Flow of a Suspension

Understanding the flow behaviour of a suspension of particles in a fluid is a large and difficult general problem of fluid mechanics. 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 problem focuses on the kinematics of erythrocytes.

Audience

The problem may appeal to a student of Engineering Physics, Mathematics or Physics.

Objective

Create a description of a set of surfaces satisfying a given value of the isoperimetric parameter (Query 320 following), which is simple enough to provide a kinematic basis for a dynamical theory. Alternatively, prove that a description satisfying specified conditions is impossible.

Methods

Any numerical or symbolic method may be used. "Numerical" includes computational. "Symbolic" includes calculation producing a closed expression, integral, series or product. Consideration should be given to the possibility of using the description in a subsequent dynamical theory. A numerical result will be less useful than a symbolic result.

Hint

In a symbolic result, we can expect to find at least one parameter ranging over a subset of R¹. Kuchel and Fackerell (citation following) used three such parameters.

Prospect

A suitable solution of this problem will eventually be the kinematic basis for a statistical and continuum mechanical derivation of the constitutive equation for a suspension of flexible objects from properties of these objects and the suspending fluid. Alternatively, a proof of the impossibility of the description, subject to specified conditions, will imply the impossibility of a derivation of the constitutive equation subject to the same conditions.

Any serious effort can produce an interesting and significant advance in understanding of flow of suspensions including blood. The problem is difficult and unlikely to be solved fully during my life.

Solution

At 2008-01-01, Query 320 has not received a response. These works may contribute to a solution. Extensive work remains.

1680 Giovanni Cassini discovered the Cassinian Ovals. They illustrate that a modestly simple expression can produce a wide variety of shapes. The Cassinias are merely curves in two dimensions whereas Query 320 asks for surfaces in 3 dimensions. The extra dimension adds significant complexity, yet the Cassinias 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

Sphericity Index = 4.84 V^2/3/A,

where V is volume and A is area." The cube of the Sphericity Index is the dimensionless ratio of Query 320.

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 36π V²/A³ ≤ 1 (A= surface area, V=enclosed volume), with equality only for the sphere. To what extent can one describe the shape of the surfaces for a given value of 36π V²/A³ (< 1)?"

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. They used Mathematica to investigate the modelling of the erythrocyte by products of elliptic functions and discussed the question of relating the "three major `shape-defining' measurements of the human erythrocyte ... to three parameters in ... curvilinear coordinates." Sphericity Indices for these surfaces have not been published.

2006 UBC Engineering Physics students Mohammad Bdair and Shymon Sumiyoshi surveyed some of the publications where erythrocyte kinematics is addressed and performed several calculations. Their work is self-published as Shape Analysis of Distorted (Biconcave) Spheres and fulfills a requirement of APSC 459.

Model erythrocytes made from a polymeric foam are available to assist in conceptualization.

An eaddress of the author can be found in the home page of the UBC Pathology Workshop.

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