Stokes flows in a two-dimensional bifurcation

• Published in: Royal Society open science Vol. 12; no. 1; pp. 241392 - 18
• Main Authors: Xue, Yidan, Payne, Stephen J., Waters, Sarah L.
• Format: Journal Article
• Language: English
• Published: England The Royal Society Publishing 01.01.2025; The Royal Society
• Subjects: Algorithms; Approximation; bifurcation; Bifurcations; biharmonic equation; Blood; Boundary conditions; Fixed objects; Flow; flow network; Geometry; lightning solver; Machine learning; Mathematical models; Mathematics; Stokes flow; Stokes law (fluid mechanics); Two dimensional analysis; Two dimensional flow; Viscosity; bifurcation; biharmonic equation; flow network; Stokes flow; lightning solver
• ISSN: 2054-5703; 2054-5703
• DOI: 10.1098/rsos.241392

The flow network model is an established approach to approximate pressure–flow relationships in a bifurcating network, and has been widely used in many contexts. Existing models typically assume unidirectional flow and exploit Poiseuille’s law, and thus neglect the impact of bifurcation geometry and finite-sized objects on the flow. We determine the impact of bifurcation geometry and objects by computing Stokes flows in a two-dimensional (2D) bifurcation using the Lightning-AAA Rational Stokes algorithm, a novel mesh-free algorithm for solving 2D Stokes flow problems utilizing an applied complex analysis approach based on rational approximation of the Goursat functions. We compute the flow conductances of bifurcations with different channel widths, bifurcation angles, curved boundary geometries and fixed circular objects. We quantify the difference between the computed conductances and their Poiseuille law approximations to demonstrate the importance of incorporating detailed bifurcation geometry into existing flow network models. We parametrize the flow conductances of 2D bifurcation as functions of the dimensionless parameters of bifurcation geometry and a fixed object using a machine learning approach, which is simple to use and provides more accurate approximations than Poiseuille’s law. Finally, the details of the 2D Stokes flows in bifurcations are presented.