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Journal of Computational Mathematics

Surface tension effects between two immiscible Stokes fluids: a computational study using unfitted hybrid high-order methods and a level-set scheme
Stefano Piccardo1; Alexandre Ern2
1 CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée 2, France; Inria, 2 rue Simone Iff, 75589 Paris, France; LaCàN, ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
2 CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée 2, France; Inria, 2 rue Simone Iff, 75589 Paris, France
The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 257-283.

We develop an unfitted Hybrid High-Order (HHO) method coupled with a level-set scheme to solve numerically the flow of two immiscible Stokes fluids separated by an unknown interface where surface tension effects are present. The interface can cut through the mesh cells and a cell-agglomeration procedure is used to prevent possible ill-conditioning caused by small cut cells. The first computational study concerns the equilibrium between pure shear flow at infinity and surface tension, leading to an interface with elliptic shape. In particular, the dependence of the capillarity number on the Taylor deformation parameter and the viscosity ratio of both fluids is investigated. The second computational study covers evolving interfaces and illustrates how an initial interface progressively relaxes toward equilibrium.

Published online:
DOI: 10.5802/smai-jcm.101
Classification: 65N30, 65N22, 65N85, 76D45
Keywords: Hybrid discretization methods, Unfitted meshes, Stokes flows, Immiscible fluids, Surface tension
Stefano Piccardo 1; Alexandre Ern 2

1 CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée 2, France; Inria, 2 rue Simone Iff, 75589 Paris, France; LaCàN, ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
2 CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée 2, France; Inria, 2 rue Simone Iff, 75589 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     author = {Stefano Piccardo and Alexandre Ern},
     title = {Surface tension effects between two immiscible {Stokes} fluids: a computational study using unfitted hybrid high-order methods and a level-set scheme},
     journal = {The SMAI Journal of computational mathematics},
     pages = {257--283},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {9},
     year = {2023},
     doi = {10.5802/smai-jcm.101},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.101/}
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Stefano Piccardo; Alexandre Ern. Surface tension effects between two immiscible Stokes fluids: a computational study using unfitted hybrid high-order methods and a level-set scheme. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 257-283. doi : 10.5802/smai-jcm.101. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.101/

[1] M. Abdelgawad; A. R. Wheeler The Digital Revolution: A New Paradigm for Microfluidics, Adv. Mater., Volume 21 (2009) no. 8, pp. 920-925 | DOI

[2] S. Adjerid; N. Chaabane; T. Lin; P. Yue An immersed discontinuous finite element method for the Stokes problem with a moving interface, J. Comput. Appl. Math., Volume 362 (2019), pp. 540-559 | DOI | MR | Zbl

[3] J. Aghili; S. Boyaval; D. A. Di Pietro Hybridization of mixed high-order methods on general meshes and application to the Stokes equations, Comput. Methods Appl. Math., Volume 15 (2015) no. 2, pp. 111-134 | DOI | MR | Zbl

[4] M. Ainsworth Pyramid algorithms for Bernstein-Bézier finite elements of high, nonuniform order in any dimension, SIAM J. Sci. Comput., Volume 36 (2014) no. 2, p. A543-A569 | DOI | Zbl

[5] R. Becker; E. Burman; P. Hansbo A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Eng., Volume 198 (2009) no. 41-44, pp. 3352-3360 | DOI | MR | Zbl

[6] E. Bezchlebová; V. Dolejší; M. Feistauer Discontinuous Galerkin method for the solution of a transport level-set problem, Comput. Math. Appl., Volume 72 (2016) no. 3, pp. 455-480 | DOI | MR | Zbl

[7] L. Botti; D. A. Di Pietro; J. Droniou A hybrid high-order method for the incompressible Navier-Stokes equations based on Temam’s device, J. Comput. Phys., Volume 376 (2019), pp. 786-816 | DOI | MR | Zbl

[8] J. U. Brackbill; D. B. Kothe; C. Zemach A continuum method for modeling surface tension, J. Comput. Phys., Volume 100 (1992) no. 2, pp. 335-354 | DOI | MR | Zbl

[9] E. Burman; M. Cicuttin; G. Delay; A. Ern An unfitted hybrid high-order method with cell agglomeration for elliptic interface problems, SIAM J. Sci. Comput., Volume 43 (2021) no. 2, p. A859-A882 | DOI | MR | Zbl

[10] E. Burman; G. Delay; A. Ern An unfitted hybrid high-order method for the Stokes interface problem, IMA J. Numer. Anal., Volume 41 (2021) no. 4, pp. 2362-2387 | DOI | MR | Zbl

[11] E. Burman; A. Ern An unfitted hybrid high-order method for elliptic interface problems, SIAM J. Numer. Anal., Volume 56 (2018) no. 3, pp. 1525-1546 | DOI | MR | Zbl

[12] E. Cáceres; J. Guzmán; M. Olshanskii New stability estimates for an unfitted finite element method for two-phase Stokes problem, SIAM J. Numer. Anal., Volume 58 (2020) no. 4, pp. 2165-2192 | DOI | MR | Zbl

[13] L. Cattaneo; L. Formaggia; G. F. Iori; A. Scotti; P. Zunino Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces, Calcolo, Volume 52 (2015) no. 2, pp. 123-152 | DOI | MR | Zbl

[14] B. Cockburn; D. A. Di Pietro; A. Ern Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM, Math. Model. Numer. Anal., Volume 50 (2016) no. 3, pp. 635-650 | DOI | Numdam | MR | Zbl

[15] R. G. Cox The deformation of a drop in a general time-dependent fluid flow, J. Fluid Mech., Volume 37 (1969) no. 3, p. 601–623 | DOI | Zbl

[16] D. A. Di Pietro; A. Ern A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Eng., Volume 283 (2015), pp. 1-21 | DOI | MR | Zbl

[17] D. A. Di Pietro; A. Ern; S. Lemaire An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., Volume 14 (2014) no. 4, pp. 461-472 | DOI | MR | Zbl

[18] D. A. Di Pietro; A. Ern; A. Linke; F. Schieweck A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Eng., Volume 306 (2016), pp. 175-195 | DOI | MR | Zbl

[19] A. Ern; J.-L. Guermond Finite Elements I: Approximation and Interpolation, Texts in Applied Mathematics, 72, Springer, 2021 | DOI

[20] C. Galusinski; P. Vigneaux On stability condition for bifluid flows with surface tension: application to microfluidics, J. Comput. Phys., Volume 227 (2008) no. 12, pp. 6140-6164 | DOI | MR | Zbl

[21] J. Gounley; G. Boedec; M. Jaeger; M. Leonetti Influence of surface viscosity on droplets in shear flow, J. Fluid Mech., Volume 791 (2016), pp. 464-494 | DOI | MR | Zbl

[22] J.-L. Guermond; M. Q. de Luna; T. Thompson A conservative anti-diffusion technique for the level set method, J. Comput. Appl. Math., Volume 321 (2017), pp. 448-468 | DOI | MR

[23] J.-L. Guermond; M. Nazarov A maximum-principle preserving C 0 finite element method for scalar conservation equations, Comput. Methods Appl. Mech. Eng., Volume 272 (2014), pp. 198-213 | DOI | MR | Zbl

[24] J.-L. Guermond; B. Popov Invariant domains and first-order continuous finite element approximation for hyperbolic systems, SIAM J. Numer. Anal., Volume 54 (2016) no. 4, pp. 2466-2489 | DOI | MR

[25] A. Hansbo; P. Hansbo An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., Volume 191 (2002) no. 47-48, pp. 5537-5552 | DOI | MR | Zbl

[26] P. Hansbo; M. G. Larson; S. Zahedi A cut finite element method for a Stokes interface problem, Appl. Numer. Math., Volume 85 (2014), pp. 90-114 | DOI | MR | Zbl

[27] F. Heimann; C. Engwer; O. Ippisch; P. Bastian An unfitted interior penalty discontinuous Galerkin method for incompressible Navier-Stokes two-phase flow, Int. J. Numer. Methods Fluids, Volume 71 (2013) no. 3, pp. 269-293 | DOI | MR | Zbl

[28] A. Huebner; S. Sharma; M. Srisa-Art; F. Hollfelder; J. B. Edel; A. J. deMello Microdroplets: A sea of applications?, Lab Chip, Volume 8 (2008), pp. 1244-1254 | DOI

[29] H. Ji; Q. Zhang A simple finite element method for Stokes flows with surface tension using unfitted meshes, Internat. J. Numer. Methods Fluids, Volume 81 (2016) no. 2, pp. 87-103 | DOI | MR

[30] A. Johansson; M. G. Larson A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary, Numer. Math., Volume 123 (2013) no. 4, pp. 607-628 | DOI | MR | Zbl

[31] M.-J. Lai; L. L. Schumaker Spline functions on triangulations, Encyclopedia of Mathematics and Its Applications, 110, Cambridge University Press, 2007, xvi+592 pages | DOI

[32] E. Olsson; G. Kreiss A conservative level set method for two phase flow, J. Comput. Phys., Volume 210 (2005) no. 1, pp. 225-246 | DOI | MR | Zbl

[33] E. Olsson; G. Kreiss; S. Zahedi A conservative level set method for two phase flow. II, J. Comput. Phys., Volume 225 (2007) no. 1, pp. 785-807 | DOI | MR | Zbl

[34] S. Popinet Numerical models of surface tension (Annual Review of Fluid Mechanics), Volume 50, Annual Reviews, Palo Alto, CA, 2018, pp. 49-75 | MR | Zbl

[35] A. Reusken Analysis of an extended pressure finite element space for two-phase incompressible flows, Comput. Vis. Sci., Volume 11 (2008) no. 4-6, pp. 293-305 | DOI | MR | Zbl

[36] W. E. H. Sollie; O. Bokhove; J. J. W. van der Vegt Space-time discontinuous Galerkin finite element method for two-fluid flows, J. Comput. Phys., Volume 230 (2011) no. 3, pp. 789-817 | DOI | MR | Zbl

[37] M. Sussman; A. S. Almgren; J. B. Bell; P. Colella; L. H. Howell; M. L. Welcome An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys., Volume 148 (1999) no. 1, pp. 81-124 | DOI | MR | Zbl

[38] G. I. Taylor The formation of emulsions in definable fields of flow, Proc. R. Soc. Lond., Ser. A, Volume 146 (1934) no. 858, pp. 501-523

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