# Poiseuille's law

The **Poiseuille's law** (or the **Hagen-Poiseuille law** also named after Gotthilf Heinrich Ludwig Hagen (1797-1884) for his experiments in 1839) is the physical law concerning the voluminal laminar stationary flow *Φ*_{V} of incompressible uniform viscous liquid (so called Newtonian fluid) through a cylindrical tube with the constant circular cross-section, experimentally derived in 1838, formulated and published in 1840 and 1846 by Jean Louis Marie Poiseuille (1797-1869), and defined by:

where

*V* is a volume of the liquid, poured in the time unit

*t*,

*v*_{s} median fluid

velocity along the axial

cylindrical coordinate *z*,

*r* internal radius of the tube, Δ

*p*^{*} the preasure drop at the two ends, η dynamic fluid viscosity and

*l* characteristic length along

*z*, a linear dimension in a cross-section (in non-cylindrical tube). The law can be derived from the

Darcy-Weisbach equation, developed in the field of

hydraulics and which is otherwise valid for all types of flow, and also expressed in the form:

where

*Re* is the

Reynolds number and ρ fluid density. In this form the law approximates the

*friction factor*, the

*energy (head) loss factor*,

*friction loss factor* or

*Darcy (friction) factor* Λ in the laminar flow at very low velocities in cylindrical tube. The theoretical derivation of slightly different Poiseuille's original form of the law was made independently by Wiedman in

1856 and Neumann and E. Hagenbach in

1858 (

1859,

1860). Hagenbach was the first who called this law the Poiseuille's law.

The law is also very important specially in hemorheology and hemodynamics, both fields of physiology.

The Poiseuilles' law was later in 1891 extended to turbulent flow by L. R. Wilberforce, based on Hagenbach's work.

The law itself shows how an interesting field this is, because the Darcy-Weisbach equation should be properly named in full as the Chézy-Weisbach-Darcy-Poiseuille-Hagen-Reynolds-Fanning-Prandtl-Blasius-von Kármán-Nikuradse-Colebrook-White-Rouse-Moody equation or CWDPHRFPBKNCWRM equation in 'short'.

Poiseuille's law corresponds to the Ohm's law for electrical circuits, where pressure drop Δ*p*^{*} is somehow replaced by voltage *V* and voluminal flow rate Φ_{V} by current *I*. According to this a term 8η *l*/π*r*^{4} is an adequate substitution for the electrical resistance *R*.