Comparison of Integral Equation Formulations for Stokesian Particulate Flow Simulations

Particulate Stokesian flows describe the hydrodynamics of rigid or deformable particles within Stokes flows, where viscous forces dominate over inertial effects. These flows are characterized by highly nonlinear fluid-structure interactions, moving interfaces, and multiple spatial and temporal scales, making numerical simulations both complex and computationally expensive. Accurately capturing these interactions requires sophisticated numerical approaches. The boundary integral equation method (BIEM) is a powerful tool for modeling such flows, as it reduces computational complexity by limiting the discretization to the immersed particle boundaries rather than the entire flow domain. This efficiency makes BIEM particularly suitable for studying systems with many particles or complex boundary geometries. In this work, we explore two fundamental BIEM formulations for Stokesian flows involving rigid particles: the first-kind and second-kind integral equations. These formulations differ in their mathematical structure and computational properties, impacting their stability, accuracy, and overall performance. By comparing these two approaches, we aim to highlight their respective advantages and limitations, providing insights into their applicability to different particulate flow scenarios. This analysis contributes to the broader understanding of numerical methods for Stokesian flows, addressing challenges inherent to fluid-structure interactions and advancing computational techniques in this field.