A Mathematical Analysis and Simulation of the F-L Effect in Two-Layered Blood Flow through the Capillaries Remote from the Heart and Proximate to Human Tissue

Abstract

In this paper, we have provided a mathematical analysis of an empirical result, namely, the Fahraeus–Lindqvist effect, a phenomenon that occurs in capillary tubes with a diameter lower than 0.3 mm. According to this effect, in capillary tubes under 0.3 mm in diameter, the apparent viscosity of blood decreases as the diameter of the tube decreases, making flow possible in these vessels. A two-phase blood flow mathematical model for human capillaries has been presented here. According to Haynes’ theory, blood is separated into two layers when it flows from the capillary. It is assumed that the first layer is plasma, and the second layer is the core layer. The plasma layer flows near the wall of the capillary, and the core layer flows along the axis of the capillary. Further, the core layer is assumed to be a mixture of two phases: one is the plasma, and the other is that of RBCs. For mathematical modeling purposes, a curvilinear coordinate system has been adopted, with physical quantities used in tensorial form. Derived equations are solved to find the effective viscosity, which depends upon the radius of the capillary; that is, it reduces viscosity to make blood flow possible. A comparative study was conducted with the experimental result of this effect, and it was observed that the proposed two-phase blood flow model is much closer to the experimental data than the single-phase blood flow model, and both have the same trends. After validation of the model with the experimental result, this model was applied to human capillaries (diameter lower than 10 μm) to show the F-L effect, and the impact of various physiological quantities that are relevant to the flow of blood into human capillaries are also discussed here. The impact of hematocrit on various parameters has been demonstrated explicitly.