The Spheroidal ellipsoids as models of regional pressure.
The strain ellipse allows one to model strain deformation in a laminar flow. The idea is to use some useful properties of an ellipse to model both rotation and strain in a circular fluid element. The assumptions are that the circle is sheared in 2 directions by a linear velocity profile which is anti symmetric . This velocity profile is a property of fluids sheared between 2 rigid boundaries, but observed in certain viscous fluids.
In fluids with such a shearing " force" the velocity profile is visualised by small bubbles flowing in and along with the fluid. The shear " profile" is seen only in a region near the boundaries and is dependent on the viscosity. Of the liquid. The greater the viscosity the larger the region in which this flow is seen.
The profile was first discussed by Newton in terms of lubricity. It seems likely that he noticed it in spinning buckets containing water. What he noticed was the slow progression of rotational motion toward the centre of the fluid mass and the rising of the mass up the side of the rotating bucket.. His comment that the velocity of the streams of water appeared to depend on the lubricity of the fluid was taken to justify that the shear " force" was equatable to some constant times the velocity gradient.
Having now found access to Newtons book 2 I can make a correction in some statements about his opinion. His opinion was that his theory in book 1 gave a better description than his theory in Book 2. By this he does not conclude that vortices are not tenable as descriptions of planetary motions, but rather they perplex the explanation, and by his premises contradict the empirical data.
It is fair to say that he gave a reasonable stab at a vorticular explanation, but several important bits of empirical data are missing from his assumptions, not the least the correct relation and propagation of pressure through resisting media.
It is also good to see that Newton proposed this standard of fluid dynamics as a hypothesis! Later he remarks that it is unquestionably a wrong hypothesis, but the best he could obtain!
Again, Newton and laminar flow are not synonymous. He expects viscosity because he uses the phrase " the want of lubricity!". Thus I feel I can now define viscosity as the want of lubricity.
This is not an inverse description, but rather a negated description: viscosity is "not lubricity", or contra lubricity. Thus Newton anticipates a fluid in which a body experiences no resistance as being perfectly lubricious. Thus no viscosity is not viscosity which is not ( not lubricity) , and not not cancels leaving lubricity.
Any proportionality is therefore entirely within the chosen descriptor. Today we use viscosity, and we may expect it to be proportional to the transfer of motion through a fluid.. The use of a velocity gradient, even in a steady flow system is therefore entirely misleading!
The analytical and observed system is a dynamic transfer of velocity through the fluid medium. Thus the medium is accelerated relative to the driving force and this is communicated to the body being moved. The velocity gradient is a visible experience of that acceleration or deceleration.
The velocity gradient therefore is not a slope in space, but a slope in sequential time. The slope in space is a kind of Hookes law of the shear force, whereas we ought to expect, and certainly of Newton, a law bases on the second derivative with respect to time!
In a steady flow, and in Eulerian reference frames we should expect to see a dynamically stable equilibrium deceleration! Or acceleration, in other words a constant acceleration. The second derivative would be a constant, which is what we see a constant change in velocity.
The fact that this change in velocity is spread throughout the medium . Instead of in a line, gives us additional information about force. The best explanation is in fact that force should be replaced by a volumetric pressure. The velocity gradient is then a consequence not of the "shear force" but rather the shearing pressure in the flow which acts multidirectionally.
Using a radiating pressure and resolving it through a surface into normal and tangential forces helps to explain fluid behaviour more empirically.