Stokes flow applied to the sedimentation of a red blood cell

Abstract

The sedimentation of a red blood cell (RBC) through the blood plasma is described as the translation of a rigid inverted prolate spheroid through a quiescent unbounded viscous fluid. The inverted spheroid is moving with constant velocity along its axis of symmetry. The physical characteristics of the RBC and the blood plasma allow us to consider this problem as a Stokes flow problem. The Kelvin inversion method and the concept of semiseparation of variables for the Stokes operators, which we used for solving the Stokes flow past an RBC, are also employed here. The stream function is then expressed as a series expansion in terms of Gegenbauer functions of even order. Through this, important hydrodynamic quantities such as the drag force, the drag coefficient and the terminal settling velocity of the RBC are calculated. The celebrated Stokes formula for the drag force exerted on a sphere is now expanded in order to account for the shape deformation of an RBC. Sample streamlines are depicted showing the dependence of these quantities on the geometrical characteristics of the RBC and also of any inverted prolate spheroid. The obtained results seem to be directly applicable in medical tests such as the Erythrocyte Sedimentation Rate (ESR).