The squeezing of red blood cells through tubes and channels of near-critical dimensions.

Abstract

The aim of this dissertation is to develop theoretical models for the motion of rigid and flexible particles through very tight spaces. The geometries of conduits which will be investigated are cylindrical tubes, parallel plane walls and rectangular channels. This work is motivated by an interest in the flow and deformation of single red blood cells in very narrow capillaries, in spleen and in bone marrow. Mammalian red cells are highly flexible, but their deformations satisfy two significant constraints. They must deform at constant volume, because the contents of the cell are incompressible, and also at nearly constant surface area, because the red cell membrane strongly resists dilation. Consequently, there exists a minimal tube diameter below which passage of intact cells is not possible. A cell in a tube with this diameter has its critical shape: a cylinder with hemispherical ends. The motion of red cells is analysed using lubrication theory. When the tube diameter is slightly larger than the minimal value, the cell shape is close to its shape in the critical case. However, the rear end of the cell becomes flattened and then concave with a relatively small further increase in the diameter. The changes in cell shape and the resulting rheological parameters are analysed using matched asymptotic expansions for the high-velocity limit and using numerical solutions. A rapid decrease in the apparent viscosity of red cell suspensions with increasing tube diameter is predicted over the range of diameters considered. The red cell velocity is found to exceed the mean bulk velocity by an amount which increases with increasing tube diameter. The same type of analysis is applied to the flow and deformation of red blood cells between two parallel plates with near-minimal spacings. First, the critical shape of the particle and the minimum gap width are determined using calculus of variations. In this case, it is a disk with a rounded edge. The flow in the plasma layers between the cell and the plates is described using lubrication theory. Approximate solutions can be obtained using a locally two-dimensional analysis at each point of the rim of the cell. Cell shapes, pressure distributions, membrane stresses and particle velocities are deduced as functions of geometrical parameters. One significant finding is that the gap width between the cell and the wall decreases with distance from the axis of symmetry parallel to the flow direction. The red cell velocity may be smaller or larger than the mean fluid velocity far from the cell, depending on the spacing of the plates, with equality when the width of the red cell is about ninety percent of the spacing between plates. The same procedure is also applied to rectangular channels.