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 or wall tension, although usage on the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall 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 is the pressure difference across the fluid interface, γ is the surface tension (or wall tension), is the unit normal pointing out of the surface, is the mean curvature, and and are the principal radii of curvature. (Some authors[who?] refer inappropriately to the factor as the total curvature.) Note that only normal stress is considered, this is because it can be shown 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.
If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface.
Note that this is not valid for a soap bubble, because its inner volume is enclosed and has a different pressure from the outside.
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
In a sufficiently narrow (i.e., low Bond number) tube of circular cross-section (radius a), the interface between two fluids forms a meniscus that is a portion of the surface of a sphere with radius R. The pressure jump across this surface is:
This may be shown by writing the Young–Laplace equation in spherical form with a contact angle boundary condition and also a prescribed height boundary condition at, say, the bottom of the meniscus. The solution is a portion of a sphere, and the solution will exist only for the pressure difference shown above. This is significant because there isn't another equation or law to specify the pressure difference; existence of solution for one specific value of the pressure difference prescribes it.
The radius of the sphere will be a function only of the contact angle, θ, which in turn depends on the exact properties of the fluids and the solids in which they are in contact:
so that the pressure difference may be written as:
In order to maintain hydrostatic equilibrium, the induced capillary pressure is balanced by a change in height, h, which can be positive or negative, depending on whether the wetting angle is less than or greater than 90°. For a fluid of density ρ:
|γ = 0.0728 J/m2 at 20 °C||θ = 20° (0.35 rad)|
|ρ = 1000 kg/m3||g = 9.8 m/s2|
— and so the height of the water column is given by:
Thus for a 2 mm wide (1 mm radius) tube, the water would rise 14 mm. However, for a capillary tube with radius 0.1 mm, the water would rise 14 cm (about 6 inches).
Capillary action in general
In the general case, for a free surface and where there is an applied "over-pressure", Δp, at the interface in equilibrium, there is a balance between the applied pressure, the hydrostatic pressure and the effects of surface tension. The Young–Laplace equation becomes:
— and characteristic pressure:
The non-dimensional equation then becomes:
Thus, the surface shape is determined by only one parameter, the over pressure of the fluid, Δp* and the scale of the surface is given by the capillary length. The solution of the equation requires an initial condition for position, and the gradient of the surface at the start point.
Application in medicine
In medicine it is often referred to as the Law of Laplace, used in the context of cardiovascular physiology, and also respiratory physiology. Arteries may be viewed as cylinders, and the left ventricle of the heart can be viewed as part cylinder, part hemisphere (a bullet shape), modeled by the Law of Laplace as T = pr / (2t), where T = wall tension, p = pressure, r = radius, t = wall thickness. For a given pressure, increased radius requires increased wall thickness to accommodate a stable wall stress; also, increased pressure requires increased thickness to maintain a stable wall stress. The latter is used to explain thickening of arteries and thickening of the left ventricle to accommodate high blood pressure. However, the thickened left ventricle is stiffer than when the thickness is normal, so it requires elevated pressures to fill, a condition known as diastolic heart failure. Note that trucks carrying pressurized gas often have multiple tubes of small radius, so that the wall tension T will be low to reduce need for thick walls to prevent pipes from bursting.
The lung contains small spherical gas-exchange chambers called alveoli, where a single alveolus can be modeled as being a perfect sphere. The Law of Laplace explains why alveoli of the lung need small radii to accommodate their thin walls for gas exchange at atmospheric pressure. The presence of numerous alveoli with small radii also results in high surface area, which is advantageous for gas exchange. 
Applying the Law of Laplace to alveoli of the lung, the pressure differential results in an external force that pushes on the surface of the alveolus, tending to decrease size during exhalation. The Law of Laplace states that pressure is inversely proportional to alveolar radius, and directly proportional to surface tension. It follows from this that if the surface tensions are equal, a small alveolus will experience a greater inward pressure than a large alveolus. 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. You can mimic this effect by connecting two inflated balloons to either ends of a plastic straw or a stiffer tube (you may need rubber bands to secure them). If the balloons are equal in thickness and radius then they can stay equally inflated, but if you squeeze one a bit to reduce its radius, the condition will be unstable, and in accord with the Law of Laplace, the smaller one will empty itself into the larger one.
The Law of Laplace also explains various phenomena encountered in the pathology of vascular or gastrointestinal walls. The wall tension in this case represents the muscular tension on the wall of the vessel. For example, if a fusiform 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.
The Law of Laplace can also be used to model transmural pressure in the heart and the rest of the circulatory system.
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.
Thomas Young laid the foundations of the equation in his 1804 paper An Essay on the Cohesion of Fluids  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  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 his book Physico-mechanical Experiments (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. Franz Ernst Neumann (1798-1895) later filled in a few details.
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- James Jurin (1718) "An account of some experiments shown before the Royal Society; with an enquiry into the cause of some of the ascent and suspension of water in capillary tubes," Philosophical Transactions of the Royal Society of London, 30 : 739–747.
- James Jurin (1719) "An account of some new experiments, relating to the action of glass tubes upon water and quicksilver," Philosophical Transactions of the Royal Society of London, 30 : 1083–1096.
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- Francis Hauksbee, Physico-mechanical Experiments on Various Subjects … (London, England: (Self-published by author; printed by R. Brugis), 1709), pages 139–169.
- Francis Hauksbee (1711) "An account of an experiment touching the direction of a drop of oil of oranges, between two glass planes, towards any side of them that is nearest press'd together," Philosophical Transactions of the Royal Society of London, 27 : 374–375.
- Francis Hauksbee (1712) "An account of an experiment touching the ascent of water between two glass planes, in an hyperbolick figure," Philosophical Transactions of the Royal Society of London, 27 : 539–540.
- [Anon.] (1911) Capillary action, Encyclopædia Britannica
- Thomas Young (1805) "An essay on the cohesion of fluids," Philosophical Transactions of the Royal Society of London, 95 : 65–87.
- Pierre Simon marquis de Laplace, Traité de Mécanique Céleste, volume 4, (Paris, France: Courcier, 1805), Supplément au dixième livre du Traité de Mécanique Céleste, pages 1–79.
- Pierre Simon marquis de Laplace, Traité de Mécanique Céleste, volume 4, (Paris, France: Courcier, 1805), Supplément au dixième livre du Traité de Mécanique Céleste. On page 2 of the Supplément, Laplace states that capillary action is due to "… les lois dans lesquelles l'attraction n'est sensible qu'à des distances insensibles; …" (… the laws in which attraction is sensible [significant] only at insensible [infinitesimal] distances …).
- In 1751, Johann Andreas Segner came to the same conclusion that Hauksbee had reached in 1709: J. A. von Segner (1751) "De figuris superficierum fluidarum" (On the shapes of liquid surfaces), Commentarii Societatis Regiae Scientiarum Gottingensis (Memoirs of the Royal Scientific Society at Göttingen), 1 : 301–372. On page 303, Segner proposes that liquids are held together by an attractive force (vim attractricem) that acts over such short distances "that no one could yet have perceived it with their senses" (… ut nullo adhuc sensu percipi poterit.).
- Carl Friedrich Gauss, Principia generalia Theoriae Figurae Fluidorum in statu Aequilibrii [General principles of the theory of fluid shapes in a state of equilibrium] (Göttingen, (Germany): Dieterichs, 1830). Available on-line at: Hathi Trust.
- Franz Neumann with A. Wangerin, ed., Vorlesungen über die Theorie der Capillarität [Lectures on the theory of capillarity] (Leipzig, Germany: B. G. Teubner, 1894).
- Rouse Ball, W. W.  (2003) "Pierre Simon Laplace (1749–1827)", in A Short Account of the History of Mathematics, 4th ed., Dover, ISBN 0-486-20630-0
- [Anon.] (1911) Capillary action, Encyclopædia Britannica
- Batchelor, G. K. (1967) An Introduction To Fluid Dynamics, Cambridge University Press
- Jurin, J. (1717/1719). "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". Philosophical Transactions of the Royal Society 30 (351–363): 739–747. doi:10.1098/rstl.1717.0026.
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