A Microscale Mathematical Blood Flow Model for Understanding Cardiovascular Diseases

Area of Science:

• Fluid Dynamics
• Biomedical Engineering
• Microcirculation

Background:

• Capillary blood flow involves complex interactions between plasma and red blood cells.
• Modeling these microfluidic systems is crucial for understanding physiological and pathological conditions.

Purpose of the Study:

• To develop an analytical model for blood plasma flow through a swarm of red blood cells in capillaries.
• To obtain analytical expansions for flow velocity components using a stream function approach.

Main Methods:

• Modeled flow as axisymmetric Stokes flow within inverted prolate spheroidal solid-fluid unitary cells.
• Introduced a stream function satisfying the biharmonic equation (E⁴ψ = 0).
• Employed Kelvin inversion to transform the problem into prolate spheroidal coordinates for solution.

Main Results:

• Derived analytical expansions for velocity components based on the stream function.
• The solution was obtained via inverse Kelvin transformation, accounting for the Stokes operator's behavior.
• The final analytical solution for the stream function is presented as an R-separable series expansion.

Conclusions:

• Successfully modeled blood plasma flow in a red blood cell swarm using a novel analytical approach.
• The method provides a detailed understanding of fluid dynamics at the microcirculatory level.
• The derived analytical solution offers a valuable tool for further research in hemorheology.