Theoretical and numerical models of depth-confined Brinkman flow

Highly depth-confined flows are a common feature of microfluidic devices. We study flows that are highly confined in the depth direction, which are often referred to as Brinkman flows. We seek a novel theoretical solution approach to depth-confined flows by the construction of an outer solution that satisfies the Brinkman equations and an inner solution that is valid near the boundaries. We use combined theoretical and numerical methods to investigate several basic cases including the flow past a depth-confined cylinder, lid-driven cavity flow, and flow over a backward-facing step. We close the theoretical flow solution using matching boundary conditions between the inner and outer flows. The resulting theoretical approach is general across a wide-range of microfluidic devices where the channel height is the smallest dimension, and significantly reduces the computational resources needed to model the system by reducing such flows to 2D flow problems. We performed 3D numerical simulations using OpenFOAM to validate our 2D theoretical formulation and provide more physical insights into the method.