Comptes Rendus
A direct relation between bending energy and contact angles for capillary bridges
Comptes Rendus. Mécanique, Volume 351 (2023) no. S2, pp. 125-137.

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.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/crmeca.200
Classification: 49N45, 53A10, 58E12, 74F10, 74G05, 74G15, 53Z05
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
Mot 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

1 LaSIE, UMR-CNRS 7356, Université de La Rochelle, avenue Michel Crépeau, 17042 La Rochelle cedex 1, France.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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  - 
%0 Journal Article
%A Olivier Millet
%A Gérard Gagneux 
%T A direct relation between bending energy and contact angles for capillary bridges
%J Comptes Rendus. Mécanique
%D 2023
%P 125-137
%V 351
%N S2
%I Académie des sciences, Paris
%R 10.5802/crmeca.200
%G en
%F CRMECA_2023__351_S2_125_0
Olivier Millet; Gérard Gagneux . A direct relation between bending energy and contact angles for capillary bridges. Comptes Rendus. Mécanique, 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] Hung-Hsi Wu Historical development of the Gauss-Bonnet theorem, Sci. China, Ser. A, Volume 51 (2008) no. 4, pp. 777-784 | DOI | MR | Zbl

[2] Robert Finn Capillary surface interfaces, Notices Am. Math. Soc., Volume 46 (1999) no. 7, pp. 770-781 | MR | Zbl

[3] Gérard Gagneux; Olivier Millet; B. Mielniczuk; M. S. El Youssoufi Theoretical and experimental study of pendular regime in unsaturated granular media, Engineering, Volume 21 (2017) no. 7-8, pp. 840-853 | DOI

[4] Pierre-Gilles de Gennes; Francoise Brochard-Wyart; David Quere Capillarity and gravity. In Capillarity and Wetting Phenomena, Springer, 2004 | DOI | Zbl

[5] Enrico Giusti Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, Birkhäuser, 1984 | DOI | Zbl

[6] F. M. Orr; L. E. Scriven; A. P. Rivas Pendular rings between solids: meniscus properties and capillary force, J. Fluid Mech., Volume 67 (1975) no. 4, pp. 723-742 | DOI | Zbl

[7] Friedrich Sauvigny Surfaces of prescribed mean curvature H(x,y,z) with one-to-one central projection onto a plane, Pac. J. Math., Volume 281 (2016) no. 2, pp. 481-509 | DOI | MR | Zbl

[8] Friedrich Sauvigny 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] Francesco Dell’Isola; A. Romano 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] Francesco Dell’Isola; Henri Gouin; Pierre Seppecher 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] Manfredo P. do Carmo Selected papers, Springer, 2012 | Zbl

[12] Manfredo P. do Carmo Differential geometry of curves and surfaces, Prentice Hall, 1976 | Zbl

[13] Manfredo P. do Carmo Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 (ISBN: 0-13-212589-7) | Zbl

[14] Alfred Gray; Elsa Abbena; Simon Salamon Modern differential geometry of curves and surfaces with Mathematica, Textbooks in Mathematics, CRC Press, 2017

[15] Ladislav Boruvka; A. W. Neumann Generalization of the classical theory of capillarity, J. Chem. Phys., Volume 66 (1977) no. 12, pp. 5464-5476 | DOI

[16] John Gaydos; Ladislav Boruvka; Yehuda Rotenberg; Pu Chen; A. W. Neumann The Generalized Theory of Capillarity, Applied Surface Thermodynamics (Surfactant science series), Volume 63, Marcel Dekker, 1996, pp. 1-52

[17] L. Scholtès; P.-Y. Hicher; F. Nicot; B. Chareyre; F. Darve 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] Ch. Delaunay Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl., Volume 6 (1841), pp. 309-315 | Numdam

[19] Dominick N. Mazzone; Gabriel I. Tardos; Robert Pfeffer 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] Gérard Gagneux; Olivier Millet 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] Gérard Gagneux; Olivier Millet An analytical framework for evaluating the cohesion effects of coalescence between capillary bridges, Granul. Matter, Volume 18 (2016) no. 2, 16 | DOI

[22] B. Mielniczuk; Olivier Millet; Gérard Gagneux; M. S. El Youssoufi 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] Hien Nho Gia Nguyen; Chao-Fa Zhao; Olivier Millet; Gérard Gagneux 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] Hien Nho Gia Nguyen; Olivier Millet; Gérard Gagneux 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] Hien Nho Gia Nguyen; Olivier Millet; Gérard Gagneux 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] Hien Nho Gia Nguyen; Olivier Millet; Gérard Gagneux 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] Hien Nho Gia Nguyen; Olivier Millet; Chao-Fa Zhao; Gérard Gagneux 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] Hien Nho Gia Nguyen; Chao-Fa Zhao; Olivier Millet; A. P. S. Selvadurai Effects of surface roughness on liquid bridge capillarity and droplet wetting, Powder Technol., Volume 378 (2021), pp. 487-496 | DOI

[29] M. A. Rodríguez-Valverde; M. A. Cabrerizo-Vílchez; R. Hidalgo-Álvarez The Young–Laplace equation links capillarity with geometrical optics, Eur. J. Phys., Volume 24 (2003) no. 2, 159 | DOI | MR

[30] Odile Carrier; Daniel Bonn Contact angles and the surface free energy of solids, Droplet, Wetting and evaporation, Academic Press Inc., 2015, pp. 15-23 | DOI

[31] Olivier Millet; Aziz Hamdouni; Alain Cimetière 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] Olivier Millet; Alain Cimetière; Aziz Hamdouni 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] Y. Okumo; Y. Takeda; M. Mano; T. Okada Design of ship hull structures: a practical guide for ingineers, Springer, 2009 | DOI

[34] C. Quilliet Depressions at the surface of an elastic spherical shell submitted to external pressure, Phys. Rev. E, Volume 74 (2006) no. 4, 046608 | DOI

[35] Gérard Gagneux; Monique Madaune-Tort Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Mathématiques et applications, 22, Springer, 1995 | Zbl

[36] A. D. Myshkis; V. G. Babskii; N. D. Kopachevskii; L. A. Slobozhanin; A. D. Tyuptsov Low-gravity fluid mechanics, Springer, 2012

[37] Philippe G. Ciarlet An Introduction to Differential Geometry, Springer, 2005 | Zbl

[38] Aziz Hamdouni; Khalid Elamri; Claude Vallée; Olivier Millet 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] Aziz Hamdouni; Olivier Millet 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] Aziz Hamdouni; Olivier Millet 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] Jérémy Hure; Benoît Roman; José Bico Wrapping an adhesive sphere with an elastic sheet, Phys. Rev., Volume 106 (2011) no. 17, 174301 | DOI

[42] Joost W. van Honschoten; Nataliya Brunets; Niels R. Tas Capillarity at the nanoscale, Chem. Soc. Rev., Volume 39 (2010) no. 3, pp. 1096-1114 | DOI

[43] Yong Jian Wang; Shuo Guo; Hsuan-Yi Chen; Penger Tong Understanding contact angle hysteresis on an ambient solid surface, Phys. Rev., Volume 93 (2016) no. 5, 052802 | DOI

[44] Erich Hartmann G 2 interpolation and blending on surfaces, Visual Comput., Volume 12 (1996) no. 4, pp. 181-192 | Zbl

[45] Mohammed Mostefa Mesmoudi; Leila De Floriani; Paola Magillo 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] R. A. Horn On Fenchel’s theorem, Am. Math. Mon., Volume 78 (1971), pp. 380-381 | DOI | MR | Zbl

Cited by Sources:

Comments - Policy