Poiseuille's Law

The Poiseuille law (sometimes the Hagen-Poiseuille law or, in another transcription, the Hagen-Poiseuille law ) is the physical law of hydrodynamics for the so-called Poiseuille flow , that is, the steady-state flow of a viscous, in the particular case incompressible, fluid in a thin cylindrical tube. It connects the flow rate of the liquid through the cross section of the pipe with a pressure drop at its ends for a given viscosity of the liquid and the geometric dimensions of the tube.

The law was empirically established in 1839 by G. Hagen , and in 1840-1841, independently by J. L. Poiseuille . Theoretically explained by J. G. Stokes in 1845.

Content

1 Wording of the law
2 Poiseuille's law for the flow of a compressible fluid in a pipe
3 Variations and generalization
4 notes
5 Literature
6 References

Law wording

With a steady laminar flow of a viscous incompressible fluid through a long (that is, with a pipe length many times greater than its diameter) straight cylindrical pipe ( capillary ) of circular cross section, the volumetric flow rate of the liquid is directly proportional to the pressure drop per unit length of the pipe and the fourth degree of radius and inversely proportional to the viscosity coefficient of the liquid .

Q={\frac {\pi R^{4}}{8\eta l}}(p_{1}-p_{2})={\frac {\pi d^{4}}{128\eta l}}\Delta p,

{\ displaystyle Q = {\ frac {\ pi R ^ {4}} {8 \ eta l}} (p_ {1} -p_ {2}) = {\ frac {\ pi d ^ {4}} {128 \ eta l}} \ Delta p,}

Where

$p_{1}-p_{2}=\Delta p$ ${\ displaystyle p_ {1} -p_ {2} = \ Delta p}$ - pressure drop at the ends of the pipe, Pa;
$Q$ ${\ displaystyle Q}$ - volumetric flow rate of the liquid, m³ / s;
$R$ ${\ displaystyle R}$ - radius of the pipe, m;
$d$ ${\ displaystyle d}$ - pipe diameter , m;
$\eta$ ${\ displaystyle \ eta}$ - coefficient of dynamic viscosity, Pa · s;
$l$ ${\ displaystyle l}$ - pipe length , m.

The formula is valid, firstly, if the fluid flow is laminar, and, secondly, the laminar flow is steady, the velocity profile in which is described by the Poiseuille flow, when the influence of the ends of the pipe can be neglected.

The phenomenon described by the formula is sometimes used to experimentally determine the viscosity of liquids. Another way to determine the viscosity of a liquid is by using the Stokes law .

Poiseuille's law for the flow of a compressible fluid in a pipe

For a compressible fluid in a pipe (gas), the volumetric flow rate and linear velocity are not constant along the pipe; at high pressures, the velocity and volumetric flow rate are lower with a constant gas flow rate reduced to normal conditions . Since the gas expands during the flow, in the general case the gas temperature changes along the pipe, i.e. the process is non-isothermal .

This means that the flow rate depends not only on the pressure in a given section of the pipe, but also on the gas temperature.

For an ideal gas in the isothermal case, when the gas temperature due to heat exchange with the pipe wall manages to equal the wall temperature and when the pressure difference between the pipe ends is small relative to the average pressure along the pipe, the volumetric flow rate at the pipe outlet is determined by the expression:

Q={\frac {\pi R^{4}}{16\eta l}}\left({\frac {P_{\mathrm {i} }^{2}-P_{\mathrm {o} }^{2}}{P_{\mathrm {o} }}}\right),

{\ displaystyle Q = {\ frac {\ pi R ^ {4}} {16 \ eta l}} \ left ({\ frac {P _ {\ mathrm {i}} ^ {2} -P _ {\ mathrm {o }} ^ {2}} {P _ {\ mathrm {o}}}} right),}

Where

P_{\mathrm {i} }

{\ displaystyle P _ {\ mathrm {i}}}

- inlet pressure, Pa;

P_{\mathrm {o} }

{\ displaystyle P _ {\ mathrm {o}}}

- outlet pressure, Pa;

l

{\ displaystyle l}

- pipe length, m;

\eta

{\ displaystyle \ eta}

- dynamic viscosity, Pa · s;

R

{\ displaystyle R}

- radius, m;

Q

{\ displaystyle Q}

- volumetric gas flow at outlet pressure, m ³ / s.

This equation can be considered as the Poiseuille law with an additional coefficient for averaging the pressure along the pipe:

{\frac {P_{\mathrm {i} }+P_{\mathrm {o} }}{2P_{\mathrm {o} }}}.

{\ displaystyle {\ frac {P _ {\ mathrm {i}} + P _ {\ mathrm {o}}} {2P _ {\ mathrm {o}}}}.

Variations and generalization

There is a generalization of the formula of the Poiseuille law for a long pipe of elliptic section. From the formula for the pipe of elliptical cross section follows the formula of the Poiseuille law for the fluid flow between two parallel planes (in the limiting case, when the semimajor axis of the ellipse tends to infinity). In the reference literature, formulas are given for the profile of fluid flow rates and for fluid flow through a unit area ^[1] ^[2] .

Notes

↑ Ebert, 1963 .
↑ Jaworski, Detlaf, 1978 .

Literature

Vishnevetsky S. M. Poiseuille Law // Physical Encyclopedia : [in 5 volumes] / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia (t. 1-2); Great Russian Encyclopedia (t. 3-5), 1988-1999. - ISBN 5-85270-034-7 .
Sutera SP, Skalak R. The history of Poiseuille's law // Annual review of fluid mechanics. - 1993 .-- T. 25 . - S. 1–19 .
Ebert G. A Brief Guide to Physics: A Reference Edition / Ed. K.P. Yakovleva. - Per. from the 2nd of it. ed. N. M. Shikunina. - M .: Fizmatgiz , 1963 .-- 552 p.
Yavorsky B.M., Detlaf A.A. Handbook of Physics. - Science , 1978.- 944 p.

Links

Original publications of Hagen and Poiseuille

[_eb709dc7d0677a3f-1] Ebert, 1963 .

[_a7d561adbab6ce3d-2] Jaworski, Detlaf, 1978 .