Reduced-order modeling of coupled transport phenomena in shallow PDMS microchannels
Yilong Zhou
This repository documents four coupled-physics simulation projects from my work on electrokinetic microfluidics. The unifying contribution is a 2D depth-averaged numerical model derived from a second-order asymptotic analysis of the full 3D governing equations — enabling accurate, computationally efficient simulation of coupled electric, thermal, flow, and species transport in shallow microchannels.
The core question each project addresses: how does a physical effect change fluid behavior at a reservoir-microchannel junction, and can a reduced-order model capture it accurately enough to be useful?
Full 3D simulation of electrokinetic microchannels is computationally expensive. A naive 2D model (infinite depth assumption) is fast but systematically wrong — it ignores the viscous drag and electroosmotic slip from the top and bottom channel walls, causing:
- 2–3× under-prediction of electrokinetic instability threshold electric fields
- Incorrect vortex size, location, and orientation in induced-charge flow
- Unphysical temperature fields requiring unrealistic fitting parameters
The depth-averaged approach treats the channel's shallow aspect ratio as a smallness parameter δ = d/H ≪ 1. Expanding the 3D governing equations asymptotically in δ and depth-averaging yields 2D equations that recover the wall effects through an additional correction term in the momentum equation:
Full derivation including the temperature equation, ICEO dual-domain electric field, and Taylor dispersion correction for species transport: theory/depth_averaging_derivation.md
| # | Physical effect | Publication | My role |
|---|---|---|---|
| 1 | Conductivity mismatch → electrokinetic instability in ferrofluid | Microfluid. Nanofluid. 2015 + Sci. Rep. 2017 | Derived depth-avg model; all experiments; Simulation (joint) |
| 2 | Joule heating → electrothermal vortices at channel entrance | Electrophoresis 2017 | Derived depth-avg model; COMSOL simulation (joint) |
| 3 | Induced charge → ICEO vortices at dielectric corners | Phys. Fluids 2017 | Derived depth-avg model; COMSOL simulation (joint) |
→ 2015 model details · → 2017 depth-averaged model
Ferrofluid and DI water co-flow through a T-shaped microchannel. Their ~180× electrical conductivity mismatch induces free charge at the interface; above a threshold DC electric field this produces instability waves and chaotic flow at Reynolds number ≈ 1.
My experimental contribution (2015): fabricated T-shaped PDMS microchannels by soft lithography, prepared ferrofluid solutions at three concentrations (0.1×, 0.2×, 0.3×), measured threshold electric fields, characterized conductivity vs. concentration.
2015 → 2017 progression: The initial regular 2D model correctly predicted the decreasing instability threshold with increasing ferrofluid concentration, but under-predicted threshold electric fields by 2–3× for all conditions. This systematic error was traced to the complete neglect of top/bottom wall stabilizing effects. The Scientific Reports follow-on developed the nonlinear depth-averaged model — extending the asymptotic analysis to include a Taylor dispersion correction in the species transport equation — and validated it across four channel depths (32–100 µm).
Three-way comparison at threshold (0.2× ferrofluid, 45 µm channel):
| Experiment | Regular 2D model | Depth-averaged model |
|---|---|---|
 |
 |
 |
| 175.0 V/cm — periodic waves, inclined upstream ← | Chaotic at 60.4 V/cm; waves inclined downstream → ✗ | Periodic waves at 202.1 V/cm; inclined upstream ← ✓ |
The depth-averaged model captures two independent failure modes of the regular 2D model: the wrong threshold electric field (60.4 vs 175.0 V/cm) and the wrong wave inclination direction. The regular 2D model predicts waves tilted downstream because it over-predicts electroosmotic velocity in the ferrofluid — a direct consequence of ignoring top/bottom wall drag. The depth-averaged model corrects both simultaneously.
The regular 2D simulation is run at a lower electric field than the experiment because the model triggers instability far too early. Labels show the actual field used in each simulation.
Quantitative summary (0.2× ferrofluid):
| Channel depth | Experiment | Depth-averaged | Error | Regular 2D | Error |
|---|---|---|---|---|---|
| 32 µm | 305.6 V/cm | 326.0 V/cm | +6.6% | 60.4 V/cm | −80% |
| 45 µm | 175.0 V/cm | 202.1 V/cm | +15.5% | 60.4 V/cm | −65% |
| 60 µm | 119.4 V/cm | 151.1 V/cm | +27% | 60.4 V/cm | −49% |
| 100 µm | 83.3 V/cm | 123.4 V/cm | +48% | 60.4 V/cm | −27% |
Depth-averaged model accuracy: d/W < 0.3 → 6–15% error; d/W > 0.5 → regular 2D becomes adequate as wall effects weaken.
DC-biased AC electric fields drive electroosmotic flow through a PDMS microchannel. Joule heating concentrates in the narrow constriction at the channel entrance, raising local temperature and creating fluid property gradients. The electric field acts on these gradients via the electrothermal body force, generating counter-rotating vortices above a threshold AC voltage.
Coupled physics: electric field → temperature (substrate thermal resistance BCs) → conductivity / permittivity / viscosity → electrothermal body force → flow
Electric field (A), temperature (B), permittivity (C), conductivity (D), electrothermal body force (E), and velocity magnitude (F) at 20 V DC-biased 600 V AC. The constriction amplifies E by ~4×, raising local temperature ~20 K, shifting σ by +50% and ε by −10%, producing the body force that drives the circulations in F.
Key result: Vortex size and location match experiment across four AC voltages (450–700 V AC). The depth-averaged thermal model replaces the unrealistic convective coefficient assumption of prior 2D models with proper substrate thermal resistance terms — no fitting parameters. Computational cost: 10 min on a laptop vs. >4 hours on a 24-core cluster for the equivalent 3D model.
In low-ionic-concentration flow, Joule heating is negligible. Instead, electric field leaks into the dielectric PDMS corners at the channel entrance, polarizing the corner surfaces and generating induced charge electroosmosis (ICEO) — counter-rotating vortices that trap and concentrate particles via ICEO + positive DEP.
Coupled physics: Laplace's equation solved simultaneously in fluid AND PDMS wall → Robin-type BC yields induced zeta potential → ICEO slip velocity → vortex flow → particle tracing (EO + EP + DEP)
The depth-averaged model (center) correctly predicts vortex location near the channel entrance corners. The regular 2D model (right) predicts the wrong location, wrong size, and wrong orientation — the largest discrepancy class in the ICEO literature — because it ignores top/bottom wall damping entirely.
Key result: Maximum induced zeta potential scales linearly with AC voltage and wall permittivity, and exponentially with decreasing corner radius. Reducing corner radius from 40 µm to 2 µm increases peak vorticity by >24× (60 s⁻¹ → 1450 s⁻¹).
multiphysics-simulation/
├── README.md
├── theory/
│ └── depth_averaging_derivation.md ← full asymptotic derivation (5 equations)
├── models/
│ ├── ferrofluid_instability_2015/
│ │ └── README.md ← regular 2D model + all experimental work
│ ├── ferrofluid_instability_2017/
│ │ └── README.md ← depth-averaged model, 3-way comparison
│ ├── joule_heating_entry_flow/
│ │ └── README.md ← coupled E/T/flow, thermal resistance BCs
│ └── induced_charge_ICEO/
│ └── README.md ← dual-domain E field, ICEO, DEP tracing
└── assets/
├── README.md
├── figures/
└── videos/
| Tool | Role |
|---|---|
| COMSOL Multiphysics 5.1/5.2 | Primary simulation platform |
| Modules | Electric Currents, Electrostatics, Heat Transfer in Fluids/Solids, Laminar Flow, Transport of Diluted Species |
| Depth-avg wall terms | Added via COMSOL "Force" and "Reaction" features |
| Mesh | Structured 4 µm square elements; triangular at T-junction fillets |
| Imaging | Nikon Eclipse TE2000U, Nikon DS-Qi1Mc CCD, NIS-Elements AR 2.30 |
| Microfabrication | Soft lithography, PDMS, SU-8 photoresist |
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Thanjavur Kumar D., Zhou Y., Brown V., Lu X., Kale A., Yu L., Xuan X. "Electric field-induced instabilities in ferrofluid microflows." Microfluidics and Nanofluidics 19, 43–52 (2015). DOI: 10.1007/s10404-015-1546-8
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Song L., Yu L., Zhou Y., Antao A.R., Prabhakaran R.A., Xuan X. "Electrokinetic instability in microchannel ferrofluid/water co-flows." Scientific Reports 7, 46510 (2017). DOI: 10.1038/srep46510
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Prabhakaran R.A., Zhou Y., Patel S., Kale A., Song Y., Hu G., Xuan X. "Joule heating effects on electroosmotic entry flow." Electrophoresis 38, 572–579 (2017). DOI: 10.1002/elps.201600296
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Prabhakaran R.A., Zhou Y., Zhao C., Hu G., Song Y., Wang J., Yang C., Xuan X. "Induced charge effects on electrokinetic entry flow." Physics of Fluids 29, 062001 (2017). DOI: 10.1063/1.4984741
Portfolio: yilong-sim.github.io