Young–Laplace equation

In physics, the Young–Laplace equation is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension. It relates the pressure difference to the shape of the surface and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness):

:

where $Delta\; p$ is the pressure difference across the fluid interface, $gamma$ is the surface tension, $hat\; n$ is a unit normal to the surface, $H$ is the mean curvature, and $R\_1$ and $R\_2$ are the principal radii of curvature. Note that only normal stress is considered, this is because it can be shown [ [http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/ Surface Tension Module] , by John W. M. Bush, at MIT OCW.] that a static interface is possible only in the absence of tangential stress.

The equation is named after Thomas Young, who developed the qualitative theory of surface tension in 1805, and Pierre-Simon Laplace who completed the mathematical description in the following year. It is sometimes also called the Young–Laplace–Gauss equation, as Gauss unified the work of Young and Laplace in 1830, deriving both the differential equation and boundary conditions using Johann Bernoulli's virtual work principles [cite paper
author = Robert Finn
title = Capillary Surface Interfaces
version =
publisher = AMS
date= 1999
url = http://www.ams.org/notices/199907/fea-finn.pdf
format =
accessdate = ] .

oap films

If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface.

Emulsions

The equation also explains the energy required to create an emulsion. To form the small, highly curved droplets of an emulsion, extra energy is required to overcome the large pressure that results from their small radius.

Capillary pressure in a tube

Axisymmetric equations

The (nondimensional) shape, "r"("z") of an axisymmetric surface can be found by substituting general expressions for curvature to give the hydrostatic Young-Laplace equations [Lamb, H. Statics, Including Hydrostatics and the Elements of the Theory of Elasticity, 3rd ed. Cambridge, England: Cambridge University Press, 1928.] :

:$frac\{r"\}\{(1+r\text{'}^2)^\{frac\{3\}\{2\}\; -\; frac\{1\}\{r(z)\; sqrt\{1+r\text{'}^2\}\; \}\; =\; z\; -\; Delta\; p^*$:$frac\{z"\}\{(1+z\text{'}^2)^\{frac\{3\}\{2\}\; +\; frac\{z\text{'}\}\{r\; sqrt\{1+z\text{'}^2\}\; \}\; =\; Delta\; p^*\; -\; z(r).$

Application in medicine

In medicine it is often referred to as the Law of Laplace, and it is used in the context of respiratory physiology, in particular alveoli in the lung, where a single alveolus is modeled as being a perfect sphere.cite book
last = Sherwood
first = Lauralee
coauthors =
editor = Peter Adams
title = Human physiology from cells to systems
edition = 6th
year =
publisher = Thomson Brooks/Cole
location =
id =ISBN 0-495-01485-0
doi = | pages =
chapter = Ch13 ]

In this context, the pressure differential is a force pushing inwards on the surface of the alveolus. The Law of Laplace states that there is an inverse relationship between surface tension and alveolar radius. It follows from this that a small alveolus will experience a greater inward force than a large alveolus, if their surface tensions are equal. In that case, if both alveoli are connected to the same airway, the small alveolus will be more likely to collapse, expelling its contents into the large alveolus.

This explains why the presence of surfactant lining the alveoli is of vital importance. Surfactant reduces the surface tension on all alveoli, but its effect is greater on small alveoli than on large alveoli. Thus, surfactant compensates for the size differences between alveoli, and ensures that smaller alveoli do not collapse.

The Law of Laplace also explains various phenomena encountered in the pathology of vascular or gastrointestinal walls. The "surface tension" in this case represents the muscular tension on the wall of the vessel. For example, if an aneurysm forms in a blood vessel wall, the radius of the vessel has increased. This means that the inward force on the vessel decreases, and therefore the aneurysm will continue to expand until it ruptures. A similar logic applies to the formation of diverticuli in the gut. [E. Goljan, "Pathology, 2nd ed." Mosby Elsevier, Rapid Review Series.]

History

Francis Hauksbee performed some of the earliest observations and experiments in 1709 and these were repeated in 1718 by James Jurin who observed that the height of fluid in a capillary column was a function only of the cross-sectional area at the surface, not of any other dimensions of the column. [Anon.] (1911) [http://www.1911encyclopedia.org/Capillary_action Capillary action] , "Encyclopedia Britannica"]

Thomas Young laid the foundations of the equation in his 1804 paper "An Essay on the Cohesion of Fluids" [Phil. Trans., 1805, p. 65] 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" [Mécanique céleste, Supplement to the tenth edition, pub. in 1806] with the formal mathematical description given above, which reproduced in symbolic terms the relationship described earlier by Young.

Laplace accepted the idea propounded by Hauksbee in the "Philosophical Transactions" for 1709, that the phenomenon was due to a force of attraction that was insensible at sensible distances. The part which deals with the action of a solid on a liquid and the mutual action of two liquids was not worked out thoroughly, but ultimately was completed by Gauss. Carl Neumann later filled in a few details. [*Rouse Ball, W. W. [1908] (2003) " [http://www.maths.tcd.ie/pub/HistMath/People/Laplace/RouseBall/RB_Laplace.html Pierre Simon Laplace (1749 - 1827)] ", in "A Short Account of the History of Mathematics", 4th ed., Dover, ISBN 0486206300]

References

Bibliography

* [Anon.] (1911) [http://www.1911encyclopedia.org/Capillary_action Capillary action] , "Encyclopedia Britannica"
*Batchelor, G. K. (1967) "An Introduction To Fluid Dynamics", Cambridge University Press
* cite journal | title=PDF| [http://www.journals.royalsoc.ac.uk/content/y5217r3860506457/fulltext.pdf An account of some experiments shown before the Royal Society; with an enquiry into the cause of the ascent and suspension of water in capillary tubes] |1.11 MiB | authorlink=James Jurin | author=Jurin, J. | journal=Philosophical Transactions of the Royal Society | pages=739–747 | volume=30 | year=1717/1719
*Tadros T. F. (1995) "Surfactants in Agrochemicals", Surfactant Science series, vol.54, Dekker