Poiseuille's Law Calculator (Laminar Pipe Flow Rate)

Flow through a narrow tube, exactly

Poiseuille’s law (sometimes Hagen-Poiseuille, after the two engineers who arrived at it independently around 1840) gives the volumetric flow rate of a thick, slow-moving fluid through a round pipe. It applies whenever the flow is laminar, meaning smooth and layered rather than turbulent. Jean Poiseuille was a physician studying blood flow through capillaries, which is exactly where the law still earns its keep today.

The formula

Q = (π × r⁴ × ΔP) / (8 × μ × L)

Where:

• Q = volumetric flow rate (volume per second)
• r = inside radius of the pipe
• ΔP = pressure difference between the two ends (what drives the flow)
• μ = dynamic viscosity of the fluid (its thickness; honey has high μ, water low)
• L = length of the pipe

The whole expression says flow rises with pressure and radius, and falls with viscosity and length. All of that is intuitive except for one explosive detail.

The fourth power of the radius is everything

Flow does not scale with radius, or with radius squared (the cross-section area). It scales with radius to the fourth power. Double the radius and flow jumps by 2⁴ = 16 times. Halve the radius and flow drops to one-sixteenth. No other term in the equation comes close to this leverage.

This single fact explains an enormous amount of the real world:

• Blood pressure and arteries. When an artery narrows by plaque (atherosclerosis), even a modest radius loss slashes flow dramatically. A 20 percent radius reduction cuts flow by about 60 percent. This is why arterial stenosis is so dangerous and why vasodilator drugs, which widen vessels only slightly, raise flow so much.
• IV drips and catheters. A wider-gauge IV catheter delivers fluid far faster. Going from a 20-gauge to a 14-gauge catheter (roughly double the radius) can increase flow more than tenfold, which is why trauma teams reach for large-bore lines.
• Plumbing and microfluidics. Slightly undersized tubing chokes flow far more than people expect, and lab-on-a-chip channels are designed around this sensitivity.

Viscosity, the other lever

Viscosity μ measures a fluid’s internal friction. Reference values in pascal-seconds (Pa·s) at about 20 degrees Celsius:

Warm a fluid and its viscosity drops, so flow rises: cold honey barely pours, warm honey runs. This is why engine oil is rated for temperature and why your car is harder to start on a freezing morning.

Hydraulic resistance

Rearranging the law into the form of Ohm’s law for fluids, Q = ΔP / R, gives the hydraulic resistance:

R = 8μL / (π r⁴)

The same r⁴ sits in the denominator, so resistance skyrockets as the tube narrows. Engineers and physiologists both use this resistance form to analyze networks of pipes or vessels in series and parallel, just like electrical circuits.

When the law stops working

Poiseuille’s law assumes:

• Laminar flow. Above a Reynolds number of roughly 2,000 the flow turns turbulent and the law no longer holds. Fast flow, wide pipes, and low viscosity all push toward turbulence.
• A Newtonian fluid, one whose viscosity does not change with flow speed. Water and air qualify. Blood does not quite, because it is a suspension of cells; its effective viscosity drops in narrow vessels (the Fahraeus-Lindqvist effect), so the law is an approximation for blood, though a useful one.
• A long, straight, rigid, round pipe. Entrance effects, bends, and flexible walls all introduce error.

Worked example, an IV line

Saline (viscosity close to water, 0.001 Pa·s) flows through an IV catheter of radius 0.5 mm and length 50 mm, driven by a pressure of 5,000 Pa (about a half-meter bag height).

r = 0.0005 m, so r⁴ = 6.25 × 10⁻¹⁴ m⁴. Q = (π × 6.25e-14 × 5000) / (8 × 0.001 × 0.05) = 9.82e-10 / 4e-4 = 2.45e-6 m³/s.

That is about 2.45 mL/s, or roughly 147 mL/min, a brisk drip. Double the catheter radius and the same pressure pushes nearly 2,400 mL/min. The fourth-power law, doing its work.