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Biomedical Engineering DISCUSSION
I am building a gravity-fed flow phantom for a biomedical lab exercise: a reservoir about 1 m above the bench feeding water through 1 m of tubing with a 1 mm inner diameter. When I swapped to 0.5 mm tubing to mimic a smaller vessel, the flow dropped to a small fraction of what it was, far more than the change in cross-sectional area.
What law governs this, what flow should I expect from these numbers, and does the same relationship hold if I later use a blood-mimicking fluid?
For steady laminar flow of a Newtonian fluid in a straight round tube, the Hagen-Poiseuille law applies: Q = π × ΔP × r^4 / (8 × µ × L). Flow goes with the fourth power of the radius. Halving the radius divides the flow by 16 for the same pressure, although the area is only divided by 4, because the mean velocity also falls by a factor of 4: more of the fluid is close to the wall, where viscous drag holds it back.
Even a 10 percent reduction in radius gives 0.9^4 = 0.66, a 34 percent drop in flow.
Your numbers: 1 m of water head is ρ × g × h = 1000 × 9.81 × 1 = 9.8 kPa. With r = 0.5 mm, L = 1 m and water at about 20 °C (µ = 1.0 mPa·s): Q = π × 9810 × (0.0005)^4 / (8 × 0.001 × 1) = 2.4 × 10^-7 m³/s, which is 0.24 mL/s or about 14 mL per minute.
With the 0.5 mm tubing (r = 0.25 mm) it is one sixteenth of that, about 0.9 mL per minute. Both figures assume all of the head is lost in the tube; fittings and any narrower connector take a share.
Check that the flow is laminar, because the law is invalid otherwise. Mean velocity in the 1 mm tube is Q / A = 2.4 × 10^-7 / (7.85 × 10^-7) = 0.31 m/s, so the Reynolds number is ρ × v × D / µ = 1000 × 0.31 × 0.001 / 0.001 = about 310, far below the transition value of around 2000 to 2300. The law also assumes fully developed flow; the entrance length is roughly 0.05 × Re × D, about 15 mm here, negligible against 1 m.
With larger tubes and higher heads the Reynolds number rises quickly and the simple law overestimates the flow.
The fourth-power dependence makes the result very sensitive to the actual bore. Tubing inner diameter tolerances of a few percent produce flow differences of 10 percent or more between nominally identical pieces, and soft tubing changes diameter under pressure or where it is clamped. Measure the flow with a graduated cylinder and a stopwatch rather than trusting the calculation, and control it with a needle valve or the reservoir height, not by changing tube size.
Temperature matters too: the viscosity of water falls from about 1.0 mPa·s at 20 °C to about 0.7 mPa·s at 37 °C, which raises the flow by over 40 percent.
With a blood-mimicking fluid the relationship holds in form, with a different viscosity. Such fluids are usually made to about 3 to 4 mPa·s to match blood at high shear rates, so expect a third to a quarter of the water flow for the same head. Real blood is not Newtonian: its apparent viscosity rises at low shear rates and changes in very small tubes, so Poiseuille's law is only an approximation for it, and physiological flow is pulsatile, not steady.
For a teaching phantom those differences are usually acceptable; state them as limitations in the write-up.
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