Pressure drop and Hagen–Poiseuille

Pressure drop is what your pump has to overcome. Get it wrong and the pump stalls, a fitting blows off, or a branch you expected to split evenly does not.

The relation

For fully developed laminar flow in a straight duct of constant cross-section, generalised to non-circular shapes:

ΔP = Po · μ · L · Q / (2 · A · Dₕ²)

Symbol	Is	Unit
ΔP	Pressure drop	Pa
Po	Poiseuille number, `f·Re` — a pure function of shape	—
μ	Dynamic viscosity	Pa·s
L	Channel length	m
Q	Volumetric flow rate	m³/s
A	Cross-sectional area	m²
Dₕ	Hydraulic diameter, `4A/P`	m

For a circular channel, Po = 64 and this reduces to the familiar Hagen–Poiseuille form, ΔP = 128 μ L Q / (π d⁴).

Why the shape matters

Po depends only on the cross-section, and the spread is large:

Shape	Po
Circle	64
Square	56.91
Rectangle, 2:1	~62
Rectangle, 4:1	~73
Rectangle, 8:1	~82
Infinite parallel plates	96

Using the circular value of 64 for a rectangular channel under-predicts the pressure drop by up to 50 % — which is the difference between a syringe pump coping and stalling.

OpenChip gives each shape its own Poiseuille number, using the Shah & London polynomial for rectangular ducts. Its endpoints — 96 for parallel plates, 56.91 for a square — are both textbook values.

A trapezoid is approximated as a rectangle of the same mean width. Its true Po sits within a few percent of that for the shallow drafts used here, and a dedicated correlation would imply a precision this whole estimate does not have.

The fourth-power rule

Hold everything else fixed and ΔP ∝ 1/Dₕ⁴, roughly.

Change	Pressure
Halve the width	×16
Reduce the width 10 %	×1.5
Double the length	×2
Double the flow rate	×2

Length and flow are linear; size is not. This is the fact behind most microfluidic surprises.

Two practical consequences:

A slightly narrow print is a much higher pressure. A channel that came out 0.05 mm under on a 0.4 mm design is 12 % narrow and about 40 % more resistive.

Branch balance is fragile. In a gradient generator, a branch 10 % wider than its neighbours carries about 46 % more flow — and the concentration series it produces is no longer linear.

Resistance, and solving a network

Expressing the relation as a hydraulic resistance — the pressure a unit of flow costs, in Pa·s/m³ — makes the whole network tractable:

R = Po · μ · L / (2 · A · Dₕ²)      ΔP = R · Q

That is Ohm’s law with pressure for voltage and flow for current. A laminar channel network is a resistor network, and OpenChip solves it as one: resistances in series add, resistances in parallel combine reciprocally, and the whole system is solved for the flows that satisfy the inlet conditions.

This is why the per-segment table can show a different flow rate in each branch without you specifying one: the geometry determines the split.

OpenChip does not evaluate R once from a segment’s average section. It integrates Po / (2 · A · Dₕ²) along the channel, using the local width and height at each point, and multiplies by the viscosity. On a uniform channel that is the formula above. On a tapered one it matters: by the fourth-power rule above, the narrow end carries most of the resistance, and an average section would hide that. Wall shear is integrated the same way, so a taper’s shear sits mostly in its narrow end too.

Reading the numbers

In the Fluid Dynamics tab:

ΔP per segment. The max figure in the summary strip is the highest gauge pressure anywhere in the network — what the pump sees — not the largest single ΔP.
The Pressure heatmap shows the gradient from inlet to outlet, which is where a single over-resistive segment becomes obvious.
Flow per segment. On a branching network, this is the column that reveals an unbalanced split.

Sanity checks

Two independent ways to be wrong-footed, both worth checking:

Compare against your pump. A typical laboratory syringe pump manages tens of kPa comfortably. If the estimate is in the hundreds, something is too narrow or too long.

Compute one segment by hand. The channel calculator on the landing page does a single segment with the same code, which makes it a quick way to check that a surprising number is the geometry rather than a mis-assigned port.