Microfluidic Network Validation of OpenFLUME
A validation report for OpenFLUME's microfluidic capability: the rectangular-channel laminar law, capillary-driven filling with displaced-gas resistance, capillary pumps and valves, and published microfluidic networks rebuilt from their papers — a serial-dilution cell, a hydrodynamic bead trap, a parallel manifold, a gravity siphon, a capillaric circuit's valve design and a bubble-damped syringe network.
Generated by scripts/microfluidics-validation-report.ts — all numbers and figures come from live solves of the current solver. The published cases are defined once in src/validation/microfluidics.ts and gated in CI by src/validation/__tests__/microfluidics.test.ts; the closed-form cases mirror src/core/__tests__/rectChannel.test.ts, capillaryChannel.test.ts and flowScaledTolerance.test.ts.
Abstract
Microfluidic networks run at flows of nanolitres to microlitres per minute through rectangular channels tens of micrometres deep, and are often driven by capillarity rather than pumps. This report checks OpenFLUME in that regime in two parts. Part A compares the rectChannel and capillaryChannel components with exact solutions: the rectangular-duct friction constant matches Shah & London to 0.002 %, capillary filling follows the Washburn law with air resistance to 1.49e-8 of the fill time at any step size, the Jurin rise is reached to 8.17e-16, and syringe-driven and resistance-coupled filling are exact to 1.08e-13. Part B rebuilds six published microfluidic networks from the parameters printed in their open-access papers. OpenFLUME reproduces the authors' COMSOL inlet pressure (85.1 vs 85.2 Pa) and design dilution ratio of a serial diluter, both trap flow ratios and the trapping outcome of a bead-trap array, the flow distribution of a 20-branch manifold, the measured flow of a gravity siphon within its stated diameter uncertainty, the tabulated capillary pressures of a capillaric circuit's valves to 1.3 %, and the exact dynamics of a bubble-damped syringe network to 9.47e-4 — including the paper's finding that responses for different bubble sizes collapse when time is scaled by V·R/P. That last case exposed, and now pins, a solver defect: the absolute Newton tolerance judged mass imbalances in kg/s, so microfluidic compliance transients stopped iterating and froze. The stopping test now weights mass-flow rows by the network's flow scale.
1. Model Summary
Rectangular channel (rectChannel, user manual §4.3.18). Fully developed laminar resistance from the exact Fourier-series solution, R = 12μL / (a³bF), F = 1 − (192a/π⁵b) Σₙ odd tanh(nπb/2a)/n⁵, with a smooth blend to Swamee–Jain on D_h above Re 2300.
Capillary channel (capillaryChannel, §4.3.19). A rectangular channel holding liquid over length s and gas over the rest: P_from − P_to = R_liq(s)·Q + R_gas(L−s)·Q − σ·P_cap + ρgΔz·s/L, with the Young–Laplace pressure P_cap = γ[(cos θ_top + cos θ_bottom)/h + 2cos θ_sides/w]. Fronts are integrated against a Thevenin equivalent of the surrounding network; stop valves use the Cho et al. (2007) burst pressure; node wetting drives trigger valves and air-trap detection.
Flow-scaled Newton tolerance. The Newton residual mixes mass rows [kg/s] with momentum rows [Pa]. With an absolute bar tol, a microfluidic mass imbalance (~10⁻¹³ kg/s) always passes, so a storage-driven step was accepted at its warm start. Below 10⁻³ kg/s the mass-flow rows (nodal balances and imposed-flow branches) are now weighted by 10⁻³ kg/s ÷ F, F the largest warm-start flow (weight ≤ 10⁹), in both the stopping test and the line-search merit. At or above 10⁻³ kg/s the weight is 1 and the solver is bit-identical to before; of the 14 shipped examples only Entrapped-air line changes, by 10⁻⁷ relative, during its at-rest phase.
Part A — Closed-Form Verification
A1. Rectangular-duct laminar law
| Aspect ratio α | Shah & London f·Re | OpenFLUME f·Re | error |
|---|---|---|---|
| 1 | 56.908 | 56.908 | 0.001 % |
| 0.5 | 62.192 | 62.192 | 0.000 % |
| 0.25 | 72.932 | 72.931 | 0.001 % |
| 0.125 | 82.34 | 82.339 | 0.002 % |
The parallel-plate limit is 96. A circular pipe sized to the hydraulic diameter of a 100 × 50 µm channel would predict 32 % less flow than the rectangle carries — the reason the dedicated component exists.
Figure 1. Laminar f·Re of a rectangular duct against Shah & London's tabulated values.
A2. Washburn filling with displaced-air resistance
A 100 × 50 µm channel, 10 mm long, θ = 30°, γ = 0.072 N/m, fills from a reservoir to a vent at equal pressure. With air viscosity included, the exact front law is r_l s²/2 + r_g(Ls − s²/2) = P_cap·t / A (r per unit length), filling in 0.0952 s.
| Time steps per fill | max |t − t_exact(s)| / t_fill | arrival-event time [s] |
|---|---|---|
| 5 | 1.49e-8 | 0.0952 |
| 50 | 1.49e-8 | 0.0952 |
| 500 | 1.31e-8 | 0.0952 |
The meniscus ODE is integrated across each step against the network's Thevenin equivalent, so the stiff early phase (velocity ∝ 1/s) costs nothing in accuracy even at 5 steps per fill.
Figure 2. Front position against the exact law with displaced-air resistance and the classic (air-free) Washburn law; markers are a 20-step OpenFLUME run.
A3. Jurin rise
A vertical 500 µm square channel, 100 mm tall, θ = 30°. Equilibrium height P_cap/(ρg) = 50.969 mm; OpenFLUME settles at 50.969 mm (error 8.17e-16), monotonically and without overshoot: yes.
Figure 3. Capillary rise approaching the Jurin height.
A4. Flow-driven and resistance-coupled filling
- Syringe pump, 1 µL/min into a hydrophobic (θ = 100°) channel: the fill follows s = Qt/A to 2.40e-14 of the channel volume, including the pinned entrance that the pump forces open.
- Reservoir behind a resistive feed, ΔP = 500 Pa through a 50 × 20 µm, 20 mm feed channel: exact law (R_u + r_g L)s + (r_l − r_g)s²/2 = (ΔP + P_cap)t/A, matched to 1.08e-13 of the fill time with 25 steps.
A5. Sequential capillary circuit
The shipped example Capillary circuit (stop + trigger valve) produces the designed event sequence (pinning at the inlet in the first step is gas back-pressure from the faster branch):
| t [s] | event | channel | node |
|---|---|---|---|
| 0.0000 | pinned | A | inlet |
| 0.0262 | nodeWetted | B1 | bend |
| 0.0542 | pinned | A | J |
| 0.0984 | nodeWetted | B2 | J |
| 0.0984 | triggered | A | J |
| 0.1441 | nodeWetted | out | pumpIn |
Part B — Published Microfluidic Networks
Every parameter below is printed in the cited open-access paper; the location is given in src/validation/microfluidics.ts. Each paper's own result type is respected: a measurement with its error bar, the authors' model value, or a statement in the text.
B1. Pressure- and flow-balanced serial-dilution cell (Vasilakis et al. 2019)
Five rectangular channels, 64 µm deep, from Table 1 ("COMSOL optimisation"), two syringe inlets at 0.4 µL/min, outlets at 0 Pa.
| Quantity | OpenFLUME | Published |
|---|---|---|
| Inlet A pressure | 85.1 Pa | 85.2 Pa (COMSOL) |
| Inlet B pressure | 85.0 Pa | 85.2 Pa (COMSOL) |
| Dilution ratio DR | 0.670 | 0.67 design; measured 0.65 ± 0.02 (0.2 µL/min), 0.70 ± 0.01 (0.4 µL/min) |
The model reproduces the design, as the authors' own optimisation did. The measurements were judged against the glucose meter's ±10 % accuracy, within which both agree; by its own standard deviation the 0.4 µL/min measurement sits about 3 σ above the design value.
B2. Hydrodynamic bead trap (Tan & Takeuchi 2007)
A trap constriction (path 1) in parallel with a loop (path 2); beads are trapped only if Q₁/Q₂ > 1. Dimensions from Table 1.
| Device | W₁/L₁/W₂/L₂/H [µm] | Q₁/Q₂ OpenFLUME | Q₁/Q₂ paper | Trapping predicted | Experiment |
|---|---|---|---|---|---|
| A | 6/5.5/25/163/28 | 0.755 | 0.76 | no | no bead trapped |
| B | 7.5/4.5/20/172.5/18 | 3.949 | 3.95 | yes | every bead trapped |
Like the paper's model, the constriction is taken as a fully developed duct of its own short length (L₁ < W₁), so entrance losses are ignored; the trapping outcome is nevertheless predicted correctly for both devices.
B3. Parallel manifold flow distribution (Guermonprez et al. 2015)
A 20-branch ladder with inlet and outlet at opposite corners; only the resistance ratio R/R_b of a manifold segment to a branch matters.
| R/R_b | first two + last two branches | spread (max − min)/Q | Paper (§IV A) |
|---|---|---|---|
| 0.43 | 83.4 % | 29.60 % | ≈ 85 % |
| 0.043 | 44.6 % | 11.16 % | ≈ 46 % |
| 0.0043 | 23.8 % | 1.79 % | spread ≤ 2 % |
Every branch flow also matches an independent dense nodal solve to 2.16e-15. The paper quotes its percentages to two figures and does not state how the manifold's end segments are counted, which this comparison cannot resolve; the remaining 1–2 point difference is within that.
Figure 4. Branch-flow fractions of the 20-branch ladder for three resistance ratios.
B4. Gravity-driven microcapillary siphon (Reis et al. 2021)
Ten parallel bores of mean hydraulic diameter 204 ± 13.8 µm under a hydrostatic head H over length L (water, μ = 1.002 mPa·s, ρ = 998.2 kg/m³).
| Quantity | OpenFLUME | Published |
|---|---|---|
| Q at H/L = 0.69, D_h = 204 µm | 2.87 µL/s | 3.63 ± 0.07 µL/s measured |
| Q at H/L = 0.69, D_h = 204 ∓ 13.8 µm | 2.17 – 3.72 µL/s | (measurement inside band) |
| Q for H = 40 mm, L = 70 mm | 2.37 µL/s | ≈ 2.5 µL/s |
The solver equals Hagen–Poiseuille to 2.51e-13. Flow scales with D⁴, so the stated ±6.8 % diameter spread is a ±27 % flow band; a mean diameter of 216 µm (within the SD) reproduces the measurement.
Figure 5. Siphon flow against head ratio with the diameter-uncertainty band and the two reported measurements.
B5. Capillaric-circuit valve and pump pressures (Olanrewaju et al. 2016)
Retention burst valves in PDMS with a hydrophobic cover; per-wall Young–Laplace with the receding angles printed in the paper (walls 31°, γ = 72 mN/m).
| Valve w × h [µm] | Table 1 [Pa] | OpenFLUME, cover 89° | error | OpenFLUME, cover 95° | error |
|---|---|---|---|---|---|
| 960 × 1000 | -194 | -192 | 1.3 % | -184 | 5.1 % |
| 670 × 750 | -271 | -268 | 1.0 % | -258 | 4.7 % |
| 480 × 650 | -358 | -354 | 1.1 % | -342 | 4.3 % |
| 380 × 550 | -441 | -439 | 0.4 % | -426 | 3.5 % |
| pump 200 × 100 (advancing 114°/45°) | −736 | -725 | 1.4 % | — | — |
The paper's RBV section states a 95° receding angle for the cover, but its Table 1 is reproduced by the 89° given in its Conclusions; 95° under-predicts every valve by 3–6 %. The full circuit's drainage sequence is not modelled: it needs receding-meniscus draining of reservoirs, which OpenFLUME does not yet represent, and the paper omits connector lengths.
Figure 6. Retention burst valve pressures: paper Table 1 against OpenFLUME with both stated cover angles.
B6. Bubble-damped syringe network (Lee et al. 2012)
Two syringe pumps, each holding an air bubble of V₀ = 2, 5 or 10 µL at 1 atm, feed resistances R₁ = 0.14 and R₂ = 0.20 psi·min/µL into a junction draining through R₃ = 0.25 psi·min/µL; Q_s1 = 1 µL/min while Q_s2 steps from 1 to 2.5 µL/min. Bubbles are isothermal (P·V = P₀V₀), as in the paper. The reference is the network's exact ODE, integrated independently.
| Bubble | tolerance | max error vs exact ODE |
|---|---|---|
| 2 µL | 10⁻⁹ | 9.47e-4 |
| 5 µL | 10⁻⁹ | 9.47e-4 |
| 10 µL | 10⁻⁹ | 9.47e-4 |
| 5 µL | 10⁻⁶ (UI default) | 9.68e-4 |
The remaining error is backward-Euler time integration: at 100 / 200 / 800 steps per t₀ it is 1.93e-3 / 9.68e-4 / 2.43e-4 — first order. The paper reports that responses for all bubble sizes collapse when time is scaled by t₀ ∼ V₀R₃/P₀; in OpenFLUME the 2 µL and 10 µL responses coincide in t/t₀ to 1.71e-12.
Measured on the previous solver (absolute tolerance), the same 5 µL case reached a maximum error of 1.2 % at tol = 10⁻⁹ — the flow froze at 0.959 µL/min while the exact solution continues to 0.971 µL/min, once the remaining mass imbalance fell below 10⁻⁹ kg/s — and 37 % at the UI default 10⁻⁶, where the syringe step never propagated to branch R1 at all. flowScaledTolerance.test.ts fails on the previous solver.
Figure 7. Flow in branch R1 after the syringe step, in scaled time: three bubble volumes on one exact curve.
Conclusions
OpenFLUME's microfluidic closures reproduce their exact solutions to solver precision, and its network solver reproduces six published microfluidic designs from their printed parameters — matching the authors' COMSOL and design values, their trapping and distribution outcomes, and their measurements within the uncertainties they state. Two limits are visible in the published cases and documented: receding-meniscus draining (needed for the full retention-burst-valve drainage sequence of capillaric circuits) and entrance effects of constrictions shorter than they are wide. The cases also exposed one solver defect in the micro-flow regime — absolute mass-row tolerance — which is fixed and regression-tested.
References
- Shah, R. K., London, A. L., Laminar Flow Forced Convection in Ducts, Academic Press, 1978.
- Bruus, H., Theoretical Microfluidics, Oxford University Press, 2008.
- Washburn, E. W., "The dynamics of capillary flow," Physical Review 17, 273 (1921).
- Cho, H., Kim, H.-Y., Kang, J. Y., Kim, T. S., "How the capillary burst microvalve works," J. Colloid Interface Sci. 306, 379–385 (2007).
- Vasilakis, N., Papadimitriou, K. I., Morgan, H., Prodromakis, T. (2019). Modular pressure and flow rate-balanced microfluidic serial dilution networks for miniaturised point-of-care diagnostic platforms. Sensors 19(4):911. doi:10.3390/s19040911
- Tan, W.-H., Takeuchi, S. (2007). A trap-and-release integrated microfluidic system for dynamic microarray applications. PNAS 104(4):1146–1151. doi:10.1073/pnas.0606625104
- Guermonprez, C., Michelin, S., Baroud, C. N. (2015). Flow distribution in parallel microfluidic networks and its effect on concentration gradient. Biomicrofluidics 9(5):054119. doi:10.1063/1.4932305
- Reis, N. M. et al. (2021). Gravity-driven microfluidic siphons: fluidic characterization and application to quantitative immunoassays. ACS Sensors 6(12):4338–4348. doi:10.1021/acssensors.1c01524
- Olanrewaju, A. O., Robillard, A., Dagher, M., Juncker, D. (2016). Autonomous microfluidic capillaric circuits replicated from 3D-printed molds. Lab Chip 16:3804–3814. doi:10.1039/c6lc00764c
- Lee, J., Rahman, F., Laoui, T., Karnik, R. (2012). Bubble-induced damping in displacement-driven microfluidic flows. Phys. Rev. E 86:026301. doi:10.1103/PhysRevE.86.026301
Nomenclature
| Symbol | Meaning |
|---|---|
| A | channel cross-section w·h |
| D_h | hydraulic diameter 2wh/(w+h) |
| f·Re | Darcy friction factor × Reynolds number (laminar constant) |
| h, w, L | channel height, width, length |
| P_cap | capillary (Young–Laplace) pressure |
| Q | volumetric flow |
| R, R_b | hydraulic resistance (manifold segment, branch) |
| r_l, r_g | liquid / gas resistance per unit length |
| s | meniscus position (liquid length) |
| t₀ | bubble time scale V₀R₃/P₀ |
| γ | surface tension |
| θ | contact angle |
| μ, ρ | viscosity, density |