The didactic object of these developments on differential geometry of curves and surfaces is to present fine and convenient mathematical strategies, adapted to the study of capillary bridges. The common thread is to be able to calculate accurately in any situation the bending stress over the free surface, represented mathematically by the integral of the Gaussian curvature over the surface (called the total curvature) involved in the generalized Young–Laplace equation. We prove in particular that the resultant of the bending energy is directly linked to the wetting angles at the contact line.
L’objet didactique de ces développements basés sur la géométrie différentielle des courbes et des surfaces est de présenter des stratégies mathématiques adaptées à l’étude des ponts capillaires. Le fil conducteur est de pouvoir calculer avec précision, dans n’importe quelle situation, la contrainte de flexion de la surface libre d’un pont capillaire, représentée mathématiquement par l’intégrale de courbure de Gauss (courbure totale) de la surface libre intervenant dans l’équation de Young–Laplace généralisée. Nous établissons en particulier un résultat très général suivant lequel la résultante de l’énergie de flexion est directement liée aux angles de mouillage au niveau de la ligne de contact.
Revised:
Accepted:
Online First:
Published online:
Keywords: Distortion of nonaxisymmetric capillary bridges, Mean and Gaussian curvatures impact, Euler characteristic, Generalized Young–Laplace equation, Bending effects, Fenchel’s theorem in differential geometry, Gauss–Bonnet Theorem, Geodesic curvature, Bending stress, Influence of the contact angles
Mots-clés : Distorsion des ponts capillaires non axisymétriques, impact des courbures moyennes et gaussiennes, caractéristique d’Euler, équation de Young–Laplace généralisée, effets de flexion, théorème de Fenchel en géométrie différentielle, théorème de Gauss–Bonnet, courbure géodésique, contrainte de flexion, influence des angles de contact
Olivier Millet 1; Gérard Gagneux 1
CC-BY 4.0
@article{CRMECA_2023__351_S2_125_0,
author = {Olivier Millet and G\'erard Gagneux },
title = {A direct relation between bending energy and contact angles for capillary bridges},
journal = {Comptes Rendus. M\'ecanique},
pages = {125--137},
publisher = {Acad\'emie des sciences, Paris},
volume = {351},
number = {S2},
year = {2023},
doi = {10.5802/crmeca.200},
language = {en},
}
TY - JOUR AU - Olivier Millet AU - Gérard Gagneux TI - A direct relation between bending energy and contact angles for capillary bridges JO - Comptes Rendus. Mécanique PY - 2023 SP - 125 EP - 137 VL - 351 IS - S2 PB - Académie des sciences, Paris DO - 10.5802/crmeca.200 LA - en ID - CRMECA_2023__351_S2_125_0 ER -
Olivier Millet; Gérard Gagneux . A direct relation between bending energy and contact angles for capillary bridges. Comptes Rendus. Mécanique, Physical Science in Microgravity within the Thematic Group Fundamental and Applied Microgravity, Volume 351 (2023) no. S2, pp. 125-137. doi : 10.5802/crmeca.200. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.200/
[1] Historical development of the Gauss-Bonnet theorem, Sci. China, Ser. A, Volume 51 (2008) no. 4, pp. 777-784 | DOI | MR | Zbl
[2] Capillary surface interfaces, Notices Am. Math. Soc., Volume 46 (1999) no. 7, pp. 770-781 | MR | Zbl
[3] Theoretical and experimental study of pendular regime in unsaturated granular media, Engineering, Volume 21 (2017) no. 7-8, pp. 840-853 | DOI
[4] Capillarity and gravity. In Capillarity and Wetting Phenomena, Springer, 2004 | DOI | Zbl
[5] Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, Birkhäuser, 1984 | DOI | Zbl
[6] Pendular rings between solids: meniscus properties and capillary force, J. Fluid Mech., Volume 67 (1975) no. 4, pp. 723-742 | DOI | Zbl
[7] Surfaces of prescribed mean curvature with one-to-one central projection onto a plane, Pac. J. Math., Volume 281 (2016) no. 2, pp. 481-509 | DOI | MR | Zbl
[8] Solution of boundary value problems for surfaces of prescribed mean curvature H (x, y, z) with 1–1 central projection via the continuity method, Lith. Math. J., Volume 58 (2018) no. 3, pp. 320-358 | DOI | MR | Zbl
[9] On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface, Int. J. Eng. Sci., Volume 25 (1987) no. 11-12, pp. 1459-1468 | DOI | MR | Zbl
[10] Radius and surface tension of microscopic bubbles by second gradient theory, C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Volume 320 (1995) no. 5, pp. 211-216 | Zbl
[11] Selected papers, Springer, 2012 | Zbl
[12] Differential geometry of curves and surfaces, Prentice Hall, 1976 | Zbl
[13] Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 (ISBN: 0-13-212589-7) | Zbl
[14] Modern differential geometry of curves and surfaces with Mathematica, Textbooks in Mathematics, CRC Press, 2017
[15] Generalization of the classical theory of capillarity, J. Chem. Phys., Volume 66 (1977) no. 12, pp. 5464-5476 | DOI
[16] The Generalized Theory of Capillarity, Applied Surface Thermodynamics (Surfactant science series), Volume 63, Marcel Dekker, 1996, pp. 1-52
[17] On the capillary stress tensor in wet granular materials, Int. J. Numer. Anal. Methods Geomech., Volume 33 (2009) no. 10, pp. 1289-1313 | DOI | Zbl
[18] Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl., Volume 6 (1841), pp. 309-315 | Numdam
[19] The effect of gravity on the shape and strength of a liquid bridge between two spheres, J. Colloid Interface Sci., Volume 113 (1986) no. 2, pp. 544-556 | DOI
[20] Analytic Calculation of Capillary Bridge Properties Deduced as an Inverse Problem from Experimental Data, Transp. Porous Med., Volume 105 (2014) no. 1, pp. 117-139 | DOI | MR
[21] An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges, Granul. Matter, Volume 18 (2016) no. 2, 16 | DOI
[22] Characterisation of pendular capillary bridges derived from experimental data using inverse problem method, Granul. Matter, Volume 20 (2018) no. 14, pp. 1-13 | DOI
[23] An original method for measuring liquid surface tension from capillary bridges between two equal-sized spherical particles, Powder Technol., Volume 363 (2020), pp. 349-359 | DOI
[24] Exact calculation of axisymmetric capillary bridge properties between two unequal-sized spherical particles, Math. Mech. Solids, Volume 24 (2019) no. 9, pp. 2767-2784 | DOI | MR | Zbl
[25] Liquid bridges between a sphere and a plane - classification of meniscus profiles for unknown capillary pressure, Math. Mech. Solids, Volume 24 (2019) no. 10, pp. 3042-3060 | DOI | MR | Zbl
[26] On the capillary bridge between spherical particles of unequal size: analytical and experimental approaches, Continuum Mech. Thermodyn., Volume 31 (2019) no. 1, pp. 225-237 | DOI | MR
[27] Theoretical and experimental study of capillary bridges between two parallel planes, European Journal of Environmental and Civil Engineering, Volume 26 (2022) no. 3, pp. 1198-1208 | DOI
[28] Effects of surface roughness on liquid bridge capillarity and droplet wetting, Powder Technol., Volume 378 (2021), pp. 487-496 | DOI
[29] The Young–Laplace equation links capillarity with geometrical optics, Eur. J. Phys., Volume 24 (2003) no. 2, 159 | DOI | MR
[30] Contact angles and the surface free energy of solids, Droplet, Wetting and evaporation, Academic Press Inc., 2015, pp. 15-23 | DOI
[31] Justification du modèle bidimensionnel non linéaire de plaque par développement asymptotique des équations d’équilibre, C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Volume 324 (1997) no. 6, pp. 349-354 | Zbl
[32] An asymptotic elastic-plastic plate model for moderate displacements and strong strain hardening, Int. J. Non-Linear Mech., Volume 22 (2003) no. 3, pp. 369-384 | DOI | MR
[33] Design of ship hull structures: a practical guide for ingineers, Springer, 2009 | DOI
[34] Depressions at the surface of an elastic spherical shell submitted to external pressure, Phys. Rev. E, Volume 74 (2006) no. 4, 046608 | DOI
[35] Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Mathématiques et applications, 22, Springer, 1995 | Zbl
[36] Low-gravity fluid mechanics, Springer, 2012
[37] An Introduction to Differential Geometry, Springer, 2005 | Zbl
[38] Compatibility of large deformations in nonlinear shell theory, Eur. J. Mech. A Solids, Volume 17 (1998) no. 5, pp. 855-864 | DOI | MR | Zbl
[39] Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part II: the strongly bent shells, Arch. Mech., Volume 55 (2003) no. 2, pp. 177-219 | Zbl
[40] An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section, Int. J. Non-Linear Mech., Volume 41 (2006) no. 3, pp. 396-416 | DOI | MR | Zbl
[41] Wrapping an adhesive sphere with an elastic sheet, Phys. Rev., Volume 106 (2011) no. 17, 174301 | DOI
[42] Capillarity at the nanoscale, Chem. Soc. Rev., Volume 39 (2010) no. 3, pp. 1096-1114 | DOI
[43] Understanding contact angle hysteresis on an ambient solid surface, Phys. Rev., Volume 93 (2016) no. 5, 052802 | DOI
[44] interpolation and blending on surfaces, Visual Comput., Volume 12 (1996) no. 4, pp. 181-192 | Zbl
[45] Discrete curvature estimation methods for triangulated surfaces, Applications of Discrete Geometry and Mathematical Morphology (Ullrich Köthe; Annick Montanvert; Pierre Soille, eds.), Springer, 2012, pp. 28-42 | DOI
[46] On Fenchel’s theorem, Am. Math. Mon., Volume 78 (1971), pp. 380-381 | DOI | MR | Zbl
Cited by Sources:
Comments - Policy