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 ΦV 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:

[itex] \Phi_{V} = {dV\over dt} = v_{s}\pi r^{2} = {\pi r^{4}\over 8 \eta} \left( - { d p^{\star} \over dz}\right) = {\pi r^{4}\over 8 \eta} { \Delta p^{\star} \over \ell} \; , [itex]

where V is a volume of the liquid, poured in the time unit t, vs 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 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:

[itex] \Lambda = {64\over {\it Re}} \; , \quad\quad Re = {2\rho v_{s} r\over \eta} \; , [itex]

where Re is the Reynolds number and ρ fluid density. In this form the law approximates the Darcy 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.

Curiosity

The law itself shows what an interesting field this is, as the Darcy-Weisbach equation was studied by a number of other scientists, including Chézy, Weisbach, Darcy, Poiseuille, Hagen, Reynolds, Fanning, Prandtl, Blasius, von Kármán, Nikuradse, Colebrook, White, Rouse, and Moody.

Also note in the formula how strongly the flow depends on the radius. If all else is held constant, a doubling of the radius of the channel results in a sixteen-fold increase in the flow.

Relation to electrical circuit

Electricity was originally understood to be a kind of fluid. This hydraulic analogy is still of some use in teaching.

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 ΦV by current I. According to this a term 8η lr4 is an adequate substitution for the electrical resistance R.

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