Poiseuille Flow

Derivation

Take a long cylindrical pipe, radius R, length L. Drive an incompressible Newtonian fluid (viscosity μ) through it with a pressure difference ΔP. Use cylindrical coordinates (r, φ, z) with z along the pipe axis. Assume steady, axially symmetric, fully developed flow: velocity has only a z-component u(r) that depends only on r.

The Navier-Stokes equation in z reduces to:

0 = -dP/dz + mu * (1/r) d/dr (r du/dr)

With constant pressure gradient dP/dz = −ΔP/L and boundary conditions u(R) = 0 (no-slip), du/dr|_(r=0) = 0 (axial symmetry):

u(r) = (DeltaP / (4 mu L)) * (R^2 - r^2)

That is the parabolic profile. The maximum velocity (centerline) is u_max = ΔP R² / (4μL).

Integrate over the cross-section:

Q = integral 0 to R of u(r) * 2*pi*r dr
  = (pi DeltaP / (2 mu L)) * integral 0 to R of r*(R^2 - r^2) dr
  = (pi DeltaP / (2 mu L)) * (R^4/2 - R^4/4)
  = pi R^4 DeltaP / (8 mu L)

The mean velocity ū = Q / (πR²) = ΔP R² / (8μL) = u_max / 2.

The r⁴ sensitivity

Doubling the radius at the same pressure drop multiplies flow by 16. This is one of the most consequential scalings in nature:

R (mm)	Relative Q	Example
0.004	1 unit	Capillary (~ 8 μm)
0.01	39 units	Arteriole, constricted
0.05	24,400	Arteriole, relaxed
0.5	2.4 × 10⁷	Aortic arch in a mouse
10	3.9 × 10¹²	Aorta in human (~ 20 mm)

Same pressure, twelve orders of magnitude in flow rate. Arterioles regulate blood distribution by changing radius modestly — the cardiovascular system gets enormous control from small geometric changes.

Worked example: blood through a capillary

Take a capillary 8 μm diameter (R = 4 μm = 4 × 10−⁶ m), length L = 0.5 mm = 5 × 10−⁴ m, pressure drop ΔP = 2 mmHg = 266 Pa. Blood plasma viscosity μ ≈ 1.2 mPa·s = 1.2 × 10−³ Pa·s.

u_max = DeltaP * R^2 / (4 mu L)
      = 266 * (4e-6)^2 / (4 * 1.2e-3 * 5e-4)
      = 266 * 1.6e-11 / 2.4e-6
      ≈ 1.77e-3 m/s ≈ 1.8 mm/s

Mean velocity ≈ 0.9 mm/s — consistent with measured blood velocity in capillaries. Transit time through a 0.5 mm capillary is about half a second — enough time for oxygen and CO₂ to diffuse across the wall. Whole-blood viscosity is higher than plasma viscosity (factor of 2-3 due to red cells), so actual capillary flow is slower, but the order of magnitude is right.

Where laminar flow breaks down

Define the Reynolds number Re = ρ ū D / μ. Below Re ≈ 2300 flow remains laminar (Poiseuille valid). Above Re ≈ 4000 flow is fully turbulent. Between them is a transition region.

System	Re	Regime
Capillary blood flow	~ 0.003	Deeply laminar
Arteriole blood flow	~ 1	Laminar
Aortic blood flow (peak systole)	~ 4000	Marginal; pulsatile
Microfluidic channel	~ 0.1 - 100	Laminar
Oil pipeline (10 cm, 1 m/s)	~ 10,000	Turbulent
Tap water in household pipe	~ 30,000	Turbulent

JavaScript — Poiseuille flow calculator

// Hagen-Poiseuille flow rate
function poiseuilleFlow({ R_m, L_m, dP_Pa, mu_Pa_s }) {
  return Math.PI * Math.pow(R_m, 4) * dP_Pa / (8 * mu_Pa_s * L_m);  // m^3/s
}

// Centerline velocity
function maxVelocity({ R_m, L_m, dP_Pa, mu_Pa_s }) {
  return dP_Pa * R_m * R_m / (4 * mu_Pa_s * L_m);  // m/s
}

// Mean velocity (Q / A)
function meanVelocity(opts) { return maxVelocity(opts) / 2; }

// Reynolds number
function reynolds({ rho_kg_m3, u_mean_m_s, D_m, mu_Pa_s }) {
  return rho_kg_m3 * u_mean_m_s * D_m / mu_Pa_s;
}

// Example: capillary
const cap = { R_m: 4e-6, L_m: 5e-4, dP_Pa: 266, mu_Pa_s: 1.2e-3 };
console.log('u_max:', maxVelocity(cap).toFixed(4), 'm/s');   // ~0.00177 m/s = 1.77 mm/s
console.log('Q:',     poiseuilleFlow(cap).toExponential(2), 'm^3/s');
console.log('Re:',    reynolds({
  rho_kg_m3: 1060, u_mean_m_s: meanVelocity(cap), D_m: 2 * cap.R_m, mu_Pa_s: cap.mu_Pa_s
}).toFixed(4));
// ~0.003 -- deeply laminar

// Verifying r^4 dependence: double the radius
const doubled = { ...cap, R_m: cap.R_m * 2 };
console.log('Q ratio (2R/R):', poiseuilleFlow(doubled) / poiseuilleFlow(cap));
// 16.0  -- the famous r^4 result

// Pressure drop needed to push 1 mL/min through a 1 mm pipe of length 1 m, water at 20C
const water = { mu_Pa_s: 1.002e-3, rho_kg_m3: 998 };
function dPForQ({ R_m, L_m, Q_m3_s, mu_Pa_s }) {
  return 8 * mu_Pa_s * L_m * Q_m3_s / (Math.PI * Math.pow(R_m, 4));
}
const Q = 1e-6 / 60;  // 1 mL/min -> m^3/s
console.log('dP:', dPForQ({ R_m: 0.5e-3, L_m: 1, Q_m3_s: Q, mu_Pa_s: water.mu_Pa_s }).toFixed(1), 'Pa');
// ~680 Pa (~ 7 cm of water)

Where Poiseuille flow matters

• Cardiovascular physiology. Arteriolar resistance regulation via smooth-muscle constriction; r⁴ dependence is the heart of blood pressure control.
• Microfluidics. Lab-on-a-chip designs operate at Re ~ 1; Poiseuille flow is the workhorse.
• Oil and gas pipelines. Initial design uses laminar formulas; corrected with turbulent friction factors for high-Re cases.
• IV drips. Drip rate set by viscosity, tubing radius, height differential -- straight Poiseuille.
• Lubrication. Oil in narrow gaps between machine parts -- Poiseuille analog.
• Capillary viscometers. Measure fluid viscosity by timing volume through a calibrated capillary at known ΔP.
• Permeability and Darcy's law. Flow through porous media is bundled Poiseuille flow in micro-channels.

Common mistakes

• Applying Poiseuille at high Re. Above Re ≈ 2300 you need turbulent flow formulas (Darcy-Weisbach + Moody chart).
• Forgetting the no-slip boundary. Velocity at the wall is zero. Without no-slip the profile would be flat -- the parabola comes from boundary friction.
• Assuming the mean equals the max. ū = u_max / 2 for parabolic profile. Get this wrong and flow estimates are off by a factor of 2.
• Using it for blood through capillaries naively. Whole blood is non-Newtonian; red cells (8 μm) are similar to capillary diameter. Poiseuille is approximate.
• Ignoring entrance length. In short pipes, the developing flow region (L_e ≈ 0.06 Re D) is a significant fraction of total length and the pressure drop exceeds the Poiseuille prediction.
• Confusing R (radius) and D (diameter). Q ∝ R⁴ means Q ∝ D⁴/16. Easy to drop the factor when reading literature.