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Corner flow

Today we examine a planar flow in the quadrant |y| < x . The boundary lines form the asymptotes of hyperbolas which are the streamlines and the flow will correspond to the usual central angle with |θ| < π/4 . A constant ρ corresponds to a streamline {(ρ cosh a , ρ sinh a) : a in R } where cosh and sinh are hyperbolic cosine and sine functions with parameter hyperbolic angle a. Note that for ρ = 1, the hyperbolic angle is twice the area of the hyperbolic sector corresponding to the angle. Therefore the area of the sector between angles a and b is the same as that between angles (a + c) and (b + c), meaning that the transformation by addition of  c  to angles leaves sector areas constant. 

Suppose there is a constant hyperbolic rotation with da/dt = κ > 0. This rate also holds for dθ/dt when θ = 0. Otherwise the hyperbolic rotation corresponds to a somewhat slower turning of the Euclidean angle θ. To obtain the exact rate of change of θ one must use differential calculus on the following structural equation of this model:  tanh a = tan θ. Differentiating gives

sech^2 a (da/dt) = sec^2 θ (dθ/dt), and sec^2 θ = 1 + tanh^2 a  then implies dθ/dt = κ/ cosh 2a . So for large a the rate of change of θ is negligible and θ is confined to an interval while a is unbounded. 

Planar cross-sections of laminar flows of  incompressible fluids correspond to equi-areal  mapping. Rather than present a velocity potential with complex number based functions, the above flow suggests the simple multiplication by a split-complex number. 

In the early days of mathematical physics a professor of astronomy, Egon R. von Oppolzer wrote that in electron theory "everything is a consequence of hydrodynamical theory". Evidently he had in mind the corner flow model discussed above. That model of fluid motion was too good to leave at the water wheel. It was Hermann Minkowski who invented "proper time" for the relativistic age: the above quadrant was re-identified with the future having Now at the corner and the ρ parameter representing proper time. The relativists saw θ as an arbitrary direction of motion into the future determined by an observer’s velocity v < c through Now satisfying tan θ = v/c .The quadrant (future) is a plenum, subject to sudden shifts according to change of observer. Instead of a steady flow model of hydrodynamics, the relativistic picture of spacetime depends on Lorentz transformations to re-set the evolution of time. Egon von Oppolzer was quoted by Lewis Peyson in his article "The relativity revolution in Germany" (page 73) which in contained in Comparative Reception of Relativity (1987) by Thomas F. Glick (pages 59 to 111). In the context of relativity studies, the hyperbolic angle parameter a is called rapidity. 

For comprehension, the corner flow can be viewed in the context of two other planar linear flows: concentric circular flow and shear flow. These other flows have been cited in connection with the development of the concepts of viscosity and Reynold’s number. For instance, in 1928 Emil Hatschek composed this sketch: "In 1890 Couette took up the system of concentric cylinders, the velocity distribution for which had already been given by Stokes, who suggested it might be studied experimentally by observing 'motes in the liquid'. Couette calculated the moment exerted by the outer cylinder on the inner one, and constructed an apparatus in which this moment could be measured and the coefficient of viscosity deduced from it; it was found to agree with Poiseuille’s values. … He showed that the velocities at which turbulence set in were approximately proportional to the kinematic viscosity."(Viscosity of  Liquids, page 13)

The laminar flow of a viscous liquid along a wall is called shear flow and corresponds to the linear transformation called shear mapping.  The physical conditions of corner flow, concentric circular flow, and shear flow differ dramatically, yet 2 x 2 real matrices, with determinant one, provide a single context for their mathematical models. In this mathematical context the corner flow is represented by squeeze mapping, where rectangles of the same area are permuted. 


Common Ground: 

When the relativity of Lorentz, Einstein, and Minkowski (LEM) was studied for foundations, it was found necessary to formulate the presumptions of earlier kinematics such as the space transformation x' = x + v t on a plane (x,t) expressing absolute space and time. Isaak Yaglom (1921 to 1988) investigated the geometric algebra associated with this classical spacetime and that associated with the LEM spacetime. However, for fundamental studies the notion of space is attenuated to a single real dimension so that the spacetimes are planar geometries. There are so many possibilities to consider algebraically for a four-dimensional spacetime that reasonable steps into spacetime algebra work first in the plane. Yaglom (1979) noted that the shear mappings of the classical view leave area invariant in the planar spacetime. Remarkably, the Lorentz transformations also leave area in spacetime invariant. These facts follow from the nature of such mappings in the group of equi-areal mappings in the plane, which consists of three types, two of which correspond to classical and LEM relativity. Such correspondence is common ground between classical and LEM physics; rather than accentuate linear differences in models for each, readers of this announcement in iMechanica can use the mutual currency to advance coherent communication.

The common physical unit is meter-second (not to be confused with meters per second). Consider the poise = kilograms per meter-second, most commonly used to express viscosity. Here the meter-second is used to provide a "density" for the massy unit, kilogram. In this blog, the unit of poise will be used to measure kinsity. Readers may exercise a bit of dimensional analysis to show that the geometric mean of a force and a mass density results in a kinsity.   

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