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, Δp^{*} the preasure drop at the two ends, η dynamic fluid viscosity and l characteristic length along z, a linear dimension in a crosssection (in noncylindrical tube). The law can be derived from the DarcyWeisbach 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 DarcyWeisbach equation should be properly named in full as the ChézyWeisbachDarcyPoiseuilleHagenReynoldsFanningPrandtlBlasiusvon KármánNikuradseColebrookWhiteRouseMoody equation or CWDPHRFPBKNCWRM equation in 'short'.
Poiseuille's law corresponds to the Ohm's law for electrical circuits, where pressure drop Δp^{*} is somehow replaced by voltage V and voluminal flow rate Φ_{V} by current I. According to this a term 8η l/πr^{4} is an adequate substitution for the electrical resistance R.
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