Poiseuille's Law Calculator

Poiseuille's Law (also called the Hagen-Poiseuille equation) describes the volumetric flow rate of a viscous, incompressible fluid through a long, straight, cylindrical pipe under laminar flow conditions. The formula is Q = (π·ΔP·r⁴) / (8·η·L), where Q is flow rate, ΔP is pressure drop, r is pipe radius, η is dynamic viscosity, and L is pipe length.

Because flow rate (Q) is proportional to the fourth power of the radius (r⁴). This means doubling the pipe radius increases flow rate by a factor of 16 (2⁴). Even small changes in radius dramatically affect how much fluid can pass through a pipe, which is why this relationship is critical in medical applications like blood vessel constriction.

Poiseuille's Law applies only under specific conditions: the flow must be laminar (Reynolds number < 2100), the fluid must be Newtonian and incompressible, the pipe must be straight, rigid, and cylindrical with a constant circular cross-section, and the flow must be fully developed (steady-state). It does not apply to turbulent flow or non-Newtonian fluids like blood at high shear rates.

Flow resistance (R) is calculated as R = 8·η·L / (π·r⁴). It is the ratio of pressure drop to flow rate (R = ΔP / Q), analogous to electrical resistance in Ohm's Law. Higher viscosity, longer pipes, and smaller radii all increase flow resistance, reducing the flow rate for a given pressure difference.

The Reynolds number (Re) is a dimensionless quantity that predicts whether flow is laminar or turbulent. It is calculated as Re = (2·ρ·v·r) / η, where ρ is fluid density and v is average velocity. For Poiseuille's Law to be valid, Re must be less than approximately 2100 (laminar regime). Values above 4000 indicate turbulent flow, and Poiseuille's equation no longer applies.

Poiseuille's Law is widely used in biomedical engineering to model blood flow in capillaries and arteries, in hydraulic system design, in microfluidics, in IV drip rate calculations, and in designing piping networks for water, oil, or chemical transport. It helps engineers select appropriate pipe diameters and predict pressure losses in fluid systems.

Common dynamic viscosity values include: water at 20°C ≈ 0.001 Pa·s (1 mPa·s), blood ≈ 0.003–0.004 Pa·s, air at 20°C ≈ 0.0000181 Pa·s, glycerin ≈ 1.5 Pa·s, and honey ≈ 2–10 Pa·s. Temperature significantly affects viscosity — most liquids become less viscous as temperature increases.

Yes, Poiseuille's Law can be applied to gases under certain conditions — specifically when the flow is laminar, the pressure changes are small relative to absolute pressure (so the gas behaves as incompressible), and the Mach number is very low. In pulmonary physiology, it is used to model airflow in small airways of the lungs where these conditions are approximately met.