Poiseuille Flow Calculator (Hagen-Poiseuille)

Flow rate, velocity profile, and Re check for fully developed laminar flow in a circular pipe.

Fluid Mechanics II

Common fluids — click to auto-fill density and viscosity

Inputs

Pipe diameter D (mm)

Pipe length L (m)

Pressure difference ΔP (Pa)

Dynamic viscosity μ (cP)

1 cP = 1 mPa·s = 0.001 Pa·s

Fluid density ρ (kg/m³)

Used only to verify Re < 2 300 (laminar condition).

Theory

Hagen-Poiseuille equation:

Q = (π × r⁴ × ΔP) / (8 × μ × L) [m³/s]

Parabolic velocity profile:

v(r) = (ΔP / 4μL) × (R² - r²)

V_max = ΔP × R² / (4μL) [at centreline, r = 0]

V_mean = V_max / 2 = Q / (πR²)

Validity condition:

Re = ρ × V_mean × D / μ < 2 300

Above Re ≈ 2 300 the flow transitions to turbulent and the parabolic profile breaks down.

Where:

Q = volume flow rate [m³/s]
r = pipe radius = D/2 [m]
R = pipe radius (used in profile equation) [m]
ΔP = pressure difference along length L [Pa]
μ = dynamic viscosity [Pa·s]
L = pipe length [m]
ρ = fluid density [kg/m³]

Flow rate scales with the fourth power of radius — halving the pipe diameter reduces Q by a factor of 16. This makes diameter the most sensitive parameter in Poiseuille flow.