On the exact solutions of Darcy–Brinkman model in rectangular Hele–Shaw channels under no-slip and slip boundary conditions

This study demonstrates the effectiveness of Darcy–Brinkman model to derive exact solutions for laminar flow in rectangular microchannels, presenting a robust alternative to traditional methods that rely on complex Fourier series expansions for velocity distributions. Derived here are analytical solutions for two specific cases: the classical Hele–Shaw configuration with no-slip boundaries and a slip-induced configuration featuring a permeable lateral wall with a square array of cubic elements. For both cases, the Darcy–Brinkman model effectively captures flow behavior including boundary layer effects and central plug flow, and the results are validated using micro-Particle Image Velocimetry measurements. For channels with high width-to-height ratios (w/h>4), the model perfectly aligns with established permeability formulations, with a mean relative error consistently remaining below 5%. In the slip-induced configuration, an effective slip length (Ls) is introduced, correlated with porosity and velocity ratios, to reflect the properties of the porous material. This slip length is expressed as a function of the Hele–Shaw cell height and porosity, modeled using an inverse error function, and demonstrates excellent agreement with experimental results, with a mean relative error below 3.5%. This study highlights the potential of Darcy–Brinkman model for exact solutions in microfluidic systems.