Comptes Rendus
no. 8
Computational modeling of material forming processes / Simulation numérique des procédés de mise en forme
Accounting for local capillary effects in two-phase flows with relaxed surface tension formulation in enriched finite elements
Loïc Chevalier1 ; Julien Bruchon1 ; Nicolas Moulin1 ; Pierre-Jacques Liotier1 ; Sylvain Drapier1
1 Industrial Chair Hexcel – Mines Saint-Étienne, Centre SMS & LGF, UMR CNRS 5307, Mines de Saint-Étienne – Université de Lyon, 158, cours Fauriel, CS 62362, 42023 Saint-Étienne cedex 2, France
Comptes Rendus. Mécanique, Volume 346 (2018) no. 8, pp. 617-633.

This paper introduces a numerical method able to deal with a general bi-fluid model integrating capillary actions. The method relies first on the precise computation of the surface tension force. Considering a mathematical transformation of the surface tension virtual work, the regularity required for the solution on the evolving curved interface is weakened, and the mechanical equilibrium of the triple line can be enforced as a natural condition. Consequently, contact angles of the liquid over the solid phase result naturally from this equilibrium. Second, for an exhaustive representation of capillary actions, pressure jumps across the interface must be accounted for. A pressure enrichment strategy is used to properly compute the discontinuities in both pressure and gradient fields. The resulting method is shown to predict nicely static contact angles for some test cases, and is evaluated on complex 3D cases.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2018.06.008
Mots clés : Fibrous media, Capillarity, Surface tension, Finite elements, Level-set method
Loïc Chevalier 1 ; Julien Bruchon 1 ; Nicolas Moulin 1 ; Pierre-Jacques Liotier 1 ; Sylvain Drapier 1

1 Industrial Chair Hexcel – Mines Saint-Étienne, Centre SMS & LGF, UMR CNRS 5307, Mines de Saint-Étienne – Université de Lyon, 158, cours Fauriel, CS 62362, 42023 Saint-Étienne cedex 2, France 
@article{CRMECA_2018__346_8_617_0,
     author = {Lo{\"\i}c Chevalier and Julien Bruchon and Nicolas Moulin and Pierre-Jacques Liotier and Sylvain Drapier},
     title = {Accounting for local capillary effects in two-phase flows with relaxed surface tension formulation in enriched finite elements},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {617--633},
     publisher = {Elsevier},
     volume = {346},
     number = {8},
     year = {2018},
     doi = {10.1016/j.crme.2018.06.008},
     language = {en},
}
TY  - JOUR
AU  - Loïc Chevalier
AU  - Julien Bruchon
AU  - Nicolas Moulin
AU  - Pierre-Jacques Liotier
AU  - Sylvain Drapier
TI  - Accounting for local capillary effects in two-phase flows with relaxed surface tension formulation in enriched finite elements
JO  - Comptes Rendus. Mécanique
PY  - 2018
SP  - 617
EP  - 633
VL  - 346
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crme.2018.06.008
LA  - en
ID  - CRMECA_2018__346_8_617_0
ER  - 
%0 Journal Article
%A Loïc Chevalier
%A Julien Bruchon
%A Nicolas Moulin
%A Pierre-Jacques Liotier
%A Sylvain Drapier
%T Accounting for local capillary effects in two-phase flows with relaxed surface tension formulation in enriched finite elements
%J Comptes Rendus. Mécanique
%D 2018
%P 617-633
%V 346
%N 8
%I Elsevier
%R 10.1016/j.crme.2018.06.008
%G en
%F CRMECA_2018__346_8_617_0
Loïc Chevalier; Julien Bruchon; Nicolas Moulin; Pierre-Jacques Liotier; Sylvain Drapier. Accounting for local capillary effects in two-phase flows with relaxed surface tension formulation in enriched finite elements. Comptes Rendus. Mécanique, Volume 346 (2018) no. 8, pp. 617-633. doi : 10.1016/j.crme.2018.06.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.06.008/

[1] R. Masoodi; K.M. Pillai Wicking in Porous Materials, Taylor & Francis, 2013

[2] D. Pino Muñoz; J. Bruchon; S. Drapier; F. Valdivieso A finite element-based level set method for fluid-elastic solid interaction with surface tension, Int. J. Numer. Methods Biomed. Eng., Volume 93 (2013) no. 9, pp. 919-941

[3] J. Bruchon; S. Drapier; F. Valdivieso 3d finite element simulation of the matter flow by surface diffusion using a level set method, Int. J. Numer. Methods Biomed. Eng., Volume 86 (2011) no. 7, pp. 845-861

[4] D. Seveno; A. Vaillant; R. Rioboo; H. Adão; J. Conti; J. De Coninck Dynamics of wetting revisited, Langmuir, Volume 25 (2009) no. 22, pp. 13034-13044

[5] P.-G. De Gennes; F. Brochard-Wyart; D. Quéré Capillarity and Wetting Phenomena – Drops, Bubbles, Pearls, Waves, 2002

[6] A. Marchand; S. Das; J.H. Snoeijer; B. Andreotti Contact angles on a soft solid: from Young's law to Neumann's law, Phys. Rev. Lett., Volume 109 (2012) no. 23

[7] T.D. Blake The physics of moving wetting lines, J. Colloid Interface Sci., Volume 299 (2006) no. 1, pp. 1-13

[8] M.V. Bruschke; S.G. Advani A finite element/control volume approach to mold filling in anisotropic porous media, Polym. Compos., Volume 11 (1990) no. 6, pp. 398-405

[9] F. Trochu; E. Ruiz; V. Achim; S. Soukane Advanced numerical simulation of liquid composite molding for process analysis and optimization, Composites, Part A, Appl. Sci. Manuf., Volume 37 (2006) no. 6, pp. 890-902

[10] H. Hu; R.G. Larson Analysis of the microfluid flow in an evaporating sessile droplet, Langmuir, Volume 21 (2005) no. 9, pp. 3963-3971

[11] H. Combeau; M. Bellet; Y. Fautrelle; D. Gobin; E. Arquis; O. Budenkova; B. Dussoubs; Y. du Terrail; A. Kumar; C.-A. Gandin; B. Goyeau; S. Mosbah; T. Quatravaux; M.A. Rady; M. Založnik Analysis of a numerical benchmark for columnar solidification of binary alloys, IOP Conf. Ser., Mater. Sci. Eng., Volume 33 (2012), pp. 12-27

[12] P. Celle; S. Drapier; J.-M. Bergheau Numerical modelling of liquid infusion into fibrous media undergoing compaction, Eur. J. Mech. A, Solids, Volume 27 (2008) no. 4, pp. 647-661

[13] G. Pacquaut; J. Bruchon; N. Moulin; S. Drapier Combining a level-set method and a mixed stabilized P1/P1 formulation for coupling Stokes–Darcy flows, Int. J. Numer. Methods Fluids, Volume 69 (2012) no. 2, pp. 459-480

[14] L. Abouorm; R. Troian; S. Drapier; J. Bruchon; N. Moulin Stokes–Darcy coupling in severe regimes using multiscale stabilisation for mixed finite elements: monolithic approach versus decoupled approach, Eur. J. Comput. Mech., Volume 23 (2014) no. 3–4, pp. 113-137

[15] M. Blais; N. Moulin; P.-J. Liotier; S. Drapier Resin infusion-based processes simulation: coupled Stokes–Darcy flows in orthotropic preforms undergoing finite strain, Int. J. Mater. Form., Volume 10 (2015) no. 1, pp. 43-54

[16] M.F. Pucci; P.-J. Liotier; S. Drapier Capillary wicking in a fibrous reinforcement – orthotropic issues to determine the capillary pressure components, Composites, Part A, Appl. Sci. Manuf., Volume 77 (2015), pp. 133-141

[17] C. Hae Park; A. Lebel; A. Saouab; J. Bréard; W.I. Lee Modeling and simulation of voids and saturation in liquid composite molding processes, Composites, Part A, Appl. Sci. Manuf., Volume 42 (2011) no. 6, pp. 658-668

[18] E. Lopez; A. Leygue; E. Abisset-Chavanne; S. Comas-Cardona; C. Aufrere; C. Binetruy; F. Chinesta Flow modeling of linear and nonlinear fluids in two and three scale fibrous fabrics, Int. J. Mater. Form., Volume 10 (2015) no. 3, pp. 317-328

[19] S. Osher; J.A. Sethian Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations, J. Comput. Phys., Volume 79 (1988) no. 1, pp. 12-49

[20] L. Benazzouk; E. Arquis; N. Bertrand; C. Descamps; M. Valat Motion of a liquid bridge in a capillary slot: a numerical investigation of wettability and geometrical effects, Houille Blanche, Volume 3 (2013), pp. 50-56

[21] J. Bruchon; N. Moulin; Y. Liu New variational formulation of the triple junction equilibrium with applications to wetting problems, Crete Island, Greece (2016)

[22] S.F. Kistler; P.M. Schweizer Liquid Film Coating: Scientific Principles and Their Technological Implications, Chapman & Hall, 1997

[23] S. Ganesan; G. Matthies; L. Tobiska On spurious velocities in incompressible flow problems with interfaces, Comput. Methods Appl. Mech. Eng., Volume 196 (2007) no. 7, pp. 1193-1202

[24] B. Lafaurie; C. Nardone; R. Scardovelli; S. Zaleski; G. Zanetti Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys., Volume 113 (1994) no. 1, pp. 134-147

[25] M.M. Francois; S.J. Cummins; E.D. Dendy; D.B. Kothe; J.M. Sicilian; M.W. Williams A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys., Volume 213 (2006) no. 1, pp. 141-173

[26] M. Discacciati; D. Hacker; A. Quarteroni; S. Quinodoz; S. Tissot; F.M. Wurm Numerical simulation of orbitally shaken viscous fluids with free surface, Int. J. Numer. Methods Fluids, Volume 71 (2013) no. 3, pp. 294-315

[27] J. Chessa; T. Belytschko An extended finite element method for two-phase fluids, J. Appl. Mech., Volume 70 (2003) no. 1, pp. 10-17

[28] P.D. Minev; T. Chen; K. Nandakumar A finite element technique for multifluid incompressible flow using Eulerian grids, J. Comput. Phys., Volume 187 (2003) no. 1, pp. 255-273

[29] R.F. Ausas; G.C. Buscaglia; S.R. Idelsohn A new enrichment space for the treatment of discontinuous pressures in multi-fluid flows, Int. J. Numer. Methods Fluids, Volume 70 (2012) no. 7, pp. 829-850

[30] H. Coppola Owen; R. Codina Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, Volume 49 (2005) no. 12, pp. 1287-1304

[31] J.E. Sprittles; Y.D. Shikhmurzaev Finite element simulation of dynamic wetting flows as an interface formation process, J. Comput. Phys., Volume 233 (2013) no. 1, pp. 34-65

[32] R.A. Sauer A contact theory for surface tension driven systems, Math. Mech. Solids, Volume 21 (2014) no. 3, pp. 305-325

[33] A. Reusken; X. Xu; L. Zhang Finite element methods for a class of continuum models for immiscible flows with moving contact lines, Int. J. Numer. Methods Fluids, Volume 84 (2017) no. 5, pp. 268-291

[34] G.C. Buscaglia; R.F. Ausas Variational formulations for surface tension, capillarity and wetting, Comput. Methods Appl. Mech. Eng., Volume 200 (2011) no. 45–46, pp. 3011-3025

[35] R.W. Style; R. Boltyanskiy; Y. Che; J.S. Wettlaufer; L.A. Wilen; E.R. Dufresne Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses, Phys. Rev. Lett., Volume 110 (2013) no. 6

[36] T.J.R. Hughes Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Eng., Volume 127 (1995) no. 1–4, pp. 387-401

[37] T.J.R. Hughes; G.R. Feijoo; L. Mazzei; J.B. Quincy The variational multiscale method – a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., Volume 166 (1998) no. 1–2, pp. 3-24

[38] R. Codina A stabilized finite element method for generalized stationary incompressible flows, Comput. Methods Appl. Mech. Eng., Volume 190 (2001) no. 20–21, pp. 2681-2706

[39] E. Hachem; B. Rivaux; T. Kloczko; H. Digonnet; T. Coupez Stabilized finite element method for incompressible flows with high Reynolds number, J. Comput. Phys., Volume 229 (2010) no. 23, pp. 8643-8665

[40] L. Abouorm; N. Moulin; J. Bruchon; S. Drapier Monolithic approach of Stokes–Darcy coupling for LCM process modelling, Key Eng. Mater., Volume 554–557 (2013), pp. 447-455

[41] Y. Liu; N. Moulin; J. Bruchon; P.-J. Liotier; S. Drapier Towards void formation and permeability predictions in LCM processes: a computational bifluid–solid mechanics framework dealing with capillarity and wetting issues, C. R. Mecanique, Volume 344 (2016) no. 4–5, pp. 236-250

[42] H. Sauerland; T.P. Fries The stable XFEM for two-phase flows, Comput. Fluids, Volume 87 (2013), pp. 41-49

[43] S. Groß; V. Reichelt; A. Reusken A finite element based level set method for two-phase incompressible flows, Comput. Vis. Sci., Volume 9 (2006) no. 4, pp. 239-257

[44] E. Bänsch Finite element discretization of the Navier–Stokes equations with a free capillary surface, Numer. Math., Volume 88 (2001) no. 2, pp. 203-235

[45] G. Dziuk An algorithm for evolutionary surfaces, Numer. Math., Volume 58 (1990) no. 1, pp. 603-611

[46] G. Dziuk; C.M. Elliott Finite elements on evolving surfaces, IMA J. Numer. Anal., Volume 27 (2007) no. 2, pp. 262-292

[47] S. Groß; A. Reusken Finite element discretization error analysis of a surface tension force in two-phase incompressible flows, SIAM J. Numer. Anal., Volume 45 (2007) no. 4, pp. 1679-1700

[48] A.N. Brooks; T.J.R. Hughes Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Eng., Volume 32 (1982) no. 1–3, pp. 199-259

[49] J. Crank; P. Nicolson A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Math. Proc. Camb. Philos. Soc., Volume 43 (1947) no. 01, pp. 50-67

[50] M. Sussman; P. Smereka; S. Osher A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., Volume 114 (1994) no. 1, pp. 146-159

[51] D. Peng; B. Merriman; S. Osher; H. Zhao; M. Kang A PDE-based fast local level set method, J. Comput. Phys., Volume 155 (1999) no. 2, pp. 410-438

[52] M.F. Pucci; P.-J. Liotier; S. Drapier Tensiometric method to reliably assess wetting properties of single fibers with resins: validation on cellulosic reinforcements for composites, Colloids Surf. A, Physicochem. Eng. Asp., Volume 512 (2017), pp. 26-33

Cité par Sources :

Commentaires - Politique