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The flow properties of aggregating red cell suspensions flowing at low rates through vertical tubes with diameters from 30 μm to 150 μm are analyzed using a theoretical model. unidirectional flow is assumed, and the distributions of velocity and red cell concentration are assumed to be axisymmetric. A three-layer approximation is used for the distribution of red cells, with a cylindrical central core of aggregated red cells moving with uniform velocity, a cell-free marginal layer near the tube wall, and an annular region located between the core and the marginal layer containing suspended non-aggregating red cells. This suspension is assumed to behave approximately as a Newtonian fluid whose viscosity increases exponentially with red cell concentration. Physical arguments concerning the mechanics of red cell attachment to, and detachment from the aggregated core lead to a kinetic equation for core formation. From this kinetic equation and the equation for conservation of red cell volume flux, a relationship between core radius and pressure gradient is obtained. Then the relative viscosity is calculated as a function of pseudo-shear rate. At low flow rates, it is shown that the relative viscosity decreases with decreasing flow and that the dependence of relative viscosity on shear rates is more pronounced in larger tubes. It is also found that the relative viscosity decreases with increasing aggregation tendency of suspension. These theoretical predictions are in good qualitative and quantitative agreement with experimental results.

A kinetic equation for rouleau formation in a simple shear flow is derived, based on several assumptions. These are (a) colliding rouleaux stick to one another with a certain probability to form a single rouleau; (b) simultaneous collisions between more than two rouleaux are negligible; (c) rouleaux are broken by a viscous force exerted by the suspending fluid on the surfaces of rouleaux; (d) when a rouleau is broken by viscous forces, only two fragments are formed. Based on a simple mathematical model, collision rate, sticking probability and degradation rate are obtained as functions of applied shear rate. From the solution of the kinetic equation, the average size of rouleaux is obtained as a function of time with shear rate as a parameter. It is shown that the average size of rouleaux increases monotonically with increasing time and tends to an equilibrium size. The average size of rouleaux in a dynamical equilibrium decreases monotonically with increasing shear rate and tends to one cell as shear rate approaches infinity. It is also found that the initial rate of rouleau formation increases with increasing shear rate at very low shear rate, but this trend is reversed at higher shear rates. The theoretical results are compared quantitatively with experimental data.

The viscoelastic properties of a concentrated red cell suspension in which red cells do not form rouleaux are discussed from a theoretical point of view. An elastic spherical shell filled with a viscous fluid is considered as a model of a normal red cell. A cell method is used to consider finite concentration of suspended particles. On the assumption that the deformation from a spherical shape is very small, the shape of a deformed shell in a shear flow and the constitutive equation of the suspension are derived.