Younglaplace equation wikipedia republished wiki 2. On the demonstration of the younglaplace equation in. The lensmakers equation can be derived using different proofs 5, 14, 15 based on geometrical constructions, ray tracing, wave optics, fermats principle, etc. Accordingly the gravitational model is expressed by. The text used in the course was numerical methods for engineers, 6th ed. The aims of this article are to study the equation of younglaplace, show how to derive the. Mukherjee, department of chemical engineering, iit kharagpur. When a liquid comes into contact with a solid in a bulk, gaseous phase, according to youngs equation, there is a relationship between the contact angle. The younglaplace equation links capillarity with geometrical optics marodrguezvalverde, m a cabrerizov lchez and rhidalgoalvarez. Young laplace equation may easily be derived either by the principle of minimum energy or by requiring a force balance. Solving a young laplace equation for a meniscus against a. The properties of surfaces necessary to derive the young laplace equation may be found explicitly by differential geometry or more indirectly by linear algebra.
The normal force balance is expressed by the younglaplace equation, where now. Lecture younglaplace and kelvin equations 1 surface. During this phase, misura will create two adimensional sample datasets which contain the solution to the younglaplace. Younglaplace equation for any curved surface with principle radii of curvature r1 and r2. Thermodynamic deviations of the mechanical equilibrium. The younglaplace 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. Pdf the classical younglaplace equation relates capillary pressure to surface tension and the principal radii of curvature of the interface between. The pressure difference is caused by the surface tension of the interface between liquid and gas. Solution of the younglaplace equation for three particles 121 figure 54. Feb 19, 20 instability and patterning of thin polymer films by dr. Derivation of the generalized younglaplace equation of curved interfaces in nanoscaled solids tungyang chen,a minsen chiu, and chungning weng department of civil engineering, national cheng kung university, tainan 70101, taiwan. Consider a small section of a curved surface with carthesian dimensions x and y.
The kelvin equation gives the vapour pressure of a curved surface, such as droplet. Crosssection of a liquid bridge between three particles for. Apr 09, 2014 hello there, can anyone explain me how to get to the young laplace equation for surfaces with two radii of curvature from the young laplace equation for axisymmetric surfaces. In physics, the younglaplace equation is a nonlinear partial differential equation that describes the equilibrium pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension. Derivation of the generalized young laplace equation of curved interfaces in nanoscaled solids tungyang chen,a minsen chiu, and chungning weng department of civil engineering, national cheng kung university, tainan 70101, taiwan.
The younglaplace equation is usually introduced when teaching surface phenomena at an elementary level young 1992. We discuss here three mechanical methods to deduce this equation, and we intend to avoid certain misunderstandings that are found in these derivations, providing thus the. The shape is prescribed by the younglaplace equation. Sep 20, 20 these videos were created to accompany a university course, numerical methods for engineers, taught spring 20. Since cylindrically symmetric problems do not depend on the azimuthal angle. A comparison of ellipsoidal and spherical harmonics for. Apr 01, 2016 based on the present results, we cannot only effectively determine the surface tension of water and the effects of temperature or electrolyte ions on the surface tension, but also show that the young laplace yl equation is valid at nanoscale. General form of laplace equation derivation by minimization of the grand potential. The younglaplace solution will not be attempted unless the contact angle between the base and the sample reaches this angle. The young laplace equation the airliquid interface, dp sk. The classical younglaplace equation relates capillary pressure to. On account of this, the analogy presented is exceptional since the involved phenomena are a priori disjoined. In equation 1, p coh stands for the local pressure difference across the liquidvapour interface, h being its local mean curvature 19. The younglaplace equation gives the pressure difference across a curved surface and its most important application is in the derivation of the kelvin equation.
The younglaplace equation can also be derived by minimizing the free energy. Young laplace equation is widely applied to predict the bubble shape, its application is limited under highly nonequilibrium conditions. The laplace pressure is determined from the younglaplace equation given as. Nondimensionalising all lengths on some problemspeci.
The young laplace equation is usually introduced using mechanical rather than thermodynamic arguments when teaching surface phenomena at an elementary level. Let us consider a twophase fluid system confined in a box of volume v, see fig. We will also convert laplaces equation to polar coordinates and solve it on a disk of radius a. Using molecular dynamics md simulations, a new approach based on the behavior of pressurized water out of a nanopore 1. Using gold nanoparticle as an example, new experimental study is conducted to reveal the particle concentration influence on the behaviour of triple line and bubble dynamics. These videos were created to accompany a university course, numerical methods for engineers, taught spring 20. The previous relation is generally known as the younglaplace equation, and is named after thomas young 17731829, who developed the qualitative theory of surface tension in 1805, and pierresimon laplace 17491827 who completed the mathematical description in the following year. Pdf this article focuses on studying about the equation and application of. We say a function u satisfying laplaces equation is a harmonic function. Index termsyounglaplace equation, contact angle, youngs. Thisexpressionis often encountered in the literature. Work of adhesion and younglaplace equationtheory of. Young laplace equation for any curved surface with principle radii of curvature r1 and r2. Pdf derivation of the generalized younglaplace equation of.
As we will see this is exactly the equation we would need to solve if we were looking to find the equilibrium solution i. Modification of the younglaplace equation and prediction. Solution of the younglaplace equation for three particles. Let there be an isothermal perturbation about this equilibrium state. Pdf derivations of the younglaplace equation researchgate. It relates the pressure difference to the shape of the surface and it is fundamentally important in the study of static capillary. Using this technique, the pressure difference across the surface. Solutions of younglaplace equation for partially saturated. Thermodynamic deviations of the mechanical equilibrium conditions for fluid surfaces. This note presents a derivation of the laplace equation which gives the rela tionship between capillary pressure, surface tension, and principal. The first equation of capillarity, also known as the younglaplace equation, is derived by analysing a small section of an arbitrarily curved surface in mechanical equilibrium adamson, 1990. Laplaces equation 3 idea for solution divide and conquer we want to use separation of variables so we need homogeneous boundary conditions.
In pendant drop experiments the bond number of a droplet is then used for the younglaplace fit. The first term is the curvature of the surface of the dropmeniscus. Younglaplace equation an overview sciencedirect topics. To obtain a better understanding of the physical meaning of the younglaplace equation we discuss three mechanical. Younglaplace equation may easily be derived either by the principle of minimum energy or by requiring a force balance. The most important equation of capillarity, the younglaplace equation, has the same structure as the gullstrand equation of geometrical optics, which relates the optic power of a thick lens to its geometry and the properties of the. The younglaplace equation is usually introduced using mechanical rather than thermodynamic arguments when teaching surface phenomena at an.
The shape of liquid drop is governed by what is known as the young laplace equation. In this section we discuss solving laplaces equation. 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. Wolfgang bernhard, in fetal and neonatal physiology fifth edition, 2017. Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces. The properties of surfaces necessary to derive the younglaplace equation may be found explicitly by differential geometry or more indirectly by linear algebra. For the earth and other bodies whose shape is nearly spherical, a natural choice is the set of spherical coordinates. Florian mieth, mike tromm, in specialized injection molding techniques, 2016. Derivation of the generalized younglaplace equation of. The young laplace equation gives the pressure difference across a fluid interface as a function of the curvatures. Assume a case where liquid such as water or isopropanol works under pressure against the hydrophobic resistance force of the capillaries arising from surface energy of the liquid. The younglaplace equation is a highly nonlinear pde. We demonstrate the decomposition of the inhomogeneous. Biocolloid and fluid physics group, department of applied physics, university of granada, e18071, granada, spain1 email.
Laplaces law, youngs equation, surface energy, surface tension. This article focuses on studying about the equation and application of young laplace in predicting the subbandage pressure. These solutions are known as harmonic functions whose forms vary according to the coordinate system used. Younglaplace equation is widely applied to predict the bubble shape, its application is limited under highly nonequilibrium conditions. General remarks on mammalian surfactant, phospholipid analysis and functions.
Take for instance the case in which a droplet is trapped between two plates, where we assume that the height between. The fundamental laws governing the mechanical equilibrium of solidfluid. Laplaces equation compiled 26 april 2019 in this lecture we start our study of laplaces equation, which represents the steady state of a eld that depends on two or more independent variables, which are typically spatial. A short derivation of this equation is presented here. As a result, a new form of youngs equation is derived, which can be reduced into the youngs equation for either a planar boundary or a vertical tube and accordingly can be considered as an extended form of youngs equation for surfaces of various forms. In this case, the surface phenomena are often described by using mechanical rather than thermodynamic arguments. A modified young laplace equation for the bubble electrospinning considering the effect of humidity by fujuan liu and hao dou national engineering laboratory for modern silk, college of textile and engineering, soochow university, suzhou, china short paper doi. I have a set of points in 2d space axisymmetry is assumed for the inner lobe. Modification of the younglaplace equation and prediction of. The younglaplace equation links capillarity with geometrical.
Since the equation is linear we can break the problem into simpler problems which do have su. Another suboption allows to tune the rungekutta integration step used by the numerical solver. Equation 117 is the fundamental equation of capillarity and is well known as younglaplace equation. The young laplace equation can also be derived by minimizing. Derivation of the generalized younglaplace equation of curved.
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