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Usually, we understand the term Capillary bridge as a minimized surface of liquid or membrane, created between two rigid bodies with an arbitrary shape. Capillary bridges also may form between two liquids.〔Ross, Sydney,The Inhibition of Foaming. II. A Mechanism for the Rupture of Liquid Films by Anti-foaming Agents,J. Phys. Chem., 1950, 54 (3), pp 429–436〕 Plateau defined a sequence of capillary shapes known as (1) nodoid with 'neck', (2) cathenoid, (3) unduloid with 'neck', (4) Cylinder, (5) Unduloid with 'haunch' (6) Sphere and (7) Nodoid with 'haunch'. The presence of capillary bridge, depending on their shapes, can lead to attraction or repulsion between the solid bodies. The simplest cases of them are the axisymmetric ones. We distinguished three important classes of bridging, depending on connected bodies surface shapes: * two planar surfaces (fig.1) * planar surface and spherical particle (fig. 2) * two spherical particles (in general, particles may not be of equal sizes, fig. 3) Capillary bridges and their properties may also be influenced by Earth gravity and properties of bridged surfaces. As a bridging substance may be used liquids or gases enclosed in a boundary, called interface (capillary surface). The interface is characterized with particular surface tension. ==History== Capillary bridges have been studied for over 200 years. The question was raised for the first time by Josef Louis Lagrange in 1760, and interest was further spread by the French astronomer and mathematician C. Delaunay. Delaunay found an entirely new class of axially symmetrical surfaces of constant mean curvature. The formulation and the proof of his theorem had a long story. It began with Euler's〔L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Opera omnia, I, 24, (1744)〕 proposition of new figure, called ''cathenoid''. Much later, Kenmotsu 〔Kenmotsu, K., Surfaces of revolution with prescribed mean curvature, Tohoku Math. J. 32(1980), 147–153.〕 solved the complex nonlinear equations, describing this class of surfaces. However, his solution is of little practical importance because has no geometrical interpretation. J. Plateau showed the existence of such shapes with given boundaries. The problem was named after him Plateau's problem. Many scientists contributed to the solution of the problem. One of them is Thomas Young.〔Young, T. An Essay on the Cohesion of Fluids. Philos. Trans. R.Soc. London 1805, 95, 65−87Soc. London 1805, 95, 65−87.〕 Pierre Simon Laplace contributed the notion of capillary tension. Laplace even formulated the widely known nowadays condition for mechanical equilibrium between two fluids, divided by a capillary surface Pγ=ΔP i.e. capillary pressure between two phases is balances by their adjacent pressure difference. A general survey on capillary bridge behavior in gravity field is completed by Myshkis and Babskii.〔A.D. Myshkis and V.G. Babskii, Low-Gravity Fluid Mechanics: Mathematical Theory of Capillary Phenomena, Springer-Verlag 1987〕 In the last century a lot of efforts were put of study of surface forces that drive capillary effects of bridging. There was established that these forces result from intermolecular forces and become significant in thin fluid gaps (<10 nm) between two surfaces.〔Nikolai V. Churaev, B.V. Derjaguin, V.M. Muller, Surface Forces, Springer Scoence and Business Media, 1987〕〔J. Israelashvilly, Intermolecular and Surface Forces, Third Edition: Revised, Elsevier, 2011〕 The instability of capillary bridges was discussed in first time by Rayleigh.〔Strut, J. W., Lord Rayleigh, On the instability of jets, Proceedings of London Mathematical Society, v.10, pp. 4-13 (1878)〕 He demonstrated that a liquid jet or capillary cylindrical surface became unstable when the ratio between its length, H to the radius R, becomes bigger than 2π. In these conditions of small sinusoidal perturbations with wavelength bigger than it perimeter, the cylinder surface area becomes larger than the one of unperturbed cylinder with the same volume and thus it becomes unstable. Later, Hove 〔Hove, W., Ph.D. Dissertation, Friendlich-Wilhelms, Universitat zu Berlin (1887)〕 formulated the variational requirements for the stability of axisymmetric capillary surfaces (unbounded) in absence of gravity and with disturbances constarined to constant volume. He first solved Young-Laplace equation for equilibrium shapes and showed that the Legendre condition for the second variation is always satisfied. Therefore the stability is determined by the absence of negative eigenvalue of the linearized Young-Laplace equation. This approach of determining stability from second variation is used now widely.〔 Pertirbation methods became very successful despite that nonlinear nature of capillary interaction can limit their application. Other methods now include direct simulation.〔Meseguer J. and A. Sanz, "Numerical and experimental study of the dynamics of axisymmetric liquid bridges," J. Fluid Mech. 153, 83 (1985)〕〔Martinez and J. M. Perales, "Liquid bridge stability data," J. Cryst, Growth 78, 369 (1986)〕 To that moment most methods for stability determination required calculation of equilibrium as a basis for perturbations. There appeared a new idea that stability may be deduced from equilibrium states.〔J. F. Padday , A. R. Pitt, The Stability of Axisymmetric Menisci, Philosophical Transactions A, (1973)〕〔E. A. Boucher; M. J. B. Evans, Pendent Drop Profiles and Related Capillary Phenomena, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 346, No. 1646. (Nov. 4, 1975), pp. 349-374〕 The proposition was further proven by Pitts〔Pitts, R.,The Stability of a Drop Hanging from a Tube, IMA J Appl Math (1976) 17 (3): 387-397〕 for axisymmetric constant volume. In the following years Vogel〔Vogel, Thomas I., Stability of a liquid drop trapped between two parallel planes, SIAM J. Appl. Math. 47 (1987), 516–525〕〔Vogel, Thomas I., Stability of a liquid drop trapped between two parallel planes II, SIAM J. Appl. Math. 49 (1989), 1009–1028〕 extended the theory. He examined the case of axisymmetric capillary bridges with constant volumes and the stability changes correspond to turning points. The recent development of bifurcation theory proved that ''exchange of stability'' between turning points and branch points is a general phenomenon.〔Michael, D. H., Annual Review of Fluid Mechanics Vol. 13: 189-216 (Volume publication date January 1981)〕〔Lowry Brian James, Paul H. Steen, Capillary Surfaces: Stability from Families of Equilibria with Application to the Liquid Bridge, Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences, (06/1995); 449:411-439〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Capillary bridges」の詳細全文を読む スポンサード リンク
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