Newton’s Principia on Fluid Mechanics


http://en.wikipedia.org/wiki/Philosophiae_Naturalis_Principia_Mathematica

The complete English Breakfast!
http://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)

http://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookII-IX
In which Newton puzzles over is concept of a Vortex but misses the motive of vorticty, that is curved or angular acceleration.

Thus let

Ω

be angular acceleration of a body about a centre, and m the measure of the quantity of density spread throughout its volume
And density be that ratio of its volume compared to pure water which balances a rotating pulley between the two;

Then let the measure called Twistorque(T) be defined thus

T = mΩ

by which we may define the measure Force that balances it

F=Tr

r being the radius at which the pressure is resolved into force.

Thus we may identify tha the lineal acceleration a is related thusly

F= ma =tr
a=Ωr.

What we can conclude is that for every lineal acceleration we may posit an identical rotational one acting at a distance r.

Seeing thereby the naturalness of circular action at a distance given a medium that transmits the pressure and resolves it there!

That a medium should exist in somewhat empty space is doubtless to be doubted; yet having such evidence as we do in that which is fluid let us not dither, but press on.

Now it seems that this fluid material is hard to get a handle on, but it is bounded by a container. The formulation that Newton devised takes the container out of the process if needs be, replacing it by a ratio. What this means is that we can talk generally about any space, but not differentially.

The calculus of differenials may allow us to get into these details, if needed, but in fact we have moved to the paradigm of Energy or work.

Work is defined as pressure acting on a surface of a mass and moving that mass a distance. But a pressure usually accelerates a mass so work is being done continuously to accelerate, Once acceleration stops, no mor work is being done, even theoug the mass is still moving!. This is considered to be the kinetic energy of the mass in motion, in other word it is the direct analogy to the celerity in an object.

Work like a mptive, increases the kinetic energy like celerity is increased by motive. the proportions of this relaton are generally specified in the energy identities.

energy is not new, but the particular formulation of it has undergone several revisions over time. The concept of it is a mysterious metaphysics that physicists rarely want to get into. Well , for what it is worth, einstein called this energy motive, and ecognised the inertial motive as well as the pressure motive. Tese correspond to the kinetic and potential energy of a system when its equilibrium is perturbed.

Returning to my force identites , the definition of twistorque is a angular force, but the definition of lineal force can be seen as an angukar energy or angular work. However this does not work dimensionally because the concept of work is a resolved one: a resolved force moves a mass an actual distance.. Thus angukar work or energy will meed to be defined by

W=TøR/r

where øR/r gives the resolving proportion; that is where the energy is actually drawn out of the system in rotation.

this is different. in the defining of work it is assumed that the mass remaons coherent and tha pressure is resolved into the coherent mass. However, in a fluid it is evident, upon inspection, that the energy is not distributed uniformly instantaneously, and in a rotating system, it is evident the energy is distributed radially. The consequence of this energy density vaiation is conflated into Newtons Third and fourth laws of motion for the solid, but for the liquid and gas situations, these laws hold only to a crtain point of coherence!

Thus in fluid dynamics, work done can result in the fluid losing its cohesiveness and coherency and we need laws of motion to describe that.

It ould seem clear that we need a form that a body transforms into. This form i propose is the ellipsoid. which further fractalises into smaller ellipsoids according to boundary conditions that must reflect conservation of spacematter,conservation of total work done, conservation of the centres of vorticity within a medium.

This last conservation law is the hardest to determone, but has to be there if we have a preservation of regional distributivity of the fractal regions within a substance.. If regions can be created and destroyed, this means that at the large scale work cannot be done to preserve spacematter! Thus points would coallesce and we would not see the deformation of matter having any effect on volume rate of flow!

The micro scale is fratally similar to the macroscale when we see turbulence. That is the conservation laws are being applied at nearly all scales. What determines the turbulent structure seems to be the compound of the mixture, At a certain work load the mixture begins to show where the boundaries of compounding are, and we see this in real time as a turbulent flow. This is the several subatances responding differently to the avalilabelenergy of excitation, or work, while also bein constrained by the energy that is compounding them.

Thus a straon ellipsoid shows what 2 centres of rotation would do if stressed apart. The strain between the 2 centres of rotation which were formally one defines an eeliptic form.

What if the centre is strained into 3 centres, what form is determined then?

We can clearly investigate forms associated with x numbers of centres of rotation casued by straining the substances apart, but the rule we need is one that tell us the likely number of centres that will appear under what work load, Thinking backwards we could determine from a chemical analysis what the elements are and at what energy they disociate, this would give us a count of the number of strained centres likely to develop at given workloads

We have moved into the area of statisitcs.

To describe a complex urbulent flow we mus know the work being done on the flowing material, the strain profile for that compound in the form of a Strain distribution for probable straincentres and then likely spatiometric forms for these transforming regional structures. These will be in dynamic rotational transformation at all scales, snd the micro spatiometric gorms will impat on the larger scale regional behaviours. We will observe Turbulence, but characterised by a definite pattern..

The overall result would be he resultant of a work or energy analysisi of the flow, accounting for kinetc and inertail /potential dynamicboth pressure anr reactive pressuresat all levels with the strain pattern in the medium dynamically, in timesteps describing turbulence.

Thus we want a set of identites that describee spatial relations across the regions for every time step or videoframe. Placing all these frames into a video should show turbulence characteristic of the material.

Since i think Universal Hyperbolic geometry is a natural spaciometry for fluid mechanics i will have to define celocity, acceleration force and work in these terms.

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