Fractal foundation of Fluid Mechanics

[jehovajah]

The text here comes after a more general introductory scholium in which he sets out his views and aims and several,advisory warnings.

http://books.google.co.uk/books?id=lSoJ2tJKfIEC&pg=PA5&dq=Motive+newton&hl=en&sa=X&ei=3IzRU-zrO4ep7AbakoFQ&ved=0CB8Q6AEwATgK#v=onepage&q=Motive%20newton&f=false

It is a debate amongs some philosophers whether Newton may have had an initial" field" concept mofivating his thinking. However it may be more enlightning to view Faradays concepts in this light, using Newtons concepts as indicative of a non newtonian(!) concept of force equivalent to Faradays rather than Maxwlls academic conception.

Then we can Compare it to one opinion about the generality of aether concepts as mysterious agents, and the specificity of measure concepts which give a quantitative description of some of these mysterious agents effects.

The central tendency behind Newtons measure concepts is indeed a vector field concept, and motive facilitates that vector summation measure. But Newton does not regard the accelerative vis as a vector field. It is a diffuse force or vis inhabiting a body from its centre and extending round abouts imparting bodily motion to bodies in its influence. You have to understand that these are not physical,explanations but mathematical measures. Absolute vis is the only philosophical principal which has causal power.

Newton thus proposed that the motive of a body be the sum of the " motion" of it's parts , and likewise the motive of a system. These systems are understood to be centrally tending, that is essentially within a great sphere towards whose centre all parts are tending.

Secondly he proposes that a body has a central pointing accelerative force / vis that causes local relative " motions" of parts or Bodies within its influence.

Thus we can look, at any body within a system or as a system and apply summation rules and product rules.

That is precisely what he carries on o do.

Now my point of interest is in fact Newtons reationship to Hooke.

There is little dispute that Newton was influenced by Hooke, and that among others he credited his insights on principles established by Hooke. Thus Hookes extensible spring laws served as the basis for his own accelerative vis descriptions and how he proposed to measure accelerative vis. His proposition was simply to balance all these accelerative vis and motive vis within a system of Bodies to achieve what really is a dynamic equilibrium, but called a static equilibrium! In so doing he could use Hookes spring law to characterise a force in a stable system.

Hookes springs revealed the dynamism in stable systems and allowed Newton to formulate a force erasure in terms of his accelerative force concept. In particular a long spring oscillates to equilibrium in an understandable way if accelerative force is quantified by Hookes spring laws.

Thus, just in balance the extension l characterises the force on the body by the general centre and the force on the body by the spring centre. Extending beyond l creates a greater acceleration back to the spring centre, shortening it means the general centre creates a greater acceleration. Without accelerative vis no terminology existed to explain this observed behaviour.

How the measure related to length and time or periodicity was crucial. And Newtons analytical definitions of the velocity which was summed by celerity of the parts and acceleration which was summed by the motive of the parts required a chain link between all three concepts. This was done empirically by measuring the displacement for each quantity of accelerative vis, , the associated velocities and celerity absorbing action of sping resistance past equilibrium, the changes in velocity and a regular periodic measure.

Tabulation of displacement against time is crucial to defining a workable concept of velocity and acceleration. That these were differentials goes without saying, but that an accelerative force should be characterised by Hookes law is apparently a concept of Newton.

The oscillating mass on a Hooke spring illustrates several key concept. Velocity does not stop when acceleration is 0. When velocity is 0 an accelerative force may become free to act. Force erasure links mass and accelerative force as an equivalent or analogous concept of force but also dynamic, resistive forces are in play in all, dynamic systems.

[jehovajah]

So briefly the dynamic inight revels that for fluids acted upon by a solid in non accelerative motion, the resistive medium would act to drain away celerity to itself. In terms of conservation of momentum we can expect the momentum to the same throughout the interaction. But because the celerity leads to an ever increasing mass of fluid we immediately expect the sold to slow down. In fact we expect the interaction to spread celerity to the whole mass of any resisting fluid making the solid come to near rest simply on conservation of momentum grounds.

Now because we are diffusing accelerative force by celerity transferring to surrounding fluid, that fluid is not going to be bound, but will spread spherically. The exact nature and veracity of that spherical fluid dynamic involves a pressure gradient between fluid elements transmitting different amounts of celerity or transferring celerity at different rates. The principles of motive and celerity are summative so a spherical superposition would be a description of the pressure gradient profile most in keeping with Newtons principles.

Newtons own investigations I have yet to explore fully but one important element missing from his vis triumvirate is the rotational moive and the rotational celerity, along with the rotational accelerative and the rotationl velocity concepts. Consequently, despite the spherical nature and veracity of his " field" concepts rotation does not have an intrinsic place in it. If it did then Magnetic fields around wires would not be misdescribed,,and vortex shedding would not be an unexpected finding .

[jehovajah]

It is probably clearer to state Newtons triumvirate of vis this way: absolute force is universal and causative and we can only deduce it from instances of it acting in centres of bodies or space/ places. This last attribute explains t
He missing "bodies " as centres of gravitational attraction required for elliptical orbits.

The instances of absolute force thus cause a local effect emanating from the centre of body or space to which bodies are tending ". This local effect is not clearly distinguished yet as centripetal or centric.ugal behaviours because these are foundational postulates which we must agree on or reject. This vis he calls accelerative and it clearly defines a causal field of action associated to the centre of a place or space. However accelerative vis is empirically observable only by the action or behaviours of bodies relative to this centre. Thus it itself is not causative, but rather it is observable by an affect on bodies within a sphere that may be called its field of influence. The only cause then is absolute vis which is universal, and this centre acts as a .conduit to an instance of universal cause in action. The observed accelerative vis emanating from this centre is quantified by the bodies impelled into relative motion, either toward or away from this local centre.

The observed motion of the bodies can be summed and these sums are called motive. Thus motive is a sum of the mass and the acceleration of the mass or masses in some combination. Experiment showed that mas and acceleration were best producted so motive is a summation of products of mass and acceleration.

However absolute vis must also be the cause of velocity, which is similarly a central emanation that moves a body as a whole . But the previous discussion interjects accelerative vis as the cause of change in velocity. It must therefore be that velocity remains in a body maintaining its constant motion. The summation of velocities in a system would then be the celerity of the system.

Again velocity is an instance of cause emanating from a centre, but this time duration of action is significant, and freedom of motion and motive is mediated by the super positions of accelerative force fields and thus motives.

As a kind of second rank behaviour or rather a necessary effect velocity, that observable behaviour of a body in motion whether accelerating, moving uniformly or relatively still , is the measure by which we link our perceptions of motion to space proper.

Celerity as a conception perhaps has its recogniseable expression as " momentum", but it would seem Newton had a much broader conception of celerity in terms of a concept called the quantity of motion measure. We might say he conceived of a body having a rest quantity of motion, a kind of kinetic energy but not as a conserved quantity. This quantity disappeared without trace in interaction.

Again, the lack of any rotational measures in this fundamental system is a serious flaw.

[jehovajah]

It occurs to me that the concept of action at a distance is not avoided by this structure of the vis, unless the bodies and space in which absolute vis finds its centres are indeed something, call it what you will from space time to aether. It is the absence of this self reflexive and self motivating substance that leads to the inescapable action at a distance.

Also, though Newton dies not make all things clear in this introductory Scholium, he by no means neglects to establish rotational motion as fundamental, and so to establish a further putative system of vis. However it is his " students" who have rejected or not understood his modifications to his principles of Astrology.

[jehovajah]

As you know, in fluid mechanics pressure takes the leading role. Because Newton actually qualified vis in this way motive vis is in fact a more flexible concept than the current bald F = ma notion!

Motive force is the sum of bodies in accelerative motion relative to a centre. Thus the complete notion is poetical in Spaciometry and entirly consistent ith bubble pressure or fluidic pressure measured at the boundary of a fom by summing the accelerated masses. In fact what is done is a bulk modulus is used to contain this motive force and like a spring this deflection once settled characterises the accelerative force.

The adjustment to surface area that is made to the f = ma concept is in fact a mathmatical method that oes not capture this very well. The introduction of density reveals the underlying philosophical difficulty.

The concept of fluid motive that I derived is exactly the reverse of Newtons concept because I though motive was causative of acceleration. In fact Newton gives no cause beyond bsolute vis. The accelerative vis thus stands on its own as a local instance of the universal cause in action . When density is introduced to distinguish various matters it also distinguishes how the accelerative vis is behaviourally modified by material density distinctions. More work needs to be done on this aspect of materiality because the quantity of motion measure seems to be a good measure of material intensity/ density as a pressure system throughout a volume.

The change in this pressure throughout the volume conveys the ability of the material to spread spherically.

High speed photography confirms this kind of behaviour and perhaps reveals the wave like deformations that are involved in establishing a bubble of a given quantity of motion!

[jehovajah]

Computational fluid Dynmics has made some remarkable progress, but it's foundations still need to be revised as Claes Johnson points out.
However here is an interesting research project into the plumes and vortices that make up a turbulent boundary layer
<a href="http://www.youtube.com/v/L9LD5eZicAg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/L9LD5eZicAg&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/L9LD5eZicAg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/L9LD5eZicAg&rel=1&fs=1&hd=1</a>

Advances in CGI and surface coloration and plotting make clear the filament awry vortices and plumes and how they decompose a spherical or spheroidal volume . These volumes are not neat spheres but deforming ones so they are morphing as they move in the flow, thus characterising turbulence

The evolution of vortices is modelled elsewhere on this site, and it is worth noticing that vortices tend to break up into regions with connections to each other that fade. This is a fluid analogue of a particulate structure, and should inform all particle theories as to the nature of the interactive behaviours of matter.

I am moving toward a fundamental fluid plasma as a primitive aether which underlies the material space. Therefore it is important to realise that this plasma is not a 4 th state of matter, but the fundamental ground of matter

The concepts of temperature and heat need to be revised so as to support endo thermic and exothermic Enthalpy in plasma behaviour.

Chemists know that endothermic reactions occur as well as exothermic ones. Physicists prefer to ignore endothermic reactions, focussing on a concept of entropy tht is biased towards exothermic reactions. It is more reasonable to accept a steady state balance between exothermic and endothermic interactions in plasma , between cold and heat, with stable matter bEing a balanced state of these cold and hot plasma interactions.

[jehovajah]

This discussion of the classic Helmholtz Kelvin flow resulting from flow over " riblets"  is flawed only by the assumption Helmholtz vorticity kinematics is correct. However the statistical modelling approach means that the results really are closer to " first principles" development rather than fully accepting Helmholtz kelvin kinematic principles. That the same results are achieved does not validate Helmholtz vorticity kinematics, rather it reveals that the boundary conditions are robust in determining the internal behaviours , thus a wide range of boundary conditions compute the same outcome because the superposition in the process tends to a fractal outcome which in these cases are logarithmic in nature.

<a href="http://www.youtube.com/v/yf4oCfdPUu8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yf4oCfdPUu8&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/yf4oCfdPUu8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yf4oCfdPUu8&rel=1&fs=1&hd=1</a>

[jehovajah]

I think I have discussed before the characterisation of shear force by a vertical velocity differential . This is supposedly based on Newtons idea of lubricity in resistive fluids, later replaced by the notion of viscosity. This initial deformation at the boundary underlies much discussion about flows around objects and between objects.

Terminal velocity analysis also links acceleration to velocity in a limiting scenario which defines an exponential velocity as a force equivalent.

My research into Newtonian Vis models or principles has revealed Newtons use of the Galilean principle derived from the observations of the Jovian system and the orbit of Venus, which essentially Galileo  adduced to a fractal system of orbits around bodies in orbits  around larger bodies.

Newton elaborated this principle to a system of absolutes : space, time, force. In particular he posited a centripetal force, as observed. This centripetal force was metaphysically structured into a triumvirate so as to develop a quantitative system of measures that linked directly to the balance of moments or the Hooke spring balance.

The absolute vis or power of a system was structured spatially by an accelerative vis or power of the system so that the absolute power was distributed according to some " law " over the region of the absolute system. This law was accelerative and affected velocity in that region over time. Finally the bodies within that region were characterised by a motive vis which was centripetal and summative , that is a vector sum was assumed for the motive of a body which itself was a product of the bulk and the accelerative velocity of the bulk, towards the centre of the absolute system. Thus weight  was the concept of motive vis Newton sought to on next to, and the velocity which products with the bulk is a velocity difference.

To make full sense of this motive Neton connects it by a system of measures to the fundamentals of bulk and displacement and time, introducing several measure the most addictive of which is the quantity of motion measure which is necessary to support the motive vis.

To this analysis Newton added the Centrifugal Anlysis of orbital motions described by Halley, and do established a centripetal centrifugal description of the accelerative force in a region counteracting the centrifugal accelerative  force in a bulk or corpus.  Thus Newyons description of orbital motion analyses the forces only in the centripetal and centrifugal case, with the resultant velocity being everywhere a consequence of competing forces or vis. Newton does not consider a ritationl vis!

The rotational vis or power is obvious to any empirical observer, but because Newton did not specifically discuss it, many have concluded it does not " exist". It is supposedly an artefact of our observations of radial accelerations, and tangential instantaneous infinitesimal impulses. Things become simpler when one realises that the Newtonisn system is a model of a centripetal / centrifugal orbital velocity system. His model was based on standard geometrical models which take a circular or elliptical locus as conditional on radial constraints. This kind of analyis does not deny the circular or elliptical motion it relates it to radial factors which the observer includes in the description of the motion. The motion itself stands alone!

The imposition of radial measures does not negate the motion that is bing measured, and thus rotational vis is not negated by centripetal and centrifugal vis!

In 1820 Örsted revealed the magnetic behaviour around a current surrounded wire.. Arago demonstrated this to the French  Academy in that year. Only Ampère realised that this was empirical evidence of a rotational force! The rest of the academy, LaPlace,LaGrange,Biot.Savant, all were incredulous. They did not believe in a rotational force.

Ampère devoted a brief but intensive period to establishing the "laws" of this rotational force using Newtonian principles. He declared this as the field of Electrodynamics. There is a ritationl force and it is buried within the obscure mathematics of Electrodynamics. It does not require all that excrudescence to be believed and understood.

In that light the rotational force as a model becomes the fundamental force of the cosmos at all scales. Lineal forces are thus derivable from it according to scales or indices based on the radii of curvature of the rotational force.and in this regard the Reynolds number is a scale factor that indexes these radii of curvature.

[jehovajah]

The Newtonian concept of Vis is thus very flexible. The kinematics that supports Newtonian mechanics goes into detail of how flexible. The concept that force is just mass by acceleration is misleading. Newtonian force is proportional to acceleration, to velocity and to displacement!.

The above statement does not directly refer to the time constants required to apply the measures because time is considered absolute in the Newtonian system. In a relativistic system time becomes relativistic!

However the vectors are usually in the same direction : acceleration, velocity,displacement. How can the power be directed in any other way? The orientation of the force in a centripetal force region is distributed by the trigonometric "laws" of the right triangle. Mechanically this was a geometric no brainier. Experiment and empirical observation showed that Pressure in the contact region was radially consistent. Thus the vector was a compound of all these radial vectors.  Because of this any vector could be resolved into radial vectors from the same point of contact.

The significance of this is that Pressure is more fundamental than lineal force in the Newtonian system.

While the system resolves the forces to relationships between straight line segments, the nature of these line segments were not explored until Hermann Grassmans Ausdehnungslehre. In particular the trig line segments are analogies not of straight line motion but rotational motion.

There are many assumptions explicit and implicit that underpin the Newtonin system, but fundamentally it uses a polar and Cartesin reference frame in combination to describe physical behaviours. Thus Newton Wren, Halley and Bernouilli may not have specified a polar reference frame, but they implicitly assumed it, and they implicitly assume a rotational force. This is only clear when the reader does not deny the centrifugal firce model of Halley. Newtons orbital system requires both centrifugal and centripetal force measures to conjoined with the motive force measure to describe the orbital motion.

In a fluid, the existence of rotational force explains how a shear force transmits orthogonally to disturb the fluid in a vertical velocity distribution which is a characteristic of the fundamental pressure in this dynamic system. Thus one can resolve the shear force by trigonometric vectors which correspond to the velocity distribution in the fluid under shear.

Now rotational force implies one other aspect of pressure promulgation. In the surface of an expanding radial or contracting radial force, a rotational force will be evident, and in a " circular" behaviour. This is what is observed in expanding bubbles, the elements in the expanding surface rotate within the surface as well as expanding or contracting. Transverse " waves" on a spherical surface thus have a chiralty of rotation, and such "waves" develop into filamentary plumes from the central spheroidal mass.

[jehovajah]

The fundamental dynAmic structure in fluid Mechanics is the plume and the vortex ring associated with it. It's a kind of " mushroom" structure. It evolves into bubbles. Spinning bubbles either expanding or contracting.

[jehovajah]

Thïs plasma demonstration of the magnetic space lead to a post on MagneticUniverse.com regarding the fluid Mechnics of plasmas as defined by high energy physics t the LHC.

<a href="http://www.youtube.com/v/ECokfl2y0Fs&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ECokfl2y0Fs&rel=1&fs=1&hd=1</a>
http://www.youtube.com/watch?b=ECokfl2y0Fs

The cathode ray is a plasma and it interacts with a dynmic equilibrium plama flux in the space of a magnet to produce a diffraction pattern. .

[jehovajah]

This is Norman at his most obscure. I am grateful that he renounces this rhetorical style early in his career and starts to develop a much more inclusive approach.
Nevertheless this abstract concept allows a mechanical description of fluid Mechanics to be connected to Lie groups which are fundamentally groups of trochoidal rotations
<a href="http://www.youtube.com/v/pqAM4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/pqAM4&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/pqAM4-pic8M&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/pqAM4-pic8M&rel=1&fs=1&hd=1</a>
 To make use of the abstract format above Norman glosses over the need for a suitable Fourier basis. The Fourier bases are the topic in the thread Twistor in the complex number forum. In that thread I explore a exponential format for rotation. The format formally introduces circular arc line segmnte, analogous to straight line segments

In this video Norman exposits Topological notions that eventually as you go through the course makes sense of Tangent planes to curved surface topologies.

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

Gradually this will lead to an algorithm to model fluid dynamics using Normans Universal Hyperbolic approach nd the chromatic algebras of matrices.
Hopefully applying Grassmans concept of extending magnitude will lead to This  the conclusion of the basic mushroom plume of fluid mechanics.
<a href="http://www.youtube.com/v/4gHhAA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/4gHhAA&rel=1&fs=1&hd=1</a>

[jehovajah]

While in the bath, and thinking about the day and bubbles, and my reply to funkyNfresh I drew a single trochoidal locus with a single curl back onto itself and then looping and flirting away, in my mind.

Let that represent a pressure or trochoidal force line segment I said.

Almost immediately I thought, that curl should be the surface tension, and then I thought of a mirror symmetry  trochoidal line segment , and the pair reminded me immediately of the plume and mushroom smoke ring, with the surface tension along the surface of the mushroom..

This surface tension comes from the trochoidal force line as  at curls .

Them I thought : what is surface tension any way? Just the region where the trochoidal force becomes curled , no longer straight out from the centre.

Then I thought about the radial expansion of a pressure/force field,  it was always a totally inadequate representation, because it left out the expansion between the radials . This trochoidal path accounts for this inter radial expansion.

Then I said: it accounts for surface tension. And then I said : Newtons tangent decomposition of a circular force represents surface tension. Until now I had to understand that as a velocity resulting  from an initial outside push or pull. But now I can conceive of it as a force resultant , the surface tension resulting in circular or spherical acceleration that makes the surface an incredibly active region!

Without further ado I realised that a pressure field must not only have radial line segments but also line segments between the tips of the radials . The centripetal and centrifugal forces may be static in dynamic equilibrium but there is nothing to stop the surface tension acceleration. So within that boundary expansion  the surface generates a rotation within the surface .

The consequence of this is that deformations called " wave" deformations are not just transverse or just longitudinal but a combination of the two which results in trochoidal deformations in space. These deformations we see as filamentation and vortex ejections if energetic enough, otherwise as an incredible surface tension

[Sockratease]

While in the bath, and thinking about the day and bubbles, and my reply to funkyNfresh I drew a single trochoidal locus with a single curl back onto itself and then looping and flirting away, in my mind.

...

Ummm...

EUREKA!

I hope you didn't run through the streets naked to go make that post   

Interesting parallels though, and somehow reminds me of certain bits of String Theory with it's curled up dimensions and hidden vibrations.

[jehovajah]

 

My wife restrained me! 

I can not express it yet, but I perceive the underlying formal structure (not space ) to be based on trochoids . I would call them Grassmann Trochoids but I have to find the direct link yet.

At this stage in my research Hermann is still using simple elements. Trochoids are the next step up from the simple circular arc segments, and they can go all the way to the n-th stage.

String theory as a model has fabulous adjustability potential, a Fourier like transformation that can reproduce any "shape" in space. But it becomes fantastically complex very quickly. That is why a fractal description is so crucially important to its application .