Hagen–Poiseuille equation
Law for pressure drop in laminar pipe flow.
The Hagen–Poiseuille equation, also known as the Poiseuille law, governs the pressure drop experienced by an incompressible, Newtonian fluid moving in laminar flow through a long cylindrical pipe with a constant cross-section. This law was derived experimentally by Jean Léonard Marie Poiseuille in 1838 and independently by Gotthilf Heinrich Ludwig Hagen, with Hagen publishing his findings in 1839 and Poiseuille in 1840–41 and 1846. A theoretical justification was later provided by George Stokes in 1845. The equation assumes the fluid is incompressible and Newtonian, the flow is laminar and fully developed through a pipe substantially longer than its diameter, and there is no acceleration of the fluid within the pipe. The pressure drop described is solely due to the fluid’s viscosity; other pressure drops, such as those from gravitational effects or changes in velocity (as described by Bernoulli’s equation), can occur simultaneously. For example, when blood flows into a narrower constriction, its speed increases and pressure decreases due to Bernoulli’s principle, while viscosity causes an additional pressure drop along the flow path proportional to the distance traveled. The equation is valid only for laminar flow; above a certain velocity or pipe diameter, turbulent flow occurs, leading to larger pressure drops than predicted. It also fails near the pipe entrance, for very low viscosity, or for wide or short pipes. The ratio of pipe length to radius must be greater than one-forty-eighth of the Reynolds number for the law to hold. If the pipe is too short, the equation can predict unphysically high flow rates, as flow is ultimately bounded by Bernoulli’s principle, which prevents negative absolute pressure. The law is fundamental in hemorheology and hemodynamics, and it was extended to turbulent flow by L. R. Wilberforce in 1891, building on earlier work by Hagenbach.
- field
- Fluid dynamics
- known_for
- Hagen–Poiseuille equation (pressure drop in laminar pipe flow)
- key_contributors
- Jean Léonard Marie Poiseuille, Gotthilf Heinrich Ludwig Hagen, George Stokes
Lore & Background
The Hagen–Poiseuille equation describes the pressure drop experienced by an incompressible, Newtonian fluid moving in laminar flow through a long cylindrical pipe with a constant cross-section. The flow must be steady, axisymmetric, and fully developed, with no radial or azimuthal velocity components. The pipe’s length must be substantially greater than its diameter, and the fluid must not accelerate. The equation fails near the pipe entrance, and it becomes invalid for low-viscosity fluids, wide pipes, or short pipes, where turbulent flow or unphysically high flow rates may occur; in such cases, the Darcy–Weisbach equation or Bernoulli’s principle provides more accurate modeling. The law was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, with Hagen publishing in 1839 and Poiseuille in 1840–41 and 1846. George Stokes provided the theoretical justification in 1845. Later, Wiedman in 1856 and Neumann and E. Hagenbach in 1858–1860 derived a slightly different form; Hagenbach first called it Poiseuille’s law. In 1891, L. R. Wilberforce extended the law to turbulent flow based on Hagenbach’s work. The equation is important in hemorheology and hemodynamics. For laminar flow in a circular pipe, the friction factor is inversely proportional to the Reynolds number, relating wall stress to pressure drop. The full solution yields a parabolic velocity profile. The pressure drop is proportional to pipe length and dynamic viscosity, and inversely proportional to the fourth power of the radius. The equation applies to air flow in lung airways, flow through a drinking straw, or through a hypodermic needle.
Reader's Guide
The Hagen–Poiseuille equation is significant because it provides a fundamental relationship for pressure drop in laminar flow through pipes, with applications ranging from airflow in lung airways to flow through drinking straws and hypodermic needles. It is derived from the Navier–Stokes equations under assumptions of steady, laminar flow of an incompressible Newtonian fluid through a long cylindrical pipe. The equation fails for turbulent flow, low viscosity, wide or short pipes, and near the pipe entrance. It relates to the Darcy–Weisbach equation through the friction factor, where for laminar flow the Darcy friction factor Λ equals 64/Re. The law is important in hemorheology and hemodynamics, fields of physiology. Its legacy includes providing the basis for understanding viscous flow in pipes and influencing later work on turbulent flow.
Did You Know?
- The law can be applied to air flow in the airways of the lungs, flow through a drinking straw, or through a hypodermic needle.
- For laminar flow in a circular pipe, the Darcy friction factor Λ equals 64 divided by the Reynolds number.
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