Jean Léonard Marie Poiseuille

From 1815 to 1816 he studied at the École Polytechnique in Paris. He was trained in physics and mathematics.[5] In 1828 he earned his D.Sc. degree with a dissertation entitled Recherches sur la force du coeur aortique (The force of the aortic heart). He was interested in the flow of human blood in narrow tubes, and invented the U-tube mercury manometer (or hemodynamometer) to measure arterial blood pressures in horses and dogs.[4]

In 1838 he experimentally derived, and in 1840 and 1846 formulated and published, Poiseuille's law (now commonly known as the Hagen–Poiseuille equation, crediting Gotthilf Hagen as well), which applies to laminar flow, that is, non-turbulent flow of liquids through pipes of uniform section, such as blood flow in capillaries and veins.[4][6] His original formulation for water of 1846[7] little resembles the present-day formulation and is given as:

${\displaystyle {\dot {m}}=\left(135.282\mathrm {\frac {mg}{mm^{3}\;s\;mmH_{2}O}} \right){\frac {\Delta Pd^{4}}{L}}\left(1+{\frac {3.36793\times 10^{-2}}{^{\circ }\mathrm {C} }}\,T+{\frac {2.209936\times 10^{-4}}{^{\circ }\mathrm {C} ^{2}}}\,T^{2}\right)}$

However it can be transformed in a more amenable form. Rewritting in more modern fashion using SI units gives:

${\displaystyle {\dot {m}}=\left(13\,795\mathrm {\frac {kg}{m^{3}\;Pa\;s}} \right){\frac {\Delta Pd^{4}}{L}}\left(1+{\frac {T}{26.6918\,^{\circ }\mathrm {C} }}+{\frac {T^{2}}{4525.02^{\circ }\mathrm {C} ^{2}}}\right)}$

Using the density of water as ${\displaystyle \mathrm {\rho ={\frac {999\,kg}{m^{3}}}} }$ and ${\displaystyle {\dot {m}}=\rho {\overline {V}}{\frac {\pi d^{2}}{4}}}$ and then solving for the pressure difference results in:

${\displaystyle \Delta P=\left({\frac {5.68769\times 10^{-2}\mathrm {Pa\,s} }{1+{\frac {T}{26.6918\,^{\circ }\mathrm {C} }}+{\frac {T^{2}}{4525.02^{\circ }\mathrm {C} ^{2}}}}}\right){\frac {L{\overline {V}}}{d^{2}}}}$

The term in parentheses, the constant and the temperature correction, are a function of viscosity. Finally using the viscosity of water at ${\displaystyle T=0^{\circ }\mathrm {C} }$, ${\displaystyle \mu =1.789548\times 10^{-3}\,\mathrm {Pa\,s} }$, allows for viscosity of different fluids to be taken into account resulting in:

${\displaystyle \Delta P=\left({\frac {5.68769\times 10^{-2}\mathrm {Pa\,s} }{1.789548\times 10^{-3}\,\mathrm {Pa\,s} }}\right){\frac {\mu L{\overline {V}}}{d^{2}}}=31.78{\frac {\mu L{\overline {V}}}{d^{2}}}}$

The equation in standard fluid dynamics notation is[8][9]

${\displaystyle \Delta P={\frac {8\mu LQ}{\pi r^{4}}},}$

or

${\displaystyle \Delta P={\frac {128\mu LQ}{\pi d^{4}}},}$

or

${\displaystyle \Delta P={\frac {32\mu L{\overline {V}}}{d^{2}}},}$

where:

${\displaystyle \Delta P}$ is the pressure loss,

${\displaystyle L}$ is the length of pipe,

${\displaystyle \mu }$ is the dynamic viscosity,

${\displaystyle Q}$ is the volumetric flow rate,

${\displaystyle r}$ is the radius,

${\displaystyle d}$ is the diameter,

${\displaystyle \pi }$ is the mathematical constant π,

${\displaystyle {\overline {V}}}$ is the average velocity.

The poise, the unit of viscosity in the CGS system, was named after him. Attempts to introduce "poiseuille" as the name of the SI unit Pa·s were unsuccessful.

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