The erythrocytes are the primary carriers of oxygen and carbon dioxide to and from the systemic tissue. The ability of these cells to deform and navigate through the capillary beds is of fundamental importance for proper functioning of the cardiovascular transport system. The erythrocyte is essentially a capsule, and flow-induced erythrocyte deformation involves the interfacial dynamics of a membrane-enclosed fluid volume stressed in a viscous flow. Elastic capsule dynamics is a complicated problem involving the coupling of fluid and membrane forces; it is also found in a variety of scientific and engineering applications. In this work, we investigate the dynamics of elastic capsules and erythrocytes using the Spectral Boundary Element (SBE) method, a high-order / high-accuracy method for capsule and cellular dynamics.

For strain-hardening Skalak elastic capsules in an extensional flow, our investigations demonstrate a shape transition in accordance with experimental observations to a cusped conformation at high flow rates, which allows the capsule to withstand the increased hydrodynamic forces. Our computational methodology reveals a region of bifurcation, in which both spindled and cusped steady-state geometries coexist for a single flow rate. The method is also used to investigate the dynamics of strain-softening Neohookean capsules in the same flow pattern. The strain-softening capsules become highly extended at weaker flow rates than strain-hardening capsules, and do not form steady-state cusped shapes.

The SBE method has been extended to model the erythrocyte by using a biconcave disc reference geometry and adaptive prestress to enforce area incompressibility. The method accurately reproduces experimental data from erythrocyte ektacytometry, but allows examination of the erythrocyte dynamics beyond the geometric constraints inherent in ektacytometry and other experimental techniques, including observation of the three-dimensional oscillatory behavior over a range of capillary numbers and viscosity ratios. Our results support a prediction by Fischer, Skalak, and coworkers that the erythrocyte shear modulus decreases at small shear deformations. Our work also suggests that cellular deformation is largely independent of the flow pattern, consistent with the findings of experimental investigators.