When Newton moved into investigating fluid dynamics he promised fluids as resistive media.

Thus fluids( liquids and gases) were characterised by their resistive effect on solid definite bodies . He coined the notion of Lubricity ( or rather defined it by a differential expression) . As you can guess it was avrathernslippery notion! Today we turn the notion tomitsbinverse and use the idea of viscosity, that is how sticky or resistive to translationl shearing or rotational shearing a material form is.

You will note that this is Newtons expressed idea of resistance and it is assigned to the medium . The idea of lubricity was assigned to the progress of the solid object passing through the medium, a crucial change in thought patterning!

Newton experimented with rotational lubricity by spinning a bucket by a twist in a ope, and further by pulling that spinning bucket inna circular( rather elliptical) trajectory around his body . However he also did experiments where he spun a long cylinder in a pool of liquid to determin how the liquid responded.

In all cases he noted a differential velocity normal to the moving solid . For. Liquid it is particularly observable in the spinning bucket that liquids tended to climb up the bucket wall when spun indicating a radial or centrifugal force was generated.

Most fluid Dynmics courses start with the differential force equation ascribed to Newtonin fluids( incompressible! That the shear force is proportional to the differential velocity profile.

Thus in an orthogonal reference plane

f

_{x}=constant*d

**v**/dy

This isvn expression, but it is often called a differential equation. Asbnnequation it makes little sense . It makes so little sense that no one bothers to explain it! It cannot be explained physically as an equation because clearly the measures are Not physically EQUAL.

It is a proportional Expressiom, like most differential "equations" derived from physical phenomena are.

If like me you were trained in lassical mathemantics you were introduced to proportions as. Kind of fractional process, and you may have wondered why the different approach to numbers. .i have dealt with the long answer to that question in the fractal foundation thread, but the hort newer is that fractions came out of performing proportions , and performing proportions is what we fundamentally hav to do to interact with our material world. More often than not a proportion captures an analogous link between non homogenous phenomena and it is this link that has.a detetminsble( sometimes irrational) quantity.

The shear force proportion holds within certain boundary conditions , and Newton then used his differential calculus approach to sum these differential proportions for the rotational hear round a cylinder. However this was not a Mathematical manipulation, but an experimental one. He measured the velocity of the fluid assumed to move circularly in bands whose lubricity was such that the circular bands slipped past each other with reducing effect . The assymptotic inverse power laws thus found and plotted are very striking and iewable in the latter part of the second book in the Prinipals for Astrologers .

So whatever the mathematical proportion was for a liquid in shear, when applied to circular or orbital motion the sum of these differential proportions was proportional to an inverse power law.

Using these experimental results Newtonntried to fit the observed orbital behaviours of planets to this mathematical fluid dynmic model. The results were disappointing. In fact his point mass model could be made to produce highly accurate and reliable results! So Newton said to following generations of Astrologers "Go Figure!"

http://m.youtube.com/watch?v=x1SgmFa0r04So D'Alembert,LaGrange, Bernoulli's,Euler all worked on the problem of using the differential proportion expressions to capture fluid dynamics, with some noteable contributions. However a paradox arose between the mathematical proportion model and the experimental results. D'Alembert showed that the models could not resolve the flow around a cylinder! That is the mathematical models were missing an essential factor. And yet the models were rigorously checked and no mistake was found in design!

So Navier and Stokes separately developed careful experimental data to check the differential proportion expressions and to complexity them not to leave anything out . In the meantime Helmholtz was applying the Lagrangian analytical method to some noteable fluid structures namely vortices and getting good results. So much so that Lord Kelvin teamed up with him to further advance the implications of their results. Both hoped that vortices would provide the answer to the Knotty mathematical problem of describing molecules .

Also at this time Maxwell was intrigued by the vorticular nature of many phenomena in space and the results of Navier and Stoked( not a great deal mathematically but experimental results were abundant) and Helmholtz , and the experimental results of Faraday.

What he hoped was a fluid dynamic breakthrough would link magnetism and electricism together and explain how strain could be passed through seemingly empty space! .

He had contributed to Boltzmann and Gauss statistical approach to gas behaviours by developing the root mean square path distance between interactions of presumably solid or elastic corpuscles . This was vital to Kelvins Kinetic theory of Gases and contributed in no small way to its adoption into the scientific mythic records! But it was inadequate to explain magnetic snd Electra magnetic behaviours.

Maxwell's plenum was filled with vortices . However he could not grasp the tru nature of fluids because he was brought up in the mechanical gears and pulleys era. His model of a fluid had to be based on gear structures! Therefore in his plenum each vortex had to be surrounded by counter rotating bearings or idler wheels . He could not rid himself of the many difficulties that model produced.

Surely a liquid could not go in two directions at once? What about the frictional. Inertial and body forces!

He learned to bury these difficulties in the constants of any proportion, and is often cited as discovering thatb2 of these constants are linked by the speed of light . In fact this is a facile observation .

The bulk properties of a material determine the wave/ deformation speed within that material . For the assumed wave nature of light and associated magnetic behaviours the speed of light in a material is the only bulk property that characterises the material , especially true when determining a material spectroscopic ally by its wavelength/ frequency spectrum'

The main point here is that from Newton onwards the behaviour of fluids has not been sufficiently described in terms of fluid characteristics: namely pressure per unit volume, unit area unit arc or line segment . Nor has this lineal approach been usefully characterised in data. That is until now and the implementation of satellite technology and lineal algebra.

The fluid paradigm means that solids no longer predominate but rather viscous material " points" or rather. Elements .

Using these viscous finite elements and a lineal algebraic approach computational models of fluids have now been made in which the fractal structure of the computation plays a key role( iteration of smaller and smaller scales of measurements) and the lineal displacements or arc segment displacements are recognised .

Thus a fluid element can translate or rotate or shear against a neighbouring fluid element without the need for lubricated intermedaries or idler wheels, and a fluid surface can move in several "directions" or modes at once!

Claes Johnson has taken this method and run with it. He has resolved D'Alemberts paradox and corrected Helmholtz and Kelvins Vortex Kinematics assumptions and eliminated Maxwells reliance on mechanical gear analogies.

Of course the finite element method relies on fractal computations something even Newton could not devote time to! Both Wallis and Newton and many noteable astrologers were phenomenal calculators of proportions often doing them in their sleep or meditation time. What they could see as they calculated and what they could express in words were often beyond the ken of men like Bishop Berkely or other critics of Leibnitz differential Geometry or Newtons Fluxions .

Today computers do the work even for the most phenomenal human calculators, and because of that our mathematical models can deal with the complexity and give us proportional results.