Young-Laplace equation

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In fluid dynamics, the Young–Laplace equation describes the pressure difference over a meniscus between two fluids,

$\Delta p = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2}\right)$

where $Δ p$ is the pressure difference over the interface, $γ$ the surface tension, and $R 1$ and $R 2$ are the principal radii of curvature at the interface.

For a spherical meniscus of radius $R$ , the equation simplifies to

$\Delta p = \gamma \frac{2}{R}.$

[edit] Application in medicine

In medicine it is often referred to as the Law of Laplace, and it is used in the context of alveoli in the lung, whereby a single alveolus is modeled as being a perfect sphere.^[1]

The Young-Laplace equation implies that a larger surface tension involves a larger force pushing inwards into the object. Also, because the relationship of $Δ p$ and r is inverse, a smaller radius would result in a larger inward force. When this is applied to the context of pulmonary systems, if two alveoli are connected to the same airway, it would be expected that the smaller alveoli would experience a greater pressure, driving its collapse, and pushing all of its gas contents into other, larger, alveoli. This however does not happen, the reason being that pulmonary surfactant, a mixture secreted in the alvoeli that decreases surface tension, has a greater effect on smaller alveoli than larger ones. Therefore while an alveoli's small radius would on its own increase pressure, this increase is offset by the extra reduction in surface tension that is the result of pulmonary surfactant function.^[1]

The Laplace pressure also causes the process of emulsification to be more thermodynamically inefficient. To form the small, highly curved droplets of an emulsion extra energy is required to overcome the large pressure that results.

[edit] History

Thomas Young laid the foundations of this equation in his 1804 paper An Essay on the Cohesion of Fluids ^[2] where he set out in descriptive terms the principles governing contact between fluids (along with many other aspects of fluid behaviour). Pierre Simon Laplace followed this up in Mécanique Céleste ^[3] with the formal mathematical description given above, which reproduced in symbolic terms the relationship described earlier by Young.

[edit] See also

[edit] References

^ ^a ^b Sherwood, Lauralee. "Ch13", in Peter Adams: Human physiology from cells to systems, 6th, Thomson Brooks/Cole. ISBN 0-495-01485-0.
^ Phil. Trans., 1805, p. 65
^ Mécanique céleste, Supplement to the tenth edition, pub. in 1806