined within this context or topological mosaic.

The Grassmnn Twistors as explained in the thread called Twistors, are Fourier transforms , they are specifically Quaternion Fouriet transforms.
The simplest expression of the idea is to : do a quaternion Fourier Anlaysis of the form or topological mosaic.
Having the opology in Grassman twistor form now confines us o setting constraints for : phase, Frequncy and Amplitude within the quaternion argumnt of each term in the Euler: Cotes expression of the Grassman Transform.

Clearly the arguments are not simple variable strings , but rather complex conditional statements . Each condition is an evaluation process so the computation is in fact a complex evaluation process rather Han an arithmetical number crunching exercise!

What numerals come out of the process relate to extension or inanity at each iteration step., and the "solution" of a differential expression or an intgrl expression for that iteration step.

The display magic comes when rendering takes those outputs and defines a surface plot for the display constraints, and the colour cycle decoders display the data in regional mosaic form highlighting extensivity and or intensivity.

Finally doing or running this many times , or rather evaluating after each iteration enables  key fram video sequence to be captured that may( or my not) give insight into internal Dynamics as expressed in the Grassmann Twistor depiction.

While we can not say that this process is how it happens, we can by taking smaller and smallet scale sizes( specifying a kind of fractal zoom!) provide more detailed tracking of th processing outcome and compare that with an ultra fast video capture of the dynamics frame by frame.

Of course, our digital signal processing allows us to use the processed data and the captured data ( after digital  signal  processing too!) as interchangeable.

While at first this sounds ideal, it poses a philosophicl problem: are we redefining reality, limiting our apprehension to the mythical output of a computational processor? Is a human. Biological computational processor, and oes this if the case, justify accepting this kind of complex out put as Truth ( Wahrheit) as Grassmann put it?

Even in 1844 Grassmann expressed his hope that these types of complex combinatorial systems( now commonly called Matrices) could be populated or internally decorated ( entdeckt) with the truth .

Should we accept this, now it has been realised In our time without philosophical debate?
Somehow as a fractal Philosopher I do not think we should just accept anything!

[jehovajah]

It has been a while since I posted here, but that is not been because I have lost interest.

In fact much has bern learned in the interim.

The main thing was to unlearn a reliance on mathematics as a revealer of physical  reality, and to put it in its place as a method of measuring by quantifying distinctions the amount and dynamic amounts over time of these differing quantities .
The establishment of princles and processes of quantification that address the spatial dynamics that we visually observe, and to apply these sketchy interpretations to the experience of proprioceptive signals as well as audio and kinaesthetic ones .

Because of the difference in general in these sensory signals no one representation is best for all, but one process applies to all and that is counting .

By counting we can use metrons  appropriate to each signal system, but the counting process itself is idealised by sequenced lists of names, and those names in sequence allied with our appropriate metrons in each ignl system allow us to transfer to the visual system a sketch of what we experience in all and every ignl system .

Fluidity and fluid mechanics requires the adoption of dynamic Arithmoi, that is specialised or normalised mosaics to support the measurement of dynamic variation over time.

And now we see through these types of accommodations that technology has enabled these mosaics to capture the fleeting ness of dynamic change, and gifted us with multiple opportunity to count the changes we discover in extension, form and dynamism.

In fluid mechanics, curvilineal surfaces are more fundamentally useful that straight lines or plane surfaces. And regional concentrations and dispersions of these curvilineal surfaces best depict the dynamics of fluids.

[jehovajah]

http://m.youtube.com/watch?v=-wokVaaGFgA
Here Norman shows how the strain ellipse and ultimately the strain ellipsoid can be encoded by a generalised dot product form
The significance of this is that currently fluid mechanics is using these forms to describe fluid behaviours in laminar flows without a great deal of intuitive background. Setting it in the format of lineal rational trigonometry and algebra gives access to Normans powerful insights.

In addition, Claes Johnson et al. Using there pioneering work in finite element modelling in fluid dynamics can now derive simpler formulae for faster calculations of the dynamic evolution of a turbulent flow, especially round aerofoils and hydrofoils general magnetic phenomena at all scales .