Poiseuille's Law
Calculate volumetric flow rate and flow resistance for laminar pipe flow using the Hagen-Poiseuille equation. Free online fluid dynamics calculator with interactive charts and common fluid presets.
About This Calculator
Poiseuille's Law (also known as the Hagen-Poiseuille equation) is a fundamental principle in fluid dynamics that describes the laminar flow of a viscous fluid through a long cylindrical pipe of constant cross-section. This calculator helps engineers, physicists, medical researchers, and students quickly compute the volumetric flow rate and flow resistance for any Newtonian fluid in laminar flow conditions.
The governing equation is Q = π·ΔP·r⁴ / (8·μ·L), where Q is the volumetric flow rate, ΔP is the pressure drop across the pipe, r is the pipe radius, μ is the dynamic viscosity of the fluid, and L is the pipe length. Flow resistance is given by R = 8·μ·L / (π·r⁴). The fourth-power dependence on radius means that small changes in pipe diameter produce dramatic changes in flow rate — a fact that is critical in fields ranging from cardiovascular physiology to industrial pipe design.
To use the calculator, select a fluid preset from the dropdown menu (water, blood, air, olive oil, honey, ethanol, glycerin, or mercury) or choose Custom Fluid to enter any dynamic viscosity value. Then enter the pipe radius, length, and pressure drop in SI units. The calculator instantly computes the flow rate in multiple units (m³/s, L/s, mL/min) and the flow resistance, with an interactive breakdown table showing all intermediate values.
Applications of Poiseuille's Law
Biomedical Engineering: Poiseuille's equation is widely used to model blood flow in blood vessels, airflow in respiratory airways, and fluid transport in dialysis machines. The strong radius dependence explains why vasoconstriction dramatically reduces blood flow.
Industrial Piping: Engineers use the law to design piping systems for transporting oils, chemicals, and other viscous fluids. It helps determine the pressure required to achieve a desired flow rate through a given pipe system.
Microfluidics: In lab-on-a-chip devices, Poiseuille flow governs the movement of fluids through microscopic channels, where laminar flow is the dominant regime.
HVAC: Airflow resistance in ducts and ventilation systems can be modeled using this equation, particularly for low-velocity laminar flow conditions.
Regional Notes
India: SI units are standard in Indian engineering and medical education. The calculator uses Pa, m, and Pa·s for all calculations.
United States: While US engineers often use imperial units, this calculator uses SI. To convert: 1 Pa = 0.000145 psi, 1 m = 3.281 ft. The viscosity of water at 20 °C is 1.002 × 10⁻³ Pa·s.
United Kingdom: SI units are the standard in UK engineering and biomedical research. The calculator follows the same convention used in UK university physics and engineering curricula.
Frequently Asked Questions
What is Poiseuille's Law?
Poiseuille's Law (also called the Hagen-Poiseuille equation) describes laminar flow of a viscous fluid through a long cylindrical pipe. The equation Q = π·ΔP·r⁴/(8·μ·L) relates volumetric flow rate (Q) to pressure drop (ΔP), pipe radius (r), fluid viscosity (μ), and pipe length (L).
What units does Poiseuille's Law calculator use?
The calculator uses SI units: dynamic viscosity in Pa·s, pipe radius in meters, pipe length in meters, pressure drop in pascals (Pa). Flow rate results are displayed in m³/s, L/s, and mL/min. Flow resistance is shown in Pa·s/m³.
When should I use Poiseuille's equation?
Use Poiseuille's equation for laminar flow (Re < 2100) in long straight cylindrical pipes of constant cross-section. Common applications include blood flow in vessels, airflow in airways, water flow in small-diameter pipes, and fluid transport in microfluidics and industrial piping systems.
Why does flow rate depend on radius to the fourth power?
In Poiseuille's equation, flow rate Q is proportional to r⁴ because the radius affects both the cross-sectional area (r²) and the velocity profile (r²). Doubling the radius increases flow rate by a factor of 16, which is why small changes in blood vessel diameter dramatically affect blood flow.
Can Poiseuille's Law be used for blood flow?
Yes, Poiseuille's Law is commonly applied to estimate blood flow in blood vessels as a first approximation. Blood exhibits non-Newtonian behavior, but the equation provides useful insights into how vessel diameter, length, and pressure changes affect blood flow. Select the Blood (37 °C) preset for blood flow calculations.
What is the difference between flow rate and flow resistance?
Flow rate (Q) is the volume of fluid passing through the pipe per unit time, measured in m³/s. Flow resistance (R = 8·μ·L/π·r⁴) is a measure of how difficult it is for fluid to flow through the pipe, measured in Pa·s/m³. Higher resistance means lower flow rate for the same pressure difference.
Is Poiseuille's Law valid for turbulent flow?
No, Poiseuille's Law only applies to laminar flow (Reynolds number below approximately 2100). For turbulent flow, the Darcy-Weisbach equation is more appropriate. The calculator assumes laminar flow conditions, so verify your Reynolds number is in the laminar regime for accurate results.
How do I select the correct fluid preset?
Choose a fluid preset that matches your working fluid and temperature. Water presets are available at 20 °C, 40 °C, and 100 °C. For blood flow applications, select Blood (37 °C). For air flow in ventilation systems, select Air (15 °C) or Air (25 °C). Use Custom Fluid for any other fluid and enter its dynamic viscosity manually.