Stokes flows in a two-dimensional bifurcation

TL;DR

This paper introduces a novel mesh-free method to compute detailed 2D Stokes flows in bifurcations, revealing the significant impact of geometry and objects on flow conductance beyond traditional Poiseuille's law approximations.

Contribution

The study develops the LARS algorithm for accurate, mesh-free 2D Stokes flow computation in bifurcations, incorporating geometry and objects, and uses machine learning for improved conductance modeling.

Findings

Flow conductances vary significantly with bifurcation geometry and objects.

Poiseuille's law approximations are often insufficient for detailed bifurcation modeling.

Machine learning models outperform traditional approximations in predicting flow conductance.

Abstract

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 LARS (Lightning-AAA Rational Stokes) algorithm, a novel mesh-free algorithm for solving 2D Stokes flow problems utilising 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's law approximations to demonstrate the importance of incorporating detailed bifurcation geometry into existing flow network models. We parameterise 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.