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Author Topic: Fractal foundation of Fluid Mechanics  (Read 15098 times)
Description: Discussion.insights, notions and paradigm shifts
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jehovajah
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« Reply #15 on: May 03, 2013, 10:28:25 AM »

This is an accessible survey of the history of Matrix algebra.

Quote
A Brief History of Linear Algebra
Jeff Christensen
April 2012
Final Project Math 2270
Grant Gustafson
University of Utah
In order to unfold the history of linear algebra, it is important that we first determine what
Linear Algebra is.  As such, this definition is not a complete and comprehensive answer, but
rather a broad definition loosely wrapping itself around the subject.  I will use several different
answers so that we can see these perspectives.  First, linear algebra is the study of a certain
algebraic structure called a vector space (BYU).  Second, linear algebra is the study of linear sets
of equations and their transformation properties.  Finally, it is the branch of mathematics charged
with investigating the properties of finite dimensional vector spaces and linear mappings
between such spaces (wiki).  This project will discuss the history of linear algebra as it relates
linear sets of equations and their transformations and vector spaces.  The project seeks to give a
brief overview of the history of linear algebra and its practical applications touching on the
various topics used in concordance with it.
Around 4000 years ago, the people of Babylon knew how to solve a simple 2X2 system
of linear equations with two unknowns.  Around 200 BC, the Chinese published that “Nine
Chapters of the Mathematical Art,” they displayed the ability to solve a 3X3 system of equations
(Perotti).  The simple equation of ax+b=0 is an ancient question worked on by people from all
walks of life.  The power and progress in linear algebra did not come to fruition until the late 17th
century. 
 The emergence of the subject came from determinants, values connected to a square
matrix, studied by the founder of calculus, Leibnitz, in the late 17th century.  Lagrange came out
with his work regarding Lagrange multipliers, a way to “characterize the maxima and minima
multivariate functions.”  (Darkwing)  More than fifty years later, Cramer presented his ideas of
solving systems of linear equations based on determinants more than 50 years after Leibnitz
(Darkwing).  Interestingly enough, Cramer provided no proof for solving an nxn system.   As we
see, linear algebra has become more relevant since the emergence of calculus even though it’s
foundational equation of ax+b=0 dates back centuries. 
Euler brought to light the idea that a system of equations doesn’t necessarily have to have
a solution (Perotti).  He recognized the need for conditions to be placed upon unknown variables
in order to find a solution.  The initial work up until this period mainly dealt with the concept of
unique solutions and square matrices where the number of equations matched the number of
unknowns.
With the turn into the 19th century Gauss introduced a procedure to be used for solving a
system of linear equations.  His work dealt mainly with the linear equations and had yet to bring
in the idea of matrices or their notations.  His efforts dealt with equations of differing numbers
and variables as well as the traditional pre-19th century works of Euler, Leibnitz, and Cramer.
Gauss’ work is now summed up in the term Gaussian elimination.  This method uses the
concepts of combining, swapping, or multiplying rows with each other in order to eliminate
variables from certain equations.  After variables are determined, the student is then to use back
substitution to help find the remaining unknown variables.   
As mentioned before, Gauss work dealt much with solving linear equations themselves
initially, but did not have as much to do with matrices.  In order for matrix algebra to develop, a
proper notation or method of describing the process was necessary.  Also vital to this process was
a definition of matrix multiplication and the facets involving it.  “The introduction of matrix
notation and the invention of the word matrix were motivated by attempts to develop the right
algebraic language for studying determinants.  In 1848, J.J. Sylvester introduced the term
“matrix,” the Latin word for womb, as a name for an array of numbers.  He used womb, because
he viewed a matrix as a generator of determinants (Tucker, 1993).  The other part, matrix
multiplication or matrix algebra came from the work of Arthur Cayley in 1855.
Cayley’s defined matrix multiplication as, “the matrix of coefficients for the composite
transformation T2T1 is the product of the matrix for T2 times the matrix of T1” (Tucker, 1993).
His work dealing with Matrix multiplication culminated in his theorem, the Cayley-Hamilton
Theorem.   Simply stated, a square matrix satisfies its characteristic equation.  Cayley’s efforts
were published in two papers, one in 1850 and the other in 1858.  His works introduced the idea
of the identity matrix as well as the inverse of a square matrix.  He also did much to further the
ongoing transformation of the use of matrices and symbolic algebra.  He used the letter “A” to
represent a matrix, something that had been very little before his works.  His efforts were little
recognized outside of England until the 1880s. 
Matrices at the end of the 19th century were heavily connected with Physics issues and for
mathematicians, more attention was given to vectors as they proved to be basic mathematical
elements.  For a time, however, interest in a lot of linear algebra slowed until the end of World
War II brought on the development of computers.  Now instead of having to break down an
enormous nxn matrix, computers could quickly and accurately solve these systems of linear
algebra.  With the advancement of technology using the methods of Cayley, Gauss, Leibnitz,
Euler, and others determinants and linear algebra moved forward more quickly and more
effective.  Regardless of the technology though Gaussian elimination still proves to be the best
way known to solve a system of linear equations (Tucker, 1993). 
The influence of Linear Algebra in the mathematical world is spread wide because it
provides an important base to many of the principles and practices.  Some of the things Linear
Algebra is used for are to solve systems of linear format,   to find least-square best fit lines to
predict future outcomes or find trends, and the use of the Fourier series expansion as a means to
solving partial differential equations.  Other more broad topics that it is used for are to solve
questions of energy in Quantum mechanics.  It is also used to create simple every day household
games like Sudoku.  It is because of these practical applications that Linear Algebra has spread
so far and advanced.  The key, however, is to understand that the history of linear algebra
provides the basis for these applications.
Although linear algebra is a fairly new subject when compared to other mathematical
practices, it’s uses are widespread.  With the efforts of calculus savvy  Leibnitz the concept of
using systems of linear equations to solve unknowns was formalized.  Other efforts from
scholars like Cayley, Euler, Sylvester, and others changed linear systems into the use of matrices
to represent them.  Gauss brought his theory to solve systems of equations proving to be the most
effective basis for solving unknowns.  Technology continues to push the use further and further,
but the history of Linear Algebra continues to provide the foundation.  Even though every few
years companies update their textbooks, the fundamentals stay the same. 
Darkwing. (n.d.). A brief history of linear algebra and matrix theory. Retrieved from
http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html
Perotti. (n.d.). History of linear algebra. Retrieved from
http://www.science.unitn.it/~perotti/History of Linear Algebra.pdf
Strang, G. (1993). The fundamental theorem of linear algebra. The American Mathematical
Monthly,100(9), 848-855. Retrieved from http://www.jstor.org/stable/2324660
Tucker, A. (1993). The growing importance of linear algebra in undergraduate mathematics. The
College Mathematics Journal, 24(1), 3-9. Retrieved from http://www.jstor.org/stable/2686426
Weisstein, E.W. Linear Algebra. From MathWorld--A Wolfram Web
Resource.http://mathworld.wolfram.com/LinearAlgebra.html

However it is noteable that the fundamental contribution of Hermann Grassman (1844)(and in part Robrt Grassmann(1862)ff) has not been noted, nor indeed the fundamental contribution of Sir William Rowan Hamilton(1834.1851). It has been claimed that Grassmann's work was obscure, and difficult to read, but i have not found it so even in the original german. It is iintoxicating and innovative, as i am sure Peano, Whitehead and others will attest.. Google "jehovajah Grassmann" if you want to know more.

i posted this to help with the matrix notion used in the ellipsoid paper.
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« Reply #16 on: May 03, 2013, 10:39:29 PM »

I have a link here to the concept of a potential flow equation in fluid dynamics.
http://en.wikipedia.org/wiki/Potential_flow.
Why describe it as a potential flow I am not yet clear on, but the gradient of a scalar field showing streamline speed as opposed to velocity is just an equation which if graphed shows the tangent to a curve or line exhibiting the relationship of speed at certain distances..
Say the gradient is everywhere one, then as one moves in the direction of the parameter increasing the speed increases. A vector determines the flow direction.
http://en.wikipedia.org/wiki/Velocity_field
http://en.wikipedia.org/wiki/Velocity_potential
http://en.wikipedia.org/wiki/Conservative_vector_field#Irrotational_vector_fields
http://en.wikipedia.org/wiki/Hamiltonian_fluid_mechanics
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« Reply #17 on: May 04, 2013, 09:05:45 AM »

In Wave mechanics there are several types of wave, but most consider the typical wave to be sinusoidal. This is not the case however. A typical wave in natural surroundings would be trochoidal, only approximating the sine form at low amplitudes.

The use of the sine function to generate and control oscillation has to be released from its usual diagrammatic constraints, For example it is usual to depict a sound wave using a transverse Fourier wave, with a warning that the oscillation is in fact longitudinal! Although it may seem a trite thing, this usual depiction, for want of rotating through π/2 radians conveys a graphical conundrum! The longitudinal wave dissipates as it travels forward. It alsos propogates in all directions in the medium. Because of these vector motives there is a natural vorticity in such wave propagation's which are completely obscured.

The unsatisfactory explanation of the mathematical description of vortex rings is down to this convention of showing longitudinal waves as transvers analogues,

Now, considereing a transverse wave motion in a medium requires the motive of propagation. A sine wave does not propagate  in space. The graph models the relative displacements of centrifugal and centripetal motives. The change in celerity or its effect velocity is crucial to the model, but it models local rotation only. Thus for a sine wave propagation there must be a net celerity  or motive that by definition is longitudinal, or in a medium radial.

What this means is that vorticity is induced by a combination and possibly a compounding of these transverse and longitudinal oscillations, and spread by the net radial or longitudinal propogation, or conversely compacted by the same.

Do we ever see a purely transverse propagation?

The skipping rope is a fundamental example of wave propogation. The tension in the rope is crucial, because it is this tension that propogates the transverse motion!

If the tension is low, the transverse motion may be damped by the longitudinal motion of the tension dissipating. If the tension is sufficient to reach to the other end, hen a transverse motion may be carried to the end, and return on the returning tension!

Newton's Third Law, although i have explained in my blog that this is an unfolding of the inertial frame concept Newton had in mind. posits this contra action in all interactions where equilibrium is disturbed.. The inertial frame maintains a static or dynamic equilibrium. In the case of a rope, the equilibrium is dynamic, even if the rope itself is static.

It is this insight about the nature of spacematter, a dynamic fluid medium in which static "motion" is a net effect of ever present dynamic ones, that Newton hoped to take forward into the study of fluid mechanics in book 2 of the Principia.

The nature of fluid spacematter was beyond his time and duties burden, and despite fundamental insights and experiments, he left the field to others to develop, secure that his principles and praxis would yield results.This in part was why he contradicted Huygens, apart from religious differences. Huygens offered no convincing empirical data, only some "mathematical and geometrical guesswork! This Hypothetical approach, Newton did not encourage, propounding that in these matters we must proceed with caution from certainty to certainty, deriving hypothesis from empirical data.

Huygens may have protested that he did  derive his solution from the empirical data of the fringe  anaomalies, but in fact, this was not demonstrated by him, just posited with no empirical evidence for the widely held belief of an extant Aether with the attributed properties.

Newton found fluid mechanics confusing/perplexing. Consequently he could say little that was certain about it. In particular he could neither demonstrate or eradicate the prevailing "guess" of DesCartes, that vortices moved the planets. For this reason he used the ill defined term gravity, in contradistinction to levity, to describe a tendency in spacematter to clump together, but as to what it was he made no hypothesis ! He had no frame for it! This from the man whose reference frame, derived from observing rigid matter, elastic matter etc, was so persuasive! Newton assessed that his rigid motion laws were a better match to empirical data than his fluid motion ones, and some have misread this as overturning a vorticular dynamic in spacematter.

In making such a comment i defer to Newton, who fully acknowledge the work of Hooke and others in his insights. It is Hooke's law, as well as the behaviour of pulleys that supported his third law of interaction.

So the propagating force is tension, and because it is opposed, when it meets a boundary the tension "reflects"! We can understand this simply as he net force being opposed by a reactive force which itself must inhere in the rope under tension. Having achieved the require tension to overcome the propagating tension through strain, the rope is now in an unstable deformation. Consequently this unstable deformation becomes a net driving pressure in the opposite direction.

In  a rope with no tension, there is no clear or transmissive boundary condition. consequently "gravitational" pressure counteracts the tension force as it propagates, but provides no unstable deformation in doing so. It acts as a restorative pressure maintaining equilibrium. The dissipation is one of our clues to the transformation of motion into other motions and forms that move away from the source through the inertial background. We tend to call this energy dissipation, which is dissipation through work. It is this work concept that allows us to link heat to mechanical actions, and underpins the mechanical description of electromagnetism, But it took Maxwell and others like Helmholtz and Kelvin to suspect that fluid mechanics would be the best reference frame in which to describe all material interactions.

The longitudinal wave, therefore is the necessary tensile support for the transverse wave. The longitudinal wave must precede the transverse wave.
The longitudinal wave is the propagation mechanism for all reflection in wave mechanics, while the transverse wave is responsible for diffraction and refraction.
While a longitudinal wave may propagate without a transverse motion. a transverse wave must always propagate within or adhering o a longitudinal one. Because a longitudinal wave propagates radially, it also contributes to refraction, but the refractive enhancement of the transverse wave is what allows a prism to spread out the colours of the spectrum.

The fact that the longitudinal wave reflects is evidence of the longitudinal oscillation. Thus as a longitudinal wave propogates it is oscillated by back pressure at each encounter in it advancement through the supporting medium. This back pressure  would be thought to be radial, but in fact it is not. The back pressure is arbitrary, defined by the response of the supporting medium. In addition if the medium has its own dynamics this acts on the longitudinal propogation. At best i would characterise the back pressure as spherical resulting in an imploding but weakening spherical wave front at the same time as an exploding weakening wave front propagates. In such a scenario i would characterise the resultant vorticity as regionalised and even turbuelent within the advancing longitudinal propagation. Add to this any transverse wave motion and i would expect trochoidal filamentation occur at some given radial distance from the source as the longitudinal reflective back pressure dissipates.

In a smoke ring or fluid behaviour around a jet we can see that this filamentation is not necessarily linear, but in fact curls of into mushroom like protrudences depending on the supporting medium's Reynolds number. This number reflects the relative effect of viscosity over inertia. That is to say the internal pressures inherent in and defining a substance which has a separate boundary to other distinguished substances with differing internal pressures. If one substance is placed within another such systems , by boundary layer interactions may generate motion which is inertial within the surrounding medium, if it is taken as a reference frame.

Thus an inertial reference frame is a matter of choice and convenience, but chiefly it encapsulates a region in which all the pressures may be considered and distinguished. I am sympathetic to the thought that vorticity may underpin viscosity/lubricity in a mechanical way similar to gears and bearings.

The longitudinal wave i feel is modeled by the strain ellipsoid, but requires a dissipative factor even as it tends to infinity. In the same way the reflected longitudinal wave would dissipate, but the superposition of components from the radial spherical reflection may be significant in the short distances from source. The quality of this reflection and thus the perceived longitudinal oscillation will depend on the reflective properties of the medium.

In passing, the electromagnetic behaviours seem so superior in this regard, that i would conclude that the basis of their efficacy is a substrate of perfectly tensioned spacematter with ideal reflection properties and very little radial dissipation. That is a focussed signal tends to remain focussed over long distances. This directional quality of strain is inherent in the parallel trammel lines along which it is defined in the strain ellipsoid.
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« Reply #18 on: May 04, 2013, 09:14:19 AM »

https://www.fractalus.com/kerry/articles/vortical_flow1.pdf
http://www.me.jhu.edu/meneveau/pubs-fractals.html

Some active links from Jules Ruis (thread earlier).
http://en.wikipedia.org/wiki/Kelvin–Helmholtz_instability
http://www.academia.edu/1253068/Electrostatica_y_Magnetostatica_de_cuerpos_elipsoidales_formalismo_del_tensor_depolarizacion
« Last Edit: May 05, 2013, 12:37:50 AM by jehovajah » Logged

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« Reply #19 on: May 06, 2013, 04:28:09 PM »



[rfun
s=imaj(z)
sc=imaj(c)
l=real(c)
m=imaj(c)
x=x#
y=y#
p=1/sin(x*pi/y)
q=1/sin(1*pi/y)
r=e^p
t=e^q
v=l+m+sc
rend
c=l+(v+l*m*sin(l))*i+(v+l^2*cos(l))*j+(v+l*sc*sin()*cos(l))*k
z=x+r*cos(y)*cos(s)*i+t*cos(y)*sin(s)*j+r*sin(y)*k+0.8*c

The full vorticity strain ellipse formula.
My strain ellipsoid has a special formulation to produce visible sculptures, but in general it can be varied. The strain ellipsoid shapes thr vorticity streams.

The combination of the Z and the C Quaternions in Quasz reveal that in this case a heightening of definition results. As these are not animated i have to look at the fluctuation as iterations increase. which i have not done yet.

I feel if the combining of different streams is not just as simple as a vector addition of the 2 stream equations, this format offers me some pathway towards conflicting 2 streams, but i do not know that yet.

Again i have to remind the viewer these ae "negatives of the actual turbulent flow which is sculpted away.
« Last Edit: May 07, 2013, 10:41:45 AM by jehovajah » Logged

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« Reply #20 on: May 08, 2013, 02:11:31 AM »

The far view of the strain ellipsoid shows me that the concept is being replicated by the fractal generator, but as i am just fooling around, i have not considered what it is i am creating and how the boundary conditions sculpt it.

The boundary conditions help me to produce a negative sculpture only in the correct fornulation. The surface produced shows me all those quaternion points that do not break the boundary condition.

Straight away  this means if i can run two motion formulae together with different boundary conditions, one inside the other  i may be able to see the combined effect of the motion formulae.

 i like to keep the idea clear and simple, and then explore it through manipulation of the idea. Although this would potentially create a "false boundary in the sculpture, i want to see what that looks like first in a simple way before complicating the boundary condition. i mean the Mandelbrot set boundary arises from the simplest boundary condition!

The other  thing is the strain ellipsoid approach relies on a straight boundary which is "scale free". The advantage of this is that the strain ellipse shape and deformation rate reflects the orientation of this straight surface to the spherical test bubble in a laminar flow. Understanding how the strain ellipse represents this interaction would help to define pressure fields at boundaries.
the advantage of this is that the notion of boundary has nowhere been defined as  rigid, and yet the paradigms are borrowed from rigid dynamics. Technically we can modify the formulae accordingly, if we know what we are doing. However this requires a greater freedom to add detail than i believei have in Quasz, so i may need to do some working round to fit it within my codelimits.
My first idea is to see how the strain ellipse/ellipsoid copes with a parabolic curved surface y=x^2
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« Reply #21 on: May 11, 2013, 09:55:00 PM »

So my strain ellipse formula is a model of pressure, not of strain. It is proactive not reactive.

Analysing the theory on strain ellipses and hoping to use the strain ellipsoid as a model of pressure lead me to explore how the background stream flow strains an ellipse. It suddenly became clear that my formula is accelerative in an ellipsoidal way. Bingo.

It was a motion formula acting like a source of pressure!  A shaped region of pressure, and if i can a shaped region of reaction with some physics might take me somewhere!
http://my.opera.com/jehovajah/blog/2013/05/05/the-problem-with-force-is-it-is-non
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« Reply #22 on: May 12, 2013, 10:57:45 AM »

The more i meditate on the ellipse. and ellipsoid the more i see its appllcability.

The strain ellipse comes out of deforming the circle. The theory starts with laminar flow, and the circle at rest in the flow. This is the same as saying that the internal points in the ellipse move with the background.

Thus i can add the background flow equation to the points in the ellipse. Adding them to the ellipse boundary should be sufficient.
Now changing the background flow changes the ellipse accordingly. By working out the position of the elliptical axes and some other orthogonal radials inside the ellipse i can read of a measurement of the train for those radials inside the ellipse.

spaciometrically we can see that there is rotational strain that is twistorque as well as axial strain that is centripetal and centrifugal . From these i can resolve into tangential strain for the selected points on the boundary where the radials fall.

Looking inside the ellipse i can see 4 regions of counter clockwise rotational strain, that is counter twistorques which are trochoidal, that is they exhibit a complex path like a trochoid rather than just rotating relative to the ellipse centre as one would unconsciously imagine. This is because as the point moves the ellipsoid rotates. To be part of an ellipsoid with any internal viscosity each internal point has to move with the appropriate rotation. The back ground motion therefore is not indicative of the motion of an elliptical strain reaction.

The difference is due to the difference between Lagrangian and Eulerian reference frames. Thus the strain on the ellipse boundary hides a simple fact: the boundary is instantaneously dynamic. It draws in from within and without fresh points of material. in a fluid this is possible, in a solid this is overlookd in the Hooke's law based strain equations.

In an elastic deformation the marked poits of the metron stretch apart. Physically this meeans that those fixed points in the laboratory reference frame (eulerian) are instantaneously replaced by new points from WITHIN the material undergoing deformation. We ignore this in simple models. but we do not in fluid models. The venturri tube which Bernoulli experimented on examines just this strain phenomenon under the guise of fluid hydro static pressure.

Applying material conservation laws leads to some explanatory principles and realations, but Bernoulli's Important observation is too simplistic to explain the fractal picture unless it is applied fractally and multi directionally in a fractal generator.

The hydrostatic pressure misleads us. A pressure system is multidirectional and thus the hydrostaic pressure is bur one net pressure reading. We need the stagnation pressure as well for a fluid flow. This is in fact the barest minimum requirements for a reasonable model, but the strain ellipsoid model actually allows us to independently measure 16 points of varying pressure on the eliptical boundary. This pressure varies because of the instantaneous motions of the strain ellipse in an instantaneously varying laminar flow field(such as near a boundary).

So Bernooulli needs to be updated by placing pressure meters at the various points around the ellipse and upstream of the flow as well as downstream.

The conservation law reveals that a cylinder of fluid is deformed longitudinally and cross sectionally. The assumption is that the material points push each other into the respective positions under the influence of an external pressure. However this cannot be the case at the boundary of the pressure systems. For example a point at the boundary has to move at zero velocity. If there is no viscosity these points are lost to the fluid body in motion, the matter is not conserved, except pragmatically.

Secondly the pressure "forces" material points into each other. Do they coalesce or do they separate a space between material points so they can "fit" in?

Logically they must coalesce, so again matter is not conserved. If we say they create "space" in a continuous or contiguous medium we have to acknowledge that work is being done in a pressure background! Thus to conserve matter we have to do work. Where does the energy to do this work come from?

<a href="http://www.youtube.com/v/QWEq3xifCDw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/QWEq3xifCDw&rel=1&fs=1&hd=1</a>

In our strain models both rigid and fluid we have not accounted for energy or work done in conserving matter. In our Thermodynamics we hav not accounted for work done in conserving matter. In our electrodynamics we have not accounted for work done in conserving matter.

We can derive Bernoulli like principles from the point of view of work done in going from one state or configuration of matter to another. The terms are recogniseable as so called energy terms. What is not realised is that these terms represent the work done to conserve matter as well as to move it and transform it.

We have a conservation rule for energy in certain mechanical systems. It is an extremely useful principle but we are not sure if energy is consrved in all interactions. We certainly know it transforms the matter involved in the interaction as just described, translating it, conserving it , rotating it, but we have to consider the triboelectric and tribomagnetic effects of "collisions" with pressure fields . in fact we have to consider whether consrvation laws apply to these transformations of form, that is into electric and magnetic and thermal pressure systems. In fact is their a conservation of "pressure" rule!

Since the Dirac Fiasco, Physics has lived with this notion that energy can be created and destroyed. This was dreamed up to avoid having negative energy, a concept that goes against the laws of thermodynamics! But since , i sayy there is no real justification empirically for excluding negative energy, maybe we ought to revise the Thermodynamical laws, and see them for what they are part of a greater role for some metaphysical notion called Energy which can transform translate and conserve matter, or rather spacematter, and it can do this dynamically in tatic or dynamic equilibrium statusee as well as in explosive dynamic statuses, but it does so because their are contra forms of Energy just as there are contra forms of everything else elemental to our reality experience.

So as you can see the strain ellipse. modified by energy or work considerations may be a way forward., providing we break free from the notion that laminar flowconserves matter within a boundary, or that Hookes law conserves matter within a boundary. Material points in a dynamic situation rotate through and conservation boundary on a trochoidal path, eventually failing because no new internal material points can rotate to the boundary and external material points then rotate into the material. This is the failure of the conservation of matter work done by energy in that material with that viscosity, The work done by the energy from that failure point onwards travels ino the breaking of electromagnetic and nuclear bonds, as we understand them, and into the radiation of "heat" and the rotational motion of the material points exhibits as sound and heat vibrations as well as light rotational disturbances in spacematter(Aether or Ether or a Lagrangian or Hamiltonian Description of material point behaviour).
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« Reply #23 on: May 12, 2013, 11:56:36 AM »

Another important aspect of modeling fluid dynamics is the model of terminal velocity.
Using the strain ellipse methods in this situation may be revealing, because terminal velocity clearly gives us a mechanism for generating the velocity gradient field close to a moving object.

This velocity gradient field is again based on a potential flow equation which may be a better model for strain ellipse method than the ubiquitous shear differential.
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« Reply #24 on: May 13, 2013, 06:07:49 AM »

Thinking about the strain ellipsoid made me realise that one solution for the flow(i think laminar) is 8 contra "eye like " forms facing each other contiguously with antysymmetric rotational flow patterns, all shaped in the ellipsoidal boundary.

 The image I have in my head is like an ellipse with each octant around 3 d axes having a flow that ranges from hyperbolic to circular as the test point moves out from the centre. These are shaped within the ellipsoid boundary. 2 of this type of strain ellipsoid form a stable orthogonal pattern that mimics " N–S" polarity mechanically for the whole system of 16 "eye-lets"

While this is not a definitive solution , being one of many, i think it is characteristic of a great many strain situations including electromagnetic. in fact i might consider a work up that replace the dipole with a strain ellipse!
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« Reply #25 on: June 05, 2013, 06:53:52 AM »

I am currently finishing off some other research while assimilating universal hyperbolic geometry.

I intuit that this is in fact the natural geometry for fluid dynamics and may provide a natural insight into the pressure ellipsoid and the strain ellipsoid for a general fluid. The natural vorticity in these forms may be sufficient to describe the centripetal centrifugal tangential frame devised by Apollonius, and adopted by Newton as the natural descriptive geometry for the action of motive as an acceleration of a body at all scales from the point to the distinguished form,

Within this geometry vectors etc are derived naturally as projected objects/entities, and boundaries arise as resultants of complex fractal iterations of projective processes. That mens they are ephemeral yet clearly experienceable, suitably described by the notion of instantaneous formations.

For me, the Elliptical error is explained best by iterative processes in universal hyperbolic geometry.

http://my.opera.com/jehovajah/blog/2012/12/10/the-elliptical-error
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« Reply #26 on: June 23, 2013, 01:38:43 PM »

Just back off holiday. Visited the Alhambra in Spain. Fractal Mosaics or what!

Anyway watching the Actual waves in the Ocean confirmed that the sine function as a wave model is totally misleading. Actual waves are trochoidal in 3d, that is they form ellipsoidal vortices that feed into each other , but whose centres do not move. The matter of which they consist rolls around these centres transferring not matter but motive, Newtonian source of acceleration!

The structure of these Actual waves is also interesting. The waves not only are ellipsoidal or trochoidal in one axis but in all axes. Thus the material rotates in a trochoidal pattern around the centres . Those who have followed my research know of the work of Lazarus Plath, a pseudonym for a creative programmer. Find him on YouTube under qqazxxsw.

<a href="http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rz8A5l_yn34&rel=1&fs=1&hd=1</a>

The Trochoids his programmes can create are deeply informative, but based on the circle. Although he provides real time variation this is only sinusoidal. However the number of trochoids that can be " layered" is phenomenal, so this is not such a limitation as at first seems.

The trochoidal wave forms in the ocean demonstrate a surface feature in which they appear to be rolling " forward"  toward the beach only. But in fact they also roll " sideways" along the beach at the same time. Thus the wave is not a parallel phenomenon but a radiative one, and the radiation is not concentric either!

These standard concepts are in fact explanatory models that mislead the student. The behaviour of actual waves in the sea are as interlocking, fractal vorticular regions. That is vorticular shells radiate out from any " motive" source in a liquid substance. The ripple effect is quite noticeable but it's fractal nature is not. It is only in the coronal display of the initial " splash down" that this ellipsoidal/ trochoidal behaviour is distinct enough to overcome the surface " tension" effects of a liquid.

<a href="http://www.youtube.com/v/VORN5SHiT7A&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/VORN5SHiT7A&rel=1&fs=1&hd=1</a>

The surface structure of the liquid is also fundamentally different. When an actual wave " breaks " on the shore the regional ellipsoid/ trochoid gains kinetic energy features that spread it out rapidly into a foam on a sandy beach or a highly projected film on a smooth surface.

This film returns into the back wash depeni g on the "gravitational" effect of its potential energy. As it returns it is actually lifted into a film of returning liquid which rides on top of the advancing ellipsoidal waves. Thus a thin layer of returning liquid is shaping the rolling advancing " wave" front. This thin layer returns into the advancing ellipsoidal spread and appears to do so smoothly. But this depends on the varying energy  of the advancing wave front. A test floater reveal the vorticity in this layer which generally follows a largely predictable path, but which also shows extreme and unexpected variety.

It was noticeable that for certain conditions this film provided a current that took the floater out into the ocean returning it to shore unexpectedly, sometimes to one side sometimes to the other side of its entry point. There was a region where the tester was never taken out to sea but merely washed in and out slowly moving along the shore line.

This set of observations was applicable to a liquid, but in a gas, the centres of the ellipsoids or trochoids themselves move arbitrarily.

The use of a background flow in which these centres are embedded is the classic description of fluid behaviours under " laminar " flow. However laminar flow is a special boundary case for objects moving in a fluid. The general motion is left uninvestigated, although interest in general turbulent regions within and outside of boundary flow is of great interest.

<a href="http://www.youtube.com/v/7aIMapx2pag&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/7aIMapx2pag&rel=1&fs=1&hd=1</a>

The fractal model provides the best in class approach to modelling these fluids, but I believe that the trochoidal formulation of the flow will provide the most general apprehension of what is the potential structure of kinetic and potential energies or Newtonian Motives in fluid mechanics.

http://www.fractalforums.com/gallery/hyperbolic-fractal-infinite-trip-animated-gif/msg63722/

Another great example by Bib
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« Reply #27 on: July 11, 2013, 11:26:11 AM »

I have 2 resources on Newtonian Fluid Mechanics.

http://my.opera.com/jehovajah/blog/2013/07/05/newtonian-fluid-motive-as-a-newyonian-active-principle

http://my.opera.com/jehovajah/blog/newtons-principia-on-fluid-mechanics

The main point being that Newton retreated back to his model in book 1 of the Principia only after considerable effort to crack the more complex fluid mechanics, in which he made not a little headway.

The fractal complexity of the situation is not what defeated him. He had the correct tools and intuitions, honed by years of meditative study of the Ancient Mechanical Philosophies. He also used uniformity as his gateway into the fractal world of nature. While I prefer to approach fractal nature from a fractal Mosaic, essentially newtons methods of fluents and Fluxions is the approach, both rhetorically and algebraically, and spaciometric ally.

The notion of infinitesimals current in Newtons day, backed by growing microscopic evidence of corpuscular forms, notably Hookes and Loewenbrooks treatises encouraged Newyon in hi application of the methods of exhaustion from the Greeks. Where he differed from the European school including Leibniz scholars, was in the rhetoric on limits. His limits never vanished. They were in fact ultimate ratios or proportions of evanescent or nascent quantities, which themselves were formed out of magnitudes by some boundary process.

Newton therefore was always pragmatic and perisos, that is approximate, rather than Artios which is precisely accurate!

What defeated Newton was the degrees of freedom in 3d space. There are 6 degrees of freedom that are orthogonal, and innumerable axes of rotation. In the planar case the 4 degrees of freedom are complemented by only one non planar axis of rotation. Thus , given a rigid body, reducing it to the planar case captures every possibility for that plane using the rectilineal axes ans the circular adjunct to them. This reference frame , in the plane gives every point 6 degrees of freedom as a basis " vector" .

Generally we split this basis into overlapping Cartesian and Polar coordinate frames in the plane.

The important notion of a spanning basis was actually first expressed by Hermann Grassmann, but it was clearly intuited by Newton, and forms part of the Toolbox Apollonius used in his work on Conics. Possibly Euclid's Conics carry the same notions. This reference frame in the plane is in fact the reference frame for universal hyperbolic geometry. I would strongly recommend viewing Norman's lectures on the subject.

So it is clear to me that Newton's general" geometry" or rather Spaciometry was that of Apollonius, and so was a hyperbolic geometry. The fuss created by Gauss, through Riemann was more an attempt to garner the intellectual high ground on the topic to himself. He did little to encourage Bolyai, and he attempted to suborn Lobachewsky by learning Russian so as to write influential commentary on Lobachewsky's work.

It is ironic, therefore that the gift he turned away, when dismissing Grassmann's work, was the very thing he was aspiring to: a great work to maintain Prussia's intellectual standing and to enhance it! And delivered by Gauss as a gift to the Prussian nation and the Emperor.

Newton's attempts to tame the surging tides of fluids by equating the fluids to infinitesimal solid cylinders go infinite length fails precisely there! The cylinders of fluids are by no means rigid. Nor is there any reason to suppose that they form independent boundaries! Thus all 6 degrees of freedom have to be used to describe the motion of a point in these cylinders and in addition an innumerable number of spherical shells representing an innumerable number of axes of rotations.

Bearing this in mind it is no wonder his work on fluid vortices failed to be accurate enough!

Universal hyperbolic geometry now offers a way to project these degrees of freedom onto suitable planes, and then to algebraically return to the 3 dimensional situation.

In addition, the advent of computing power makes this a much more manageable but still overwhelming task.. Computational Fluid Mechanics attempts to do this task, but I believe using a less elegant approach than UHG.

We shall see.

In addition, the Newtonian notion of motive has clarified for me the structural dynamics of vortices in the UHG and leads me to posit a 3 region structure to a fluid element, which will not necessarily be a cube, but more generally a bubble with an interior spiral, a bubble skin with a trochoidal dynamic and an outer region where high energy interactions take place with the bubble skin driving the internal dynamics of both skin and bubble interior, or transmitting the internal dynamics of both interior and bubble skin.

This skin interaction is studied under surface tension, and that needs to be fully exposited in its transmissible nd receptive roles.

Finally, while this naturally applies to liquids, fluids in fact are a more nebulous aspect of SpaceMatter. Consequently the principles discovered in the liquid case will also have analogous implications for the more ephemeral and nebulous gases, plasmas and electric and magnetic "fluids".

I would also add that in none of these concepts will mass be defined by gravity, nor will gravity be defined by mass. The notions of Mass will be entirely Newtonian, and defined by the action of a body under the influence of motive. Consequently, as we shall see Gravity will be the resultant action of a number of motives which will all be physical in the classical sense of Phusis. That means they will all have opposing motives!
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« Reply #28 on: July 17, 2013, 09:02:46 AM »



This is a trochoidal description of a fluid . In its simplest form it shows many interesting structures, which are constructs. They construct the spatial regions where strain is in equilibrium and thus potentially at its greatest. The dark areas show where stress has broken the equilibrium in the many directional attitude of pressure.

the effect of a global pressure may be evident but it is very slight, so no clear vorticity streans are evident.

The next stage is to explore many motive laws to see what boundary conditions they generate!

Note the difference. the boundary conditions within the fluid are generated by the motive laws. We only observe them when they achieve some kind of equilibrium, otherwise they are beyond our senses.
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« Reply #29 on: July 26, 2013, 09:48:36 AM »

I have some interesting structural results , but the margin here is too small to write them in.
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