Laminar flow and the Reynolds number

The Reynolds number is the ratio of inertial forces to viscous forces in a flow. It is dimensionless, and it is the single number that tells you what kind of flow you have.

Re = ρ · v · Dₕ / μ

Symbol	Is	Unit
ρ	Fluid density	kg/m³
v	Mean velocity	m/s
Dₕ	Hydraulic diameter, `4A/P`	m
μ	Dynamic viscosity	Pa·s

OpenChip computes it per segment and shows it in the Fluid Dynamics table.

What the value means

Re	Regime	What it looks like
< 2000	Laminar	Smooth parallel layers. No cross-mixing.
2000–4000	Transitional	Intermittent, unpredictable.
> 4000	Turbulent	Chaotic, strongly mixing.

These are the conventional pipe-flow thresholds, and they are what OpenChip labels against.

Microfluidic values

Work an example. Water at 20 °C, a 0.8 × 0.5 mm rectangular channel, 10 µL/min:

Cross-sectional area: 0.4 mm² = 4 × 10⁻⁷ m²
Volumetric flow: 10 µL/min ≈ 1.67 × 10⁻¹⁰ m³/s
Mean velocity: ≈ 4.2 × 10⁻⁴ m/s — about 0.42 mm/s
Hydraulic diameter: 2wh/(w+h) ≈ 0.62 mm
ρ = 998.2 kg/m³, μ = 1.002 × 10⁻³ Pa·s

Re = (998.2 × 4.2e-4 × 6.2e-4) / 1.002e-3 ≈ 0.26

0.26. Roughly four orders of magnitude below the turbulent threshold. You could raise the flow rate a thousandfold and still be laminar.

This is typical, not a contrived example. It is what the counter-flow exchanger reports, and it is why every microfluidic design problem is a laminar design problem.

What follows from it

Flow is predictable. Same inputs, same result, every time. No intermittency, no chaotic transitions. This is why an analytical model is useful here at all — see The limits of analytical estimates.

Flow is reversible. Run a laminar flow backwards and it retraces its path. The famous demonstration is dye stirred into glycerol and then un-stirred; the same principle applies in your channel.

Nothing mixes on its own. Two streams stay side by side and cross only by diffusion. Every mixer in the template library is a way of working around this.

Velocity is not uniform across the channel. The profile is parabolic-ish — zero at the walls, maximum in the middle. Fluid at the centre arrives sooner than fluid at the edge, so a “residence time” is really a mean, and there is a distribution around it. This matters for reaction chemistry and for anything where the slowest fluid element determines the outcome.

The hydraulic diameter

Dₕ = 4A/P — four times cross-sectional area over wetted perimeter. It lets a non-circular channel be compared with a round one.

Shape	Dₕ
Circle, diameter d	d
Rectangle, w × h	`2wh/(w+h)`
Square, side a	a

For the 0.8 × 0.5 mm channel above: 2 × 0.8 × 0.5 / 1.3 ≈ 0.62 mm.

What OpenChip does with it

It computes Re per segment, labels the regime, and reports both — even when the answer is always “laminar”.

That is deliberate. You might be designing for a transition, and hiding a number because it usually looks boring would hide exactly the thing you are hunting for.

The heatmap’s Reynolds metric colours the network by it, which makes it easy to see where a narrow segment has pushed the local velocity up.

Assumptions

The Reynolds number OpenChip reports assumes:

Fully developed flow. The velocity profile has settled. Near an inlet it has not — see the entrance-length discussion on The limits of analytical estimates.
A Newtonian fluid, with one viscosity independent of shear rate.
A constant cross-section over the segment. Where a channel tapers, the reported value is for the geometry the solver sampled.