Live Visualization
Velocity Profile & Particle Flow
Warm = fast · Cool = slow
Parabolic velocity profile
Parabolic velocity profile
Inner Radius (r)
0.51 mm
The r⁴ relationship makes this the most powerful variable
Pressure Gradient (ΔP)
100 mmHg
Gravity IV ≈ 73 mmHg · Pressure bag ≈ 150–300 mmHg
Viscosity (η)
1.0 cP
NS ≈ 1 cP · Blood ≈ 3–4 cP · Contrast ≈ 5–8 cP
Catheter Length (L)
32 mm
PIV ≈ 32 mm · Midline ≈ 200 mm · PICC ≈ 450 mm
Flow Rate (Q)
—mL/hr
Max Velocity (center)
—cm/s
Parabolic peak at vessel axis
Hydraulic Resistance (R)
—
Pa·s/m³ × 10¹²
Reference Table
Catheter Gauge Flow Comparison
| Gauge | Inner Radius | Flow @ 100 mmHg, NS | Relative to 18G | Visual |
|---|
The Fourth Power Law
A 14G catheter (r = 0.80mm) flows ~7.1× more than an 18G (r = 0.51mm).
That's (0.80/0.51)⁴ = 6.1. Radius is everything.
Blood vs Saline
Blood is 3–4× more viscous than normal saline. The same catheter,
pressure, and length will flow 3–4× less blood than saline — critical
for trauma resuscitation planning.
Length Matters Less
A PICC (450mm) vs a PIV (32mm) has ~14× the length, reducing flow ~14×.
But changing from 20G to 18G radius (0.40→0.51mm) increases flow ~2.6×.
Gauge beats length.
Pressure Bag Rationale
Pressure bags at 300 mmHg provide ~4× the pressure of gravity (73 mmHg).
Flow scales linearly with ΔP — unlike radius, so it's a useful adjunct
but can't compensate for a small catheter.
Historical Origin
The math was already medical.
Gotthilf Hagen (1839) and Jean Léonard Marie Poiseuille (1838–42) derived this equation independently — but Poiseuille was a physician specifically studying blood flow in capillaries. Unlike most physics-to-medicine translations, the Poiseuille equation started at the bedside.
The clinical application to vascular access — catheter selection, pressure bags, large-bore doctrine — came from anesthesiology and trauma medicine in the 1960s–90s.
Intracav Opportunity
Real-Time Ultrasound Flow Estimation
Ultrasound makes the Poiseuille equation actionable at the bedside. Diameter is directly measurable. Depth determines angle. Compressibility is a proxy for pressure state.
Proposed overlay:
US diameter → estimate flow rate (Poiseuille-based)
Show "optimal catheter size" recommendation
Flag turbulence risk before insertion
Surface as green / amber / red risk signal
US diameter → estimate flow rate (Poiseuille-based)
Show "optimal catheter size" recommendation
Flag turbulence risk before insertion
Surface as green / amber / red risk signal