Physics Foundations · Topic 6 of 6

Level 116 min read

The dimensionless numbers that run microfluidics

A field guide to Re, Pe, Ca, We and Bo — reading each as a contest between two effects, so you can tell which physics actually governs your device.

If you only remember three things

A dimensionless number answers one question: which physical effect dominates here? It is a ratio of two competing effects, so its value tells you which one wins.
Each number can be read two equivalent ways — as a ratio of forces, or as a ratio of timescales. Both give the same value; the timescale view is often the more useful one for transport and mixing.
There are no universal magic thresholds. 'High' and 'low' depend on your choice of characteristic length and velocity, on the geometry, the fluid, and what you are trying to achieve.
The five that run microfluidics: Re (inertia vs viscosity), Pe (advection vs diffusion), Ca (viscous vs interfacial), We (inertia vs interfacial), Bo (gravity vs interfacial).

What you’ll learn

What a dimensionless number is, and why nondimensionalising the physics produces them
The five key numbers — Re, Pe, Ca, We, Bo — with their formulas, units and meaning
How to read each one as either a force ratio or a timescale ratio
Why thresholds are not universal, and how to choose a characteristic length and velocity
Which number dominates in real microfluidic situations, and how the numbers relate

Intuition

Think of every microfluidic situation as a tug-of-war between two physical effects — say inertia pulling one way and viscosity the other. A dimensionless number is the scoreboard: it divides one effect by the other, so a value far above 1 means the first effect is winning, far below 1 means the second is, and near 1 means it is a genuine contest.

Because the number is a pure ratio, the units cancel — which is exactly why it travels across scales. The same Reynolds number describes a swimming bacterium and a scale model in a wind tunnel, as long as the ratio of inertia to viscosity matches.

A second, equivalent picture is a race between two clocks. Instead of asking which force is bigger, ask which process finishes first: does the flow carry a molecule downstream before diffusion can spread it sideways? The Péclet number is that race written as a single number.

The concept

When you write the governing equations of a flow and rescale every variable by a characteristic value — a length L, a velocity U, and so on — the equations reorganise into a dimensionless form whose coefficients are pure numbers. Those coefficients are the dimensionless numbers, and each one measures the relative size of two terms in the equations, i.e. two competing physical effects. That is why they are diagnostic tools rather than mere formulas: the value tells you which term you may neglect and which one governs the behaviour.

Every number here can be read as a ratio of forces or as a ratio of timescales, and the two readings are algebraically identical. The Reynolds number, for example, is both the ratio of inertial to viscous forces and the ratio of the time for momentum to diffuse across the channel (L²/ν) to the time for the flow to carry fluid along it (L/U). The force picture is intuitive; the timescale picture is often more useful for transport and mixing, where the real question is which process happens first.

The five competitions share a structure. Re weighs inertia against viscosity; Pe weighs advection against diffusion; Ca weighs viscous forces against interfacial (surface) tension; We weighs inertia against interfacial tension; Bo weighs gravity against interfacial tension. Notice that three of them — Ca, We and Bo — measure something against interfacial tension, which is why surface tension is such a recurring character at the microscale.

Because they share variables, the numbers are related. Pe = Re·Sc, where the Schmidt number Sc = ν/D compares momentum diffusion to mass diffusion; We = Re·Ca; and We/Bo = U²/(gL), a Froude number comparing inertia to gravity. One number stands apart: the capillary number Ca = μU/γ contains no length at all, so, unlike the others, it does not depend on device size — one reason it is the natural parameter for droplet break-up regardless of scale.

It is tempting to memorise cut-offs — 'turbulence above Re ≈ 2000', say — but those values are tied to a specific geometry (there, a long circular pipe) and to specific definitions of L and U. Change the cross-section, choose the hydraulic diameter instead of the width, use the mean instead of the maximum velocity, or switch fluids, and the meaningful boundary moves. Treat thresholds as calibrated guidelines for a stated configuration, never as universal constants; what is robust is the comparison the number expresses, not a magic value.

Why it matters

Computing the right dimensionless number is often faster and more revealing than a full simulation: it tells you at a glance whether your flow is laminar, whether it will mix on its own, whether droplets will form cleanly, and whether gravity matters — and therefore which equations and design rules actually apply.

How it works

Reynolds, Re = ρUL/μ — inertia vs viscosity (the template above). Density ρ [kg/m³], velocity U [m/s], length L [m], viscosity μ [Pa·s]; Re is dimensionless. Timescale view: momentum-diffusion time L²/ν over advection time L/U. High Re → inertial, possibly turbulent; low Re → laminar. Microfluidic reality: Re is typically ≪ 1 to ~100, so flow is laminar and predictable. Example: water at 10 mm/s in a 100 µm channel gives Re ≈ 1. Limitation: the turbulent threshold is geometry-specific, and L is a choice (use the hydraulic diameter for non-circular channels).
Péclet, Pe = UL/D — advection vs diffusion. Velocity U [m/s], length L [m], molecular diffusion coefficient D [m²/s]; dimensionless. Timescale view: diffusion time L²/D over advection time L/U. High Pe → advection dominates, so co-flowing streams stay unmixed for a long distance; low Pe → diffusion keeps up and mixing is fast. Microfluidic reality: for typical molecules Pe across a channel width is large, which is why microscale mixing is hard and needs long paths or engineered (e.g. chaotic) mixers. Relation: Pe = Re·Sc. Example: U = 1 mm/s, L = 100 µm, D = 10⁻⁹ m²/s → Pe ≈ 100. Limitation: D depends strongly on molecule size and temperature, and Pe uses whichever L matters for the transport in question (channel width for cross-stream mixing), which may differ from the L used for Re.
Capillary, Ca = μU/γ — viscous forces vs interfacial tension. Viscosity μ [Pa·s], velocity U [m/s], interfacial tension γ [N/m]; dimensionless — and note there is no length, so Ca is scale-independent. High Ca → viscous shear dominates and stretches interfaces (jetting, thread break-up); low Ca → interfacial tension dominates and interfaces stay compact and rounded (dripping, well-defined droplets). Microfluidic reality: Ca selects droplet-generation regimes, and typical operation sits at low-to-moderate Ca (often ~10⁻³–10⁻¹). Example: μ = 10⁻³ Pa·s, U = 10 mm/s, γ = 0.03 N/m → Ca ≈ 3×10⁻⁴. Limitation: the dripping↔jetting value depends on geometry (T-junction vs flow-focusing) and on the viscosity ratio of the two phases — it is not universal — and in confined junctions at very low Ca break-up is driven by pressure, not shear.
Weber, We = ρU²L/γ — inertia vs interfacial tension. Density ρ [kg/m³], velocity U [m/s], length L [m], interfacial tension γ [N/m]; dimensionless. High We → inertia overwhelms surface tension and interfaces deform and break (splashing, atomisation); low We → surface tension holds the interface together. Microfluidic reality: because velocities and lengths are small, We is usually small, so surface tension (not inertia) shapes interfaces — though We rises in high-speed step-emulsification or jetting. Relation: We = Re·Ca. Example: ρ = 1000, U = 0.1 m/s, L = 100 µm, γ = 0.05 N/m → We ≈ 0.02. Limitation: which velocity and length you use (drop diameter? channel width? relative velocity of the phases?) changes the value, so define them.
Bond, Bo = ρgL²/γ — gravity vs interfacial tension (also called the Eötvös number). Density ρ [kg/m³], gravity g [m/s²], length L [m], interfacial tension γ [N/m]; dimensionless. High Bo → gravity dominates and interfaces flatten while buoyancy and sedimentation matter; low Bo → surface tension dominates and gravity is negligible. Microfluidic reality: Bo is almost always ≪ 1 at the microscale — equivalently L is far below the capillary length ℓ_c = √(γ/ρg) (~2.7 mm for water) — so orientation usually does not matter and drops stay spherical. Example: ρ = 1000, L = 100 µm, γ = 0.072 N/m → Bo ≈ 1.4×10⁻³. Limitation: Bo can matter for large density mismatches, long residence times (slow settling still happens) or millimetre-scale features; 'gravity is negligible' is a consequence of small L, not a law.

The equation

Re = ρUL/μ = UL/ν

The template reading. Reynolds number: inertial ÷ viscous forces, equivalently the momentum-diffusion time (L²/ν) ÷ the advection time (L/U).

Variables

Symbol	Variable	Unit
Re	Reynolds number	—
ρ	Fluid density	kg/m³
U	Characteristic velocity (e.g. mean flow speed)	m/s
L	Characteristic length (e.g. hydraulic diameter)	m
μ	Dynamic viscosity	Pa·s
ν	Kinematic viscosity, ν = μ/ρ	m²/s

Assumptions

The equation above assumes:

L and U are deliberate choices: in a non-circular channel L is usually the hydraulic diameter Dₕ = 4A/P, and U is usually the mean velocity — always state which you mean.
The fluid is Newtonian with constant properties; strongly shear-thinning or variable-viscosity fluids need extra care.
The flow is single-phase and incompressible — the same precondition applies before reading any of these numbers.
The number describes a regime, not a hard threshold; the laminar–turbulent boundary in particular is geometry-specific.
Entrance and development effects mean the effective number can differ near inlets, bends and junctions.

Reading the number

High value

Re ≫ 1: inertia dominates viscosity. Flow can become unstable and, high enough, turbulent, so mixing by chaotic velocity fluctuations becomes possible. Rare inside microchannels.

Low value

Re ≪ 1: viscosity dominates inertia. Flow is smooth, laminar and reversible; streams run side by side and mix only by diffusion. This is the everyday microfluidic world.

Re compares inertial to viscous forces, Re = ρUL/μ, or equivalently the viscous (momentum-diffusion) time L²/ν to the advection time L/U. The familiar pipe-flow figures (laminar below Re ≈ 2000) are calibrated for a circular pipe and are not universal — they shift with geometry and with how L and U are defined.

Worked example

Illustrative calculation (round numbers, to show the method — not measured data). Water in a straight 100 µm channel at U = 1 mm/s. Take ρ = 1000 kg/m³, μ = 10⁻³ Pa·s (so ν = 10⁻⁶ m²/s) and a small-molecule D ≈ 10⁻⁹ m²/s. Which effects dominate?

Re = UL/ν = (10⁻³)(10⁻⁴)/10⁻⁶ = 0.1 Pe = UL/D = (10⁻³)(10⁻⁴)/10⁻⁹ = 100

Re ≈ 0.1 (≪ 1) → firmly laminar: viscosity wins, with no turbulence to help. Pe ≈ 100 (≫ 1) → advection wins over diffusion: the stream is carried downstream faster than it can mix sideways. The same channel is simultaneously 'low Re' and 'high Pe' — which is exactly why microfluidic flows are orderly yet hard to mix.

Microfluidic example

Straight microchannel (pressure-driven flow). The contest is inertia vs viscosity — the Reynolds number. At the micron scale Re is tiny, so viscosity dominates: the flow is laminar, steady and reversible, with a predictable parabolic profile. Which dominates? Viscosity.

Mixing two co-flowing streams. Now the question is advection vs diffusion — the Péclet number. Pe is large, so advection dominates and the streams refuse to mix over short distances. Which dominates? Advection — which is why you need a long serpentine path or an active/chaotic mixer to force diffusion to catch up (in a chaotic mixer the required length grows only slowly, roughly logarithmically, with Pe).

Droplet formation at a T-junction or flow-focuser. Here viscous shear competes with interfacial tension — the capillary number. At low Ca interfacial tension wins and you get clean, monodisperse dripping; as Ca rises, viscous forces win and the system shifts to jetting with a thin thread. Which dominates? It depends on Ca — the knob you turn (via flow rate, viscosity or surfactant) to choose the regime.

Capillary-driven filling. A hydrophilic channel fills itself; whether that matters against body forces is gravity vs interfacial tension — the Bond number. Bo ≪ 1, so surface tension dominates gravity and the filling is set by capillary pressure and viscous drag, not by height or orientation. Which dominates? Interfacial tension.

Gravity-sensitive systems (sedimenting cells or beads, density-mismatched phases, larger chambers). Gravity re-enters when L grows or densities differ — again the Bond number, plus slow settling over long times. In a millimetre-scale reservoir or during a long incubation, Bo is no longer negligible and cells or beads sediment. Which dominates? It can tip toward gravity — a reminder that 'the microscale ignores gravity' is a statement about small L and short times, not a universal truth.

Practical design implications

Pick — and state — your characteristic length and velocity before quoting a number. For Re in a non-circular channel use the hydraulic diameter; for cross-stream mixing use the channel width in Pe. The same flow gives different numbers under different (equally valid) choices.
Use the number to decide what to neglect. Low Re lets you drop inertial terms (Stokes flow); low Bo lets you ignore gravity; high Pe warns that diffusion will not mix for you. That is the practical payoff of computing them.
Change a number by changing the physics you control. Ca and the droplet regime move with flow rate, viscosity and surfactant (γ); Pe and mixing move with velocity and channel width; Re rarely leaves the laminar range at these scales.
Watch the interfacial trio together. Ca, We and Bo all measure something against surface tension, so a surfactant that lowers γ raises all three at once — helpful for one goal, harmful for another.
Do not design to a borrowed threshold. Validate the actual transition for your geometry and fluids rather than assuming a textbook cut-off transfers to your device.

Try it yourself

Put these numbers into the reynolds number calculator and see the result for your own channel.

Open the Reynolds number calculator →

Common mistakes

Watch out for:

Treating thresholds as universal. 'Turbulence above Re ≈ 2000' is a circular-pipe result; the meaningful boundary depends on geometry and on how L and U are defined.
Using the wrong (or an unstated) characteristic length. Radius vs diameter vs hydraulic diameter, or width vs length, can shift a number by large factors — and make two people's 'Pe' incomparable.
Assuming microfluidics means high speed and inertia. Almost all microfluidic flow is low-Re; reaching for turbulent-mixing intuition is a classic error.
Forgetting that Ca has no length. Ca does not scale with device size, so shrinking a device does not change Ca at fixed velocity — a point that trips up scaling arguments.
Conflating the force-ratio and timescale readings carelessly. They are equal for a given number, but mixing up which L or U belongs to which process yields the wrong timescale.

Researcher notes(advanced)

The numbers arise formally from nondimensionalising the governing equations (Navier–Stokes, the advection–diffusion equation, the interfacial stress balance); each appears as the coefficient of a specific term, which is why setting it small justifies dropping that term (e.g. Stokes flow as Re → 0). See Bruus (2008).
Interrelations worth remembering: Pe = Re·Sc (Sc = ν/D), We = Re·Ca, and We/Bo = U²/(gL) (a Froude number). These let you convert between numbers when one variable is hard to measure directly.
Characteristic-length choice is not cosmetic: Re commonly uses the hydraulic diameter, Pe the transverse dimension relevant to mixing, and Bo the dimension along gravity. A bare number reported without its L and U is ambiguous.
In confined T-junctions, droplet break-up at low Ca is dominated by the pressure build-up across the forming droplet rather than by viscous shear (Garstecki et al., 2006), so the naïve 'shear vs tension' Ca picture is incomplete in that regime.
Watch for name and definition variants: the Bond number is also the Eötvös number, and the Péclet number can be defined for heat (using thermal diffusivity α) rather than mass (D). Always check which diffusivity and which length a source uses before comparing values.

Related concepts

Related tools