Microfluidics
Theory and Physics
Fundamentals of Microfluidics
Professor, what's the difference between microfluidics and macro-scale fluid dynamics? Is it just the smaller scale?
Changing the scale changes the "ruler" of physics. At the macro scale, inertial forces dominate, but at the micro scale (characteristic length $L \sim 1\text{--}100\,\mu\text{m}$), viscous forces and surface tension become dominant.
Let's compare with specific numbers. For water flow velocity $U = 1\,\text{mm/s}$ and channel width $D = 100\,\mu\text{m}$:
Since Re << 1, inertial forces can be neglected, and Stokes flow (creeping flow) dominates.
Re = 0.1 means it's completely viscosity-dominated. No worries about turbulence.
Correct. Furthermore, the importance of surface tension is indicated by the Capillary number:
Since Ca << 1, the shape of droplets/bubbles is determined by surface tension.
Stokes Equations
In the limit of Re << 1, the inertial term in the Navier-Stokes equations vanishes, leaving the Stokes equations.
The Stokes equations have the following important properties:
- Linear: Superposition of solutions is possible.
- Time-reversible: Reversing the applied force returns the fluid to its original state (the reason mixing is difficult).
- Instantaneous response: Without inertia, pressure changes propagate instantly throughout the domain.
Time-reversible means that at the micro scale, stirring doesn't cause mixing?
It won't mix with conventional methods. This is why designing micro-mixers is challenging. You must rely on diffusion or utilize chaotic advection (e.g., zigzag channel structures).
Physics Unique to Microfluidics
Let's organize the physical phenomena that become important at the micro scale.
| Phenomenon | Governing Parameter | Influence at Macro Scale | Influence at Micro Scale |
|---|---|---|---|
| Surface Tension | Ca, We, Bo | Usually negligible | Dominant |
| Electroosmotic Flow (EOF) | $\zeta$ potential, Debye length | Negligible | Used as a driving force |
| Slip Flow | Knudsen number Kn | no-slip | Slip occurs for Kn > 0.01 |
| Diffusive Mixing | Peclet number Pe | Convection-dominated | Diffusion-dominated |
| Contact Angle Hysteresis | Advancing/receding contact angle | Not critical | Governs device operation |
The Bond number $\text{Bo} = \rho g L^2 / \sigma$ also becomes very small at the micro scale ($\text{Bo} \sim 10^{-6}$ for $L = 100\,\mu\text{m}$). This means gravity is completely negligible, and behavior is the same whether in space or on Earth.
Gravity having no effect is counterintuitive, but it's a scale effect, right?
The Birth of Microfluidics—The μTAS Revolution and Lab-on-a-Chip in the 1990s
The dawn of microfluidics began in the early 1990s when Manz & Widmer (1990) at ETH Zurich proposed the concept of "μTAS (Micro Total Analysis System)". This involved fabricating channels tens to hundreds of micrometers wide on glass substrates using semiconductor manufacturing technology (photolithography), creating the concept of "Lab-on-a-Chip" where mixing, separation, and detection of reagents are completed on a single chip. At this channel scale, Stokes flow with Reynolds numbers less than 1 dominates, and since turbulent mixing cannot be used, passive mixing structures like T-junctions or helical shapes are necessary. Today, this technology is utilized in the microchannels of COVID-19 rapid test kits and DNA sequencers, with CFD-based channel design optimization being at the core of product development.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—these are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost drops significantly, solving first with a steady-state assumption is a basic CFD strategy.
- Convection term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They're completely different! Convection is transport by flow, conduction is transfer by molecules. There's an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever left milk in coffee without stirring? Even without mixing, after a while, they naturally blend. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re flows, convection overwhelmingly dominates, and diffusion plays a minor role.
- Pressure term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow arises where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- Source term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it upward. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force applied to molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq approximation (natural convection): Density variation is considered only in the buoyancy term; constant density is used in other terms.
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock wave capturing required), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units. |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis. |
| Density $\rho$ | kg/m³ | Air: ~1.225 kg/m³ @20°C, Water: ~998 kg/m³ @20°C |
| Viscosity coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition. |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
CFD Methods for Microfluidics
Can regular CFD solvers be used for microfluidics simulations?
Within the range where the continuum approximation holds (Kn < 0.01), standard Navier-Stokes-based CFD can be used. However, there are numerical challenges specific to the micro scale.
Two-Phase Flow (Droplet/Bubble) Interface Tracking
Droplet generation and T-junction merging are common in microfluidic devices. Let's compare the main interface tracking methods.
| Method | Advantages | Disadvantages | Application Examples |
|---|---|---|---|
| VOF (Volume of Fluid) | Good mass conservation, low cost | Interface diffusion, contact line issues | Droplet generation, slug flow |
| Level-Set | Smooth interface shape | Mass non-conservation | Merging, splitting |
| CLSVOF | Combines advantages of VOF+LS | Complex implementation | Problems requiring high accuracy |
| Phase-Field | Natural description of contact line | Strict mesh requirements | Wetting phenomena, contact angle control |
| Front-Tracking | Explicit interface tracking | Weak against topology changes | Single droplet deformation |
The CSF (Continuum Surface Force) model is standard for numerical calculation of surface tension, but at the micro scale, spurious currents become a problem. These are non-physical flows arising from the discretization of surface tension, becoming more pronounced as Ca decreases.
So if surface tension is large but the mesh is coarse, non-physical flow occurs.
Countermeasures include:
- Ensure at least 10 cells per interface for resolution.
- Curvature calculation via the Height Function method (significantly more accurate than CSF).
- Sharp Surface Force method.
- Using the Phase-Field model (reduces spurious currents via thermodynamic description of the interface).
Electroosmotic Flow (EOF) Calculation
Electroosmotic flow, where fluid is driven by an electric field, is often used in microchannels. The governing equations are:
$\rho_e$ is charge density, $\varepsilon$ is permittivity, $\mathbf{E} = -\nabla\phi$ is the electric field. The Debye length region $\lambda_D \sim 1\text{--}100\,\text{nm}$ must be resolved, requiring extremely fine meshing.
If the Debye length is on the order of nm, the mesh scale differs by more than 3 orders of magnitude from the entire channel.
Correct. In practice, the Helmholtz-Smoluchowski slip velocity condition:
is often applied as a wall boundary condition, using an approach that does not resolve the interior of the EDL (electrical double layer). $\zeta$ is the zeta potential, $E_t$ is the tangential electric field at the wall.
"Capillary Force" in Microchannels—A World Where Surface Tension Rules Over Gravity
When channel width falls below 100μm, gravity ceases to be a dominant force for flow. Instead, surface tension takes the lead, with capillary pressure ΔP=4γcosθ/D (D is channel diameter) drawing in liquid. In fact, in paper-based microfluidics for disease diagnosis, blood flows via capillary force without any pump. To handle this numerically, the usual Navier-Stokes equations must be combined with methods like "VOF" or "Lattice Boltzmann Method" to track the gas-liquid interface, and the choice of method greatly influences result accuracy.
Upwind Scheme
1st-order upwind: Large numerical diffusion but stable. 2nd-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
2nd-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is typically judged when residuals for continuity, momentum, and energy equations drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2–0.3, Velocity: 0.5–0.7 are typical initial values.
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