In physics, the
Δ p = − γ ∇ ⋅ n ^ = 2 γ H = γ ( 1 R 1 + 1 R 2 ) displaystyle begin aligned Delta p&=-gamma nabla cdot hat n \&=2gamma H\&=gamma left( frac 1 R_ 1 + frac 1 R_ 2 right)end aligned where Δ p displaystyle Delta p is the pressure difference across the fluid interface, γ displaystyle gamma is the surface tension (or wall tension), n ^ displaystyle hat n is the unit normal pointing out of the surface, H displaystyle H is the mean curvature, and R 1 displaystyle R_ 1 and R 2 displaystyle R_ 2 are the principal radii of curvature. (Some authors[who?] refer inappropriately to the factor 2 H displaystyle 2H as the total curvature.) Note that only normal stress is considered,
this is because it can be shown[1] 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
Contents 1 Soap films
2 Emulsions
3
4.1
5 Application in medicine 6 History 7 References 8 Bibliography Soap films[edit]
Main article: Soap film
If the pressure difference is zero, as in a soap film without gravity,
the interface will assume the shape of a minimal surface.
Emulsions[edit]
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.
Spherical meniscus with wetting angle less than 90° 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: Δ p = 2 γ R . displaystyle Delta p= frac 2gamma R . This may be shown by writing the
R = a cos θ displaystyle R= frac a cos theta so that the pressure difference may be written as: Δ p = 2 γ cos θ a . displaystyle Delta p= frac 2gamma cos theta a . Illustration of capillary rise. Red=contact angle less than 90°; blue=contact angle greater than 90° 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: h = 2 γ cos θ ρ g a . displaystyle h= frac 2gamma cos theta rho ga . — where g is the gravitational acceleration. This is sometimes known
as the Jurin rule or Jurin height[3] after
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: h ≈ 1.4 × 10 − 5 a displaystyle happrox 1.4times 10^ -5 over a m. 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[edit]
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
Δ p = ρ g h − γ ( 1 R 1 + 1 R 2 ) displaystyle Delta p=rho gh-gamma left( frac 1 R_ 1 + frac 1 R_ 2 right) The equation can be non-dimensionalised in terms of its characteristic length-scale, the capillary length: L c = γ ρ g , displaystyle L_ c = sqrt frac gamma rho g , — and characteristic pressure: p c = γ L c = γ ρ g . displaystyle p_ c = frac gamma L_ c = sqrt gamma rho g . For clean water at standard temperature and pressure, the capillary length is ~2 mm. The non-dimensional equation then becomes: h ∗ − Δ p ∗ = ( 1 R 1 ∗ + 1 R 2 ∗ ) . displaystyle h^ * -Delta p^ * =left( frac 1 R_ 1 ^ * + frac 1 R_ 2 ^ * right). 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. A pendant drop is produced for an over pressure of Δp*=3 and initial condition r0=10−4, z0=0, dz/dr=0 A liquid bridge is produced for an over pressure of Δp*=3.5 and initial condition r0=0.25−4, z0=0, dz/dr=0
r ″ ( 1 + r ′ 2 ) 3 2 − 1 r ( z ) 1 + r ′ 2 = z − Δ p ∗ displaystyle frac r'' (1+r'^ 2 )^ frac 3 2 - frac 1 r(z) sqrt 1+r'^ 2 =z-Delta p^ * z ″ ( 1 + z ′ 2 ) 3 2 + z ′ r ( 1 + z ′ 2 ) 1 2 = Δ p ∗ − z ( r ) . displaystyle frac z'' (1+z'^ 2 )^ frac 3 2 + frac z' r(1+z'^ 2 )^ frac 1 2 =Delta p^ * -z(r). Application in medicine[edit]
In medicine it is often referred to as the Law of Laplace, used in the
context of cardiovascular physiology, and also respiratory
physiology[6]
History[edit]
^ Surface Tension Module, by John W. M. Bush, at MIT OCW. ^ Robert Finn (1999). "Capillary Surface Interfaces" (PDF). AMS. ^ "Jurin rule". McGraw-Hill Dictionary of Scientific and Technical Terms. McGraw-Hill on Answers.com. 2003. Retrieved 2007-09-05. ^ a b See:
^ Lamb, H. Statics, Including Hydrostatics and the Elements of the
Theory of Elasticity, 3rd ed. Cambridge, England: Cambridge University
Press, 1928.
^ Basford, Jeffrey R. (2002). "The Law of Laplace and its relevance to
contemporary medicine and rehabilitation". Archives of Physical
Francis Hauksbee, Physico-mechanical Experiments on Various Subjects
… (London, England: (Self-published by author; printed by R.
Brugis), 1709), pages 139–169.
^ a b Maxwell, James Clerk; Strutt, John William (1911).
"Capillary Action". Encyclopædia Britannica. 5 (11th ed.).
pp. 256–275.
^ 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,
Bibliography[edit] Maxwell, James Clerk; Strutt, John William (1911). "Capillary Action". In Chisholm, Hugh. Encyclopædia Britannica. 5 (11th ed.). Cambridge University Press. pp. 256–275. Batchelor, G. K. (1967) An Introduction To Fluid Dynamics, Cambridge University Press Jurin, J. (1716). "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. Tadros T. F. (1995) Surfactants in Agrochemicals, Surfactant Science series, vo |