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The flow of a fluid containing particles is a large and difficult general problem of fluid mechanics. The problem may be less difficult than turbulence. Nevertheless it offers interesting challenges to the theoretician.
Particles take many forms. Ore particles in a slurry are rigid and irregularly shaped. Erythrocytes in blood are flexible and more regularly shaped. Adhesion forces can be significant between organic particles whereas adhesion between inorganic particles can be small. This discussion focuses on the kinematics of erythrocytes.
A particle moving in a fluid is a dynamical system. For a given value of the isoperimetric parameter, the kinematics of the surface of a particle can be represented by a category where each object is a configuration of the surface and each arrow is a deformation taking one configuration to another. For the erythrocyte the obvious name for this category is "erythrocyte category". Allowing for continuous deformation of the erythrocyte, the category contains an infinite number of objects and an infinite number of arrows. The very large size of the category necessitates machine assistance for any explicit representation and calculation.
1680 Giovanni Cassini discovered the Cassinian
Ovals.
They illustrate that a modestly simple expression
can produce a wide variety of shapes. The ovals are merely
curves in two dimensions whereas Query 320 asks for surfaces
in 3 dimensions. The extra dimension adds significant complexity,
yet Cassini's ovals might inspire an interesting approach to the problem.
1968 Peter Canham and Alan Burton published "Distribution of Size and Shape in Populations of Normal Human Red Cells" in Circulation Research, Volume 22, p. 405. By numerical integration they calculated surface area and volume for many erythrocytes. "We introduced the term sphericity index to provide a comparison between the shape of a cell and a sphere. It is defined by
1972 Evan Evans and Y.-C. Fung published "Improved measurements of the erythrocyte geometry" in Microvascular Research, Volume 4, pages 335-342, Elsevier. In this work the shape of the surface was adjusted by four parameters.
1985 Query 320, Notices, Amer. Math. Soc. 1985, 32, 9. "For a surface of the type of the sphere in R3, the isoperimetric inequality states [0 ≤] 36π V2/A3 ≤ 1 (A= surface area, V=enclosed volume), with equality only for the sphere. For a given value of [the isoperimetric parameter] 36π V2/A3 (< 1), to what extent can the shape of the surfaces be described?"
1989 "A 3-dimensional Dyadic Walburn-Schneck Constitutive Equation For Blood", Biorheology, 26, pp. 37-44.
1999 Philip Kuchel and Edward Fackerell published "Parametric-Equation Representation of Biconcave Erythrocytes" in Bulletin of Mathematical Biology, Volume 61, pp. 209-220. Mathematica was used to investigate the modelling of the erythrocyte by products of elliptic functions. The question of relating the "three major `shape-defining' measurements of the human erythrocyte ... to three parameters in ... curvilinear coordinates." was discussed. Sphericity Indices for these surfaces have not been published.
2006 UBC Engineering Physics students Mohammad Bdair and Shymon Sumiyoshi surveyed publications addressing erythrocyte kinematics and performed several calculations. The work was self-published as Shape Analysis of Distorted (Biconcave) Spheres and fulfilled a requirement of APSC 459. The course has been superseded by ENPH 459.
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