Thin-gap approximations for microfluidic device design

Over 125 years ago, Henry Selby Hele-Shaw realised that the depth-averaged flow in thin-gap geometries can be closely approximated by two-dimensional (2-D) potential flow, in a surprising marriage between the theories of viscous-dominated and inviscid flows. Hele-Shaw approximation allows visualisation of potential flows over 2-D aerofoils and also undergirds important discoveries in the dynamics of interfacial instabilities and convection, yet it has found little use in modelling flows in microfluidic devices, although these devices often have thin-gap geometries. Here, we derive a Hele-Shaw approximation for the flow in the kinds of thin-gap geometries created within microfluidic devices. Using the method of weighted residuals, we reinterpret the Hele-Shaw approximation as the leading term of an orthogonal polynomial expansion that can be systematically extended to higher-order corrections. The resulting leading-order equation coincides with the previously derived 2-D approximations, but our derivation is shorter and more direct. By extending the expansion beyond leading order, we obtain a new reduced model that captures non-parabolic gapwise velocity profiles and out-of-plane flow effects. We provide substantial numerical evidence showing that approximate equations can successfully model real microfluidic and inertial-microfluidic device geometries. By reducing three-dimensional flows to 2-D models, our validated model will allow for accelerated device modelling and design.