Non-Convex Particle-in-Cell Model for the Mathematical Study of the Microscopic Blood Flow

Abstract

The field of fluid mechanics was further explored through the use of a particle-in-cell model for the mathematical study of the Stokes axisymmetric flow through a swarm of erythrocytes in a small vessel. The erythrocytes were modeled as inverted prolate spheroids encompassed by a fluid fictitious envelope. The fourth order partial differential equation governing the flow was completed with Happel-type boundary conditions which dictate no fluid slip on the inverted spheroid and a shear stress free non-permeable fictitious boundary. Through innovative means, such as the Kelvin inversion method and the R-semiseparation technique, a stream function was obtained as series expansion of Gegenbauer functions of the first and the second kinds of even order. Based on this, analytical expressions of meaningful hydrodynamic quantities, such as the velocity and the pressure field, were calculated and depicted in informative graphs. Using the first term of the stream function, the drag force exerted on the erythrocyte and the drag coefficient were calculated relative to the solid volume fraction of the cell. The results of the present research can be used for the further investigation of particle–fluid interactions.

Keywords

• Stokes flow
• red blood cell
• particle-in-cell model
• Kelvin inversion
• Happel-type boundary conditions