The fundamental axiom of fluid mechanics has to be space is fluid.
If space is fluid it must also be flowing in arbitrary motion Panta Rhei. .
If space is in motion then rest must be relative to some observer and his local reference frame or some observable and it's reference frame.
If space is fluid then rigidity must be relative to some observer and his measure of rigidity or some observables reference frame and it's relative constancy of form.
If space is fluid then the quantity of space shall be the volume of fluid conjuncted with its density, and it's density will be how rigid and consistent it may be under pressure.
A fluid space my receive and exert pressure throughout its volume, and such a pressure may divide itself proportionately to each volume of space according to its density. In this case the greater density may shear from the lesser in proportion to the rigidity of the consistency and in relation to the lubricity that exists between each volume as a region, so that the greater the lubricity the greater the shear.
If space is of sufficient rigidity the shear may occasion a break between regions of hig rigidity through sudden and impulsive introduction of a thin region of low rigidity and high lubricity, or permanent deformation through the rigidity of the volume temporarily changing its nature and attendant properties, or permanently changing its nature and attendant properties.
In a fluid space a instantaneous and extremely large pressure for each volume in an exhaustively small region may exist.
The quantity of pressure will be measured by what quantity of fluid is accelerated.
The quantities of velocity and acceleration of fluid will be measured by a stochastic process of assaying the varying displacements of test volumes paced in the fluid.
The motion of the fluid will be modelled by the motion of the test volumes as current streams and the vibration of the volumes as pressure cavities..
A pressure cavity is a region of fluid with its own local reference frame relative to which the fluid motion of the region is measured as a result of applied patterns of pressure.
Fluid mechanics is a paradigm shift that is unacknowledged. Newton's laws of motion were specifically formulated to enable him to tackle fluid dynamical issues, namel the vortex. Despite a valiant effort he ended up admitting confusion. The reason is the Arithmos that he relied upon.
An Arithmos is a mosaic made up of monads or Metrons. The Metron Neeton unconsciously chose was that of a corpuscle. Despite the inherent flexibility of the corpuscle he could not help but give it a rigid perfect form, one of the platonic solids in 3d space. His sketchbook shows he was capable of investigating any solid shape built of these forms, and his method of fluents gave him the dynamical modelling capability he would need, but he could not see the Arithmos he needed to tackle fluid motion
The fundamental Arithmos require is that of a Fractal bubble structure with flexible and compressible oscillatory boundaries.
It is normal in fluid mechanics to deal with a cube, as a truncated part of a stream. The cube is rigid , uniform and part of a uniform streame .
I replace the cube with a spheroidal bubble, part of a fractal stream with non uniform structure, and with a surface that is flexible and oscillatory thus oscillating the spheroidal volume and perturbing the pressure internally but maintaining the total spheroid pressure as constant. The dynamics of the boundary are that of a surface tension hich is equal in total pressure to the internal pressure.
The simplest stream therefore will be such a bubble taking an elongated form. The flow of the stream will be the translation of this bubble relative to its contiguous streams.
The choice of monad is crucial to making a mosaic of the form.by using fractal spheroidal bubbles with local behaviours which can accommodate pressure, strain, shear and oscillation at the boundary we develop a more accurate fluid model.
Right from the outset we must clarify shear. The notion of friction is important. When friction is experienced it is called a shear force. A shear force is indirect to the obstruction theoretically. Thus a direction of force is usually indicated by a line. The direction of force is not helpful.
The direction of force is part of Newtons first law of motion, but this law does not give a direction of force, it gives a direction of Reaction.
Thus a pressure is placed on an object . It's reaction is instantaneously in a linear direction( by a modelling assumption)". This reaction line defines the action of the force in that instant.
The third law of motion now instantly comes into play dynamically determining the out come.
So now we have several aspects to observe ib shear flow. If the pressure forces the fluid against a boundary , the direction of instantaneous motion is into the boundary . If the boundary holds the liquid then it does not slip .
However the rest of the stream bubble surface is able to slide leading to a rotational torque in the stream bubble. Each stream bubble rotates or slips depending on the pressure interaction normal to the boundary layer interactions.
The Newtonian Shear is often illustrated as a plate moving on a stream bubble horizontally. In the analysis pressure is not introduced, instead some notion called shear is used. It is not properly defined, instead a velocity gradient is drawn and the differential of the velocity gradient is introduced. Then this is said to be some how perpendicular to the flow and the friction coefficient is thus defined, called viscosity!
This friction coefficient will be used to define a friction force In line with the flow. Thus enabling a net force between each stream layer to be defined in laminar flow.
What is aimed at is making a velocity gradient into an acceleration gradient, but by confusingly having a force acting perpendicularly to the flow.
The problem arises in this initial introduction because pressure is left out.
Pressure of the top plate is transmitted through the fluid. On moving the top plate the pressure profile remains constant , however the stream bumbles begin to interact. The stream bubble under the top plate begins to roll. The stream bubble under Neath provides a frictional reaction and it begins to roll. Eventually the bottom stream bubble is interacted and it reacts by remaining stationary. This provides the necessary frictional reaction to deform the whole bubble system.
Now it requires the surface tensions of the rolling bubbles to pull the trialling ede away from the vertical as the leading edge rolls forward into surface contact. Therefore we do not have a confusing play of forces , but a rotational system in which pressure and friction make sense.mviscosity is now how easily the bubbles deform while rolling.
The model developed up till now was a stacked set of cards moel . In such a model the frictional force must increase with depth and the translation force required to overcome it must increase, thus the net force diminishes with depth eventually becoming zero at the bottom plate. The rate of change of friction now becomes a measure of viscosity.
The Spaciometry of change of friction is a Spaciometry of change of force . The Spaciometry of change of force is a Spaciometry of change of acceleration. If the Spaciometry of change of acceleration is constant then we can plot a constant acceleration change slope. If the Spaciometry of change of acceleration is zero we can plot a constant velocity change slope. For each height In the traditional introduction the graph is assumed to be a constant velocity change graph for each of these constant velocity change slopes This implies no changing acceleration in time only in space . Thus no changing friction. The system is therefore not physical as shown.
The interpretation is that it is wrong to assume this situation can exist. We have been fooled by an inadequate analysis of the forces and the given geometry.
To rectify this we have to take the differential in time between successive layers. This gives a measure of the inter layer slip and thus the inter layer acceleration( potential ). Having this potential or fictional acceleration allows us to model a fictional or potential net force gradient which compared to the actual force at the boundary should enable us to calculate a friction or viscosity for the fluid.
The introduction of the differential velocity gradient is misleading as explained above,,and consequently shear strain and shear stress are still unphysically related.
Terminal velocity and velocity gradients in boundary layers.
The introduction of tau as a quantity of shear is misleadingly introduced as a concept of Newton. Newton 's concept was lubricity or slipperiness. It actually makes sense to say that the velocity gradient is a measure of viscosity, but then viscosity must be read as drag on an impressing body which has achieved terminal velocity under the impression of a force. At terminal velocity the net force is zero and the velocity gradient will be characteristic of the lubricity or viscosity.
When I push on the surface of a pool of water, the characteristic velocity gradients are perceptibly different if I push down compared with pushing horizontally . The terminal velocities are different, and thus so is the viscosity or slipperiness. The pressure I apply deforms the water column or the water slab at the boundary of the impressing area depending on slipperiness. In highly slippery liquids I can push a slab of liquid through the surface layer creating turbulence at the leading edge due to pressure instabilities, but the surface flow around the driving face and the edge wii retain a stable velocity gradient related to the terminal velocity issue.
This next video makes it clear without ever looking at the boundary layer the kind of velocity gradient to be expected in the layer at terminal velocity.
And one also needs to keep in mind the laws of buoyancy. In that case we recognise that the displacement of mass is what creates lifting force, but this becomes drag when the mass displacement is not sufficient to counteract the falling force. In this case the mass displacement becomes a continuing process dependent on velocity of the falling mass and the viscosity of the displaced mass . The slipperiness determines how quickly the fluid disperses. This is an accelerative process, but due to the short travel distances this acceleration also attains a dispersal terminal velocity.
This dispersal terminal velocity actually dynamically increases to a maximum governed by the edge boundary pressures. The dynamics of this system provides the overall effect of a terminal velocity.
However, we have to consider the effect of this dynamic mass displacement on the fluids, with their attendant viscosities, and tis requires us to look at the issue of turbulent flow, but not as mere turbulence. Turbulent flow has to be seen as the causative principle of what we hitherto have distinguished as heat, electricity and magnetism.