Thin gap approximations for microfluidic device design

TL;DR

The paper addresses efficient modeling of thin-gap microfluidic flows where classical Hele-Shaw theory omits inertia and boundary layers. It introduces a modified Method of Weighted Residuals to derive a 2D model by expanding the vertical velocity profile in a Gegenbauer basis and deriving an optimal closure parameter $a$, yielding a refined 2D Darcy–Brinkman–type equation that matches 3D simulations for both Stokes and finite-$\mathrm{Re}$ flows. The authors validate the model on Poiseuille and coaxial geometries and in centrifuge-on-a-chip devices, showing relative errors below roughly $12\%$ and accurate prediction of interface shapes and separation bubbles. This framework enables rapid device design and can be extended to higher-order corrections and other fields.

Abstract

Over 125 years ago, Henry Selby Hele-Shaw realized that the depth-averaged flow in thin gap geometries can be closely approximated by two-dimensional (2D) potential flow, in a surprising marriage between the theories of viscous-dominated and inviscid flows. Hele-Shaw flows allow visualization of potential flows over 2D airfoils and also undergird important discoveries in the dynamics of interfacial instabilities and convection, yet they have found little use in modeling 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. Although these equations have been reported before, prior work used a less direct derivation. Here, we obtain them via a modified Method of Weighted Residuals (MWR), interpreting the Hele-Shaw approximation as the leading term of an orthogonal polynomial expansion that can be systematically extended to higher-order corrections. We provide substantial numerical evidence showing that approximate equations can successfully model real microfluidic and inertial-microfluidic device geometries. By reducing three-dimensional (3D) flows to 2D models, our validated model will allow for accelerated device modeling and design.

Figures (5)

• Figure 1: The Hele–Shaw approximation models thin-gap flows via 2D potential theory. (a) Streamlines around a model airfoil: top—experiment werle1973hydrodynamic; bottom—2D potential flow computed in COMSOL 5.4. (b) Thin-gap geometry as a testbed for Saffman–Taylor instability, where a low-viscosity fluid (blue-dyed water) displaces a more viscous one (glycerol). (c) Example microfluidic device with thin gap: a centrifuge-on-a-chip hur2011high, where a narrow inlet channel feeds a larger rectangular chamber; the assumed parabolic gapwise velocity profile is shown.
• Figure 2: Validating a 2D approximation for the pressure-driven flow in a rectangular channel. (a) Flow profile at $z = 0$: exact solution (black curve), approximation from Eq. \ref{['eq:pressureDriven weight function 1']} (red curve), and approximation from Eq. \ref{['eq:pressureDriven ultraspherical weight function']} (blue dashed curve), for $L = 1$. (b) Relative error, measured by the $L^\infty$ norm $\frac{\lVert u - u_{\text{app}} \rVert_{\infty}}{\lVert u \rVert_{\infty}}$, as a function of aspect ratio $L$. Red curve: Eq. \ref{['eq:pressureDriven weight function 1']}, blue dashed curve: Eq. \ref{['eq:pressureDriven ultraspherical weight function']}.
• Figure 3: Coaxial flow. (a) Simulation geometry based on anna2003formation: left— inlets, right- outlet. Outer inlets have mass flux $Q$, the inner inlet has flux $1$. Inlet and outlet lengths are extended by 20 to ensure fully developed flow. (b) Dividing streamlines for $Q=1.7$: classical Hele–Shaw (red), derived $a=6/5$ approximation (blue dotted), and 3D simulation (black dashed).
• Figure 4: 2D approximation for finite $\mathrm{Re}$ flows in a centrifuge-on-a-chip geometry. Comparison of the separation bubble width in 3D simulation (black), 2D approximation (blue, Eq. \ref{['eq:partb']}), and unweighted residual approximation (red, Eq. \ref{['eq:parta']}). Right panels: Streamline patterns colored by velocity magnitude (red = larger, blue = smaller), comparing the 2D approximation (top) with the full 3D simulation (bottom).
• Figure 5: Fidelity of the numerically computed 3D flow field to the Hele-Shaw (parabolic flow profile) for a centrifuge-on-a-chip flow. (a) $r$ (defined in Eq. \ref{['eq:non-parabolicity']}) at $\mathrm{Re}=200$ across the centrifuge chamber. (b,c) $u-$ profiles against $z$ sampled on two different transects (locations shown in (a)). To compare shapes, all profiles are normalized so that $u=1$ at $z=0$ (b) shows a small $r$-region, where the Hele-Shaw approximation is valid, and (c) a region on the boundary of the inlet jet, where it is not. Line colors give $x$-ordinates. (d) Adding $Q_3$ to our basis allows non-Hele-Shaw features to be rendered. Two specific non-parabolic velocity profiles from the $y=6$ transect are shown (solid lines) along with their projections onto $Q_1$ and $Q_3$ (dotted line). Simulations were performed in COMSOL Multiphysics 6.2, with P2+P2 elements and the extra fine mesh further refined to ensure a maximum cell length of 0.1.