Stress and Strain in Fluid Mecanics

In an earlier post I state that a fluid is not a continuum, but a fractal disposition of contiguous regions at all scales. Elsewhere I state that a region is continuous until it is not! The reason for these statements is to highlight the abstract nature of the notion of continuous, and the linking to differentiability.

2 lines may be discontinuous and discrete, or continuous and discrete. We call the second case contiguous. We may make these abstract forms collinear etc but that does not make them identical unless we define that case. In fact I have had to define collinearity as identity in order to justify that a constructed diameter to a circle, that is rigorously a constructed set of dual points, passing through the centre of a circle actually goes through a unique dual point on either side of the circle. Unless collinearity is defined as identity I can only point to the case by case verification by drawing.

Thus a contiguous continuity may be collinear with a continuous line . If I make both lines identical I am blurring the distinction between contiguous and continuos. I prefer to call it what it is, an imposed description on the same line.

Contiguity for a surface or a solid is easier to distinguish. A line crossing an imposed boundary reveals whether the surface is continuous or contiguous by where it continues after the boundary! Surfaces and solids can be contiguous along a line or plane, but shifted along the boundary of contact. In a solid this is associated with a break in the medium, allowing some other substance to intersperse, but in a fluid, such a break is not necessary. A fluid is to be characterised by these instantaneous boundary shifts where contiguity is preserved but continuity is not.

http://www.math.harvard.edu/~knill/history/matrix/bell/index.html

How does this relate to stress and strain? As usual these notions are derived from rigid or semi rigid models. They imply an elasticity to the rigid models, and elasticity is taken to imply continuous deformation under the load( stress).

There are tensile ( pulling apart) and compressive( pushing together) stresses and these dorm the rigid bodies shape or and size.

How are shear forces and viscosity related to this model and where does vorticity fit in?

The most important conceptual change is from force to pressure. One has to derive force from pressure,

Pressure acts in a volume, and it I here's in a volume? We measure pressure by how that voluminous potential acts on a surface.
Pressure as a potential acts radially, but it does not inherent uniformly. Thus the pressure in a volume , if it is in some external pressure gradient tends to reflect that pressure gradient.

A pressure gradient is measured by the force on a unit area or a unit volume if small enough at each position in a reference frame. The force is the acceleration of some object, and is measured by netting this acceleration out, that is to say what counter acceleration is needed to keep the test area or volume static.

The kinematics of hydrostatics usually define the notion of pressure well.

So now we have to understand how fluid dynamic conditions enlarge the notions of stress and strain and shear both tensile and compressive.

The most important phenomenon I I want to get across is that in a fluid these pressures are dynamically transmitted by strain waves or undulations, and that these oscillations propagate in definite velocities that distribute the strain throughout any given region. In particular, these strain oscillations reflect at boundaries, refract and diffract in ways analogous to the so called "wave" properties of light.

One of the mysteries of diffraction is how waves bend after going through a narrow slit. The answer is given by the venturriflow effect, and the Bernoulli principle. In the case of light we have to consider the effect of super luminal flow acting in less than a pico second in a venturri flow tube., but my main interest is in the initial shear that a pressure system drives through a bounded region with a small gap, even if this boundaries is practically 2 dimensional. The idea is to model the dielectric flow around a battery with 2 wires attached to a capacitor.

http://cnx.org/content/m42206/latest/?collection=col11406/latest

http://youtu.be/9fIrmjxlT18

Notice how the strain ellipse is like a balloon expanding and stretching and elongating.
Using the strain ellipsoid I believe I can account for dark energy in a fluid dynamic universe in which the fluid is SpaceMatter.. The notions of SpaceMatter also account for dark energy. The notions of the thermodynamical laws are revised , particularly the second law which applies to closed systems only, and takes no account of a fractal disposition of regions within a fluidic SpaceMatter, or a scale free almost self similarity. Fractal entrainment is not included, and so the fractal disposition of motion in a turbulent fluid flow is left out of the enthalpic description.

Concentrating on entropy instead of Enthalpy is the misleading emphasis of moderns thermodynamics.

The empirical data is clear, forms are " created" and" destroyed" at all scales. The observation Panta Rhei expresses this obliquely. Modern diencephalic expresses it clearly, but has lost site of the bigger picture in academic argument and subject boundary wars.

To understand dark matter I simply observe that SpaceMatter is apprehended by substances, which are defined as perceptions of space matter. Our current understanding allows us to develop a fractal structure to these substance perceptions . My understanding is, in the broad substance category of plasma an electric substance sheaths a more ponderous magnetic substance. We may also perceive , at any rate by energy statuses, more substances sheathed by these electric and magnetic substances, forming a fractal substructure within the magnetic within the electric substance(s).

Given this fractal distribution we have to ask , at least under current formulations, what is the density of these substances. The problem is our notions of density are based on a comparison with the density of water. We cannot accurately weigh these finer substances because we cannot actually isolate them. Yet we can apprehend them and distinguish them. The underlying difficulty, the elephant in the room is the misapprehension of mass, which makes gravity, an unknown force, the dominant force for measurement.

The obvious and possibly simplest thing to do is to scrap mass, return to the notion of density as a comparison of substances in a force field and choose a clearer , stronger force in which to measure, either electric force or magnetic force would be candidates. The only reason this has not been done I suppose is because of the logical tautology( I am discounting vested interests which may be a more significant reason). However , logical tautology is how we define things in any case, despite pundits trying to argue to the contrary. Iteration and recursion allow tautology to differentiate qualities distinctly and clearly, because it is the ralmost self similarity of tautology that is ignored. The same object viewed by 2 observers, by the iterative processes within them develop a similar but distinctive description. This is a well attested fact of evidential reporting by witnesses. A judge is appointed to give a singular decision based on both evidences, and usually this is done by comparing and contrasting. What the judge decides is relevant can be contested, but we usually consent to the decision because the purpose of the process is to get a single consensus view.

The use of consensus clearly leaves out many original apprehensions but apparently this is how societies work. The individual, on the other hand is often free( but not always encouraged to) to research outside the consensus box. This can lead to upset in the consensus view which, despite worthy statements to the contrary, is not always welcome.

Once we have defined the pressure /potential "field" in which we will compare the density of substances against a standard substance in a standard volume, we may be able to then use newtons definition of the quantity of matter appropriately. I am sure that this will reveal the analogous nature of dark matter and substantially throw light onto the arcane processes by which we come up with such ideas.

Will this remove dark matter and dark energy? Not really. The notions themselves arise out of what we do not know, and that will always be larger than what we do know, but for the current generation, as Feynman said, "it will be good enough!".