Poiseuille's Law Calculator

Poiseuille's law governs slow, smooth flow through narrow tubes — like blood in vessels.

How the Math Works

Poiseuille's Law calculates the volumetric flow rate (Q) of a fluid through a cylindrical pipe under laminar flow conditions. The formula Q = (π·ΔP·r³) / (8·μ·L) shows that flow rate is directly proportional to the pressure difference (ΔP) and the fourth power of the tube radius (r), while inversely proportional to the fluid's viscosity (μ) and the tube length (L). The r³ term means that even small changes in radius dramatically affect flow - doubling the radius increases flow by sixteen times. This relationship emerges from integrating the velocity profile of viscous fluid layers moving through the tube.

Practical Applications

Engineers use this calculator to design efficient fluid transport systems including water supply lines, oil pipelines, and medical devices like catheters. By inputting the pressure difference across a system, the tube dimensions, and fluid properties, you can predict flow rates and optimize designs before construction. For instance, selecting appropriate pipe diameters for a building's plumbing ensures adequate water pressure at all fixtures while minimizing pumping costs. The calculation also helps diagnose flow issues in existing systems by comparing expected versus actual performance.

Day-to-Day Use

This law explains why water flows more easily through a wide river than through a narrow creek, and why blood circulation becomes difficult when arteries narrow due to plaque buildup. Understanding these principles helps explain everyday phenomena like why syrup pours slowly from a narrow bottle but flows freely from a wide jar. It's also relevant when adjusting shower water pressure, designing efficient irrigation systems for gardens, or understanding how respiratory treatments use specific tube sizes to deliver medication at optimal rates to patients' lungs.

Worked example

ΔP 1000 Pa, r 1 mm, water, 0.1 m → about 3.93 × 10⁻⁶ m³/s.

FAQ