Beyond Hele-Shaw: Systematic Reduced Modeling of Flow in Thin-Gap Microfluidics

Thin-gap geometries are common in microfluidic devices, arising naturally from the constraints of fabrication techniques. However, simulating the resulting flows in three dimensions is computationally intensive, and classical two-dimensional (2D) approximations—such as the Hele-Shaw model—fail to capture essential physics, including inertial effects and near-wall dynamics. We present a systematic approach for deriving accurate 2D equations that approximate the full three-dimensional (3D) Navier–Stokes equations and their coupled scalar counterparts. The method follows the framework of the Method of Weighted Residuals, with physically informed choices of basis and weight functions guided by asymptotic analysis of the relevant flow quantities in the thin-gap limit. We validate the derived two-dimensional model against full three-dimensional simulations across a range of representative microfluidic and inertial flow configurations, including coaxial flow devices and centrifuge-on-a-chip geometries used for particle separations. Our results show that the model maintains high fidelity even beyond the classical Hele-Shaw regime and provides a robust foundation for higher-order corrections and multiphysics extensions, such as electrokinetic transport and magnetohydrodynamic flows. This framework enables efficient and accurate simulation of thin-gap flows and offers immediate utility for the design and optimization of microfluidic systems involving particles and bubbles.