Abstract
At small scales, the interaction of multicomponent fluids and solids can be dominated by capillary forces giving rise to elastocapillarity. Surface tension may deform or even collapse slender structures and thus, cause important damage in microelectromechanical systems. However, under control, elastocapillarity could be used as a fabrication technique for the design of new materials and structures. Here, we propose a computational model for elastocapillarity that couples nonlinear hyperelastic solids with two-component immiscible fluids described by the Navier–Stokes–Cahn–Hilliard equations. As fluid–structure interaction computational technique, we employ a boundary-fitted approach. For the spatial discretization of the problem we adopt a NURBS-based isogeometric analysis methodology. A strongly-coupled algorithm is proposed for the solution of the problem. The potential of this model is illustrated by solving several numerical examples, including, capillary origami, the static wetting of soft substrates, the deformation of micropillars and the three dimensional wrapping of a liquid droplet.
Similar content being viewed by others
Change history
03 July 2017
An erratum to this article has been published.
References
Aarts DGAL, Lekkerkerker HNW, Guo H, Wegdam GH, Bonn D (2005) Hydrodynamics of droplet coalescence. Phys Rev Lett 95:164503
Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 199(58):229–263
Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput Mech 43(1):3–37
Bazilevs Y, Hsu M-C, Scott MA (2012) Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput Methods Appl Mech Eng 249–252:28–41
Bazilevs Y, Takizawa K, Tezduyar TE, Hsu M-C, Kostov N, McIntyre S (2014) Aerodynamic and FSI analysis of wind turbines with the ALE-VMS and ST-VMS methods. Arch Comput Methods Eng 21(4):359–398
Bazilevs Y, Takizawa K, Tezduyar TE (2013) Computational fluid–structure interaction. Methods and applications. Wiley, London
Bazilevs Y, Takizawa K, Tezduyar TE, Hsu M-C, Kostov N, McIntyre S (2014) Aerodynamic and FSI analysis of wind turbines with the ALE-VMS and ST-VMS methods. Arch Comput Methods Eng 21(4):359–398
Beirao da Veiga L, Buffa A, Sangalli G, Vazquez R (2013) Analysis suitable T-splines of arbitrary degree: definition, linear independence, and approximation properties. Math Models Methods Appl Sci 23(11):1979–2003
Bico J, Roman B, Moulin L, Boudaoud A (2004) Adhesion: elastocapillary coalescence in wet hair. Nature 432(7018):690–690
Bostwick JB, Daniels KE (2013) Capillary fracture of soft gels. Phys Rev E 88(4):042410
Brennen CE (2005) Fundamentals of multiphase flow. Cambridge University Press, Cambridge
Bueno J, Bazilevs Y, Juanes R, Gomez H (2017) Droplet motion driven by tensotaxis. Extrem Mech Lett 13:10–16
Bueno J, Bona-Casas C, Bazilevs Y, Gomez H (2015) Interaction of complex fluids and solids: theory, algorithms and application to phase-change-driven implosion. Comput Mech 55(6):1105–1118
Bueno J, Gomez H (2016) Liquid-vapor transformations with surfactants. Phase-field model and isogeometric analysis. J Comput Phys 321:797–818
Casquero H, Bona-Casas C, Gomez H (2015) A NURBS-based immersed methodology for fluid-structure interaction. Comput Methods Appl Mech Eng 284:943–970
Casquero H, Bona-Casas C, Gomez H (2017) NURBS-based numerical proxies for red blood cells and circulating tumor cells in microscale blood flow. Comput Methods Appl Mech Eng 316:646–667 (2017 Special Issue on Isogeometric Analysis: Progress and Challenges)
Casquero H, Lei L, Zhang J, Reali A, Gomez H (2016) Isogeometric collocation using analysis-suitable T-splines of arbitrary degree. Comput Methods Appl Mech Eng 301:164–186
Casquero H, Lei L, Zhang Y, Reali A, Kiendl J, Gomez H (2017) Arbitrary-degree T-splines for isogeometric analysis of fully nonlinear Kirchhoff–Love shells. Comput Aided Design 82:140–153
Casquero H, Liu L, Bona-Casas C, Zhang Y, Gomez H (2016) A hybrid variational-collocation immersed method for fluid–structure interaction using unstructured T-splines. Int J Numer Methods Eng 105(11):855–880
Cerda E, Mahadevan L (2003) Geometry and physics of wrinkling. Phys Rev Lett 90(7):074302
Chakrapani N, Wei B, Carrillo A, Ajayan PM, Kane RS (2004) Capillarity-driven assembly of two-dimensional cellular carbon nanotube foams. Proc Natl Acad Sci 101(12):4009–4012
Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-\(\alpha\) method. J Appl Mech 60:371–375
Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis toward integration of CAD and FEA. Wiley, London
de Gennes PG (1985) Wetting: statics and dynamics. Revi Mod Phys 57:827–863
DeVolder M, Hart AJ (2013) Engineering hierarchical nanostructures by elastocapillary self-assembly. Angew Chem Int Ed 52(9):2412–2425
Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, London
Donea J, Huerta A, Ponthot J-Ph, Rodrguez-Ferran A (2004) Encyclopedia of computational mechanics. Arbitrary Lagrangian–Eulerian methods, chapter 14, vol 1. Wiley, London
Duprat C, Bick AD, Warren PB, Stone HA (2013) Evaporation of drops on two parallel fibers: influence of the liquid morphology and fiber elasticity. Langmuir 29(25):7857–7863 PMID: 23705986
Duprat C, Protiere S, Beebe AY, Stone HA (2012) Wetting of flexible fibre arrays. Nature 482(7386):510–513
Eggers J, Lister JR, Stone HA (1999) Coalescence of liquid drops. J Fluid Mech 401:293–310
Gomez H, Calo VM, Bazilevs Y, Hughes TJR (2008) Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput Methods Appl Mech Eng 197:43334352
Gomez H, Hughes TJR (2011) Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J Comput Phys 230(13):5310–5327
Gomez H, Reali A, Sangalli G (2014) Accurate, efficient, and (iso) geometrically flexible collocation methods for phase-field models. J Comput Phys 262:153–171
Gomez H, van der Zee K (2016) Encyclopedia of computational mechanics. Computational phase-field modeling. Wiley, London
Hsu M-C, Akkerman I, Bazilevs Y (2014) Finite element simulation of wind turbine aerodynamics: validation study using nrel phase vi experiment. Wind Energy 17(3):461–481
Huang J, Juszkiewicz M, de Jeu WH, Cerda E, Emrick T, Menon N, Russell TP (2007) Capillary wrinkling of floating thin polymer films. Science 317(5838):650–653
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195
Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian–Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29(3):329–349
Jansen KE, Whiting CH, Hulbert GM (2000) A generalized-\(\alpha\) method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190(34):305–319
Jeong JH, Goldenfeld N, Dantzig JA (2001) Phase field model for three-dimensional dendritic growth with fluid flow. Phys Rev E 64:041602
Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput Methods Appl Mech Eng 119:73–94
Kamensky D, Hsu MC, Schillinger D, Evans JA, Aggarwal A, Bazilevs Y, Sacks MS, Hughes TJR (2015) An immersogeometric variational framework for fluid–structure interaction. Comput Methods Appl Mech Eng 284:1005–1053
Kamensky D, Hsu M-C, Yu Y, Evans JA, Sacks MS, Hughes TJR (2017) Immersogeometric cardiovascular fluid–structure interaction analysis with divergence-conforming b-splines. Comput Methods Appl Mech Eng 314:408–472
King RJ (1982) Pulmonary surfactant. J Appl Physiol 53(1):1–8
Liu J, Landis CM, Gomez H, Hughes TJR (2015) Liquid-vapor phase transition: thermomechanical theory, entropy stable numerical formulation, and boiling simulations. Comput Methods Appl Mech Eng 297:476–553
Lorenzo G, Scott MA, Tew K, Hughes TJR, Zhang YJ, Liu L, Vilanova G, Gomez H (2016) Tissue-scale, personalized modeling and simulation of prostate cancer growth. Proc Natl Acad Sci 113(48):E7663–E7671
Moure A, Gomez H (2016) Computational model for amoeboid motion: coupling membrane and cytosol dynamics. Phys Rev E 94(4):042423
Prosperetti A, Tryggvason G (2009) Comput methods for multiphase flow. Cambridge University Press, Cambridge
Py C, Reverdy P, Doppler L, Bico J, Roman B, Baroud CN (2007) Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. Phys Rev Lett 98:156103
Raccurt O, Tardif F, d’Avitaya FA, Vareine T (2004) Influence of liquid surface tension on stiction of SOI MEMS. J Micromech Microeng 14(7):1083
Roman B, Bico J (2010) Elasto-capillarity: deforming an elastic structure with a liquid droplet. J Phys Condens Matter 22(49):493101
Shao D, Levine H, Rappel W-J (2012) Coupling actin flow, adhesion, and morphology in a computational cell motility model. Proc Natl Acad Sci 109(18):6851–6856
Sigrist J-F (2015) Fluid–structure interaction. Wiley, London
Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New Yoirk
Stein K, Benney R, Kalro V, Tezduyar TE, Leonard J, Accorsi M (2000) Parachute fluid-structure interactions: 3-D computation. Comput Methods Appl Mech Eng 190(3):373–386
Stein K, Tezduyar T, Benney R (2003) Mesh moving techniques for fluid–structure interactions with large displacements. J Appl Mech 70:58–63
Style RW, Boltyanskiy R, Che Y, Wettlaufer JS, Wilen LA, Dufresne ER (2013) Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses. Phys Rev Lett 110:066103
Style RW, Jagota A, Hui C-Y, Dufresne ER (2016) Elastocapillarity: surface tension and the mechanics of soft solids. arXiv preprint arXiv:1604.02052
Takizawa K (2014) Computational engineering analysis with the new-generation space-time methods. Comput Mech 54:193–211
Takizawa K, Bazilevs Y, Tezduyar TE (2012) Space-time and ALE-VMS techniques for patient-specific cardiovascular fluid–structure interaction modeling. Arch Comput Methods Eng 19(2):171–225
Takizawa K, Bazilevs Y, Tezduyar TE, Long CC, Marsden AL, Schjodt K (2014) ST and ALE-VMS methods for patient-specific cardiovascular fluid mechanics modeling. Math Models Methods Appl Sci 24:2437–2486
Takizawa K, Tezduyar TE, Terahara T, Sasaki T (2016) Heart valve flow computation with the integrated space–time VMS, slip interface, topology change and isogeometric discretization methods. Comput Fluids. doi:10.1016/j.compfluid.2016.11.012
Tanaka T, Morigami M, Atoda N (1993) Mechanism of resist pattern collapse during development process. Jpn J Appl Phys 32(12S):6059
Taroni M, Vella D (2012) Multiple equilibria in a simple elastocapillary system. J Fluid Mech 712:273–294
Tawfick SH, Bico J, Barcelo S (2016) Three-dimensional lithography by elasto-capillary engineering of filamentary materials. MRS Bull 41(02):108–114
Tezduyar TE (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch Comput Methods Eng 8:83–130
Tezduyar TE, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26(10):27–36
Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space-time finite elements: solution techniques. Int J Numer Methods Fluids 54:855–900
Travasso RDM, Poiré EC, Castro M, Rodrguez-Manzaneque JC, Hernández-Machado A (2011) Tumor angiogenesis and vascular patterning: a mathematical model. PloS ONE 6(5):e19989
Vahidkhah K, Balogh P, Bagchi P (2016) Flow of red blood cells in stenosed microvessels. Sci Rep 6:28194. doi:10.1038/srep28194
Vilanova G, Colominas I, Gomez H (2017) A mathematical model of tumour angiogenesis: growth, regression and regrowth. J R Soc Interface 14(126):20160918
Wei X, Zhang YJ, Hughes TJR, Scott MA (2015) Truncated hierarchical Catmull–Clark subdivision with local refinement. Comput Methods Appl Mech Eng 291:1–20
Wei X, Zhang YJ, Hughes TJR, Scott MA (2016) Extended truncated hierarchical Catmull–Clark subdivision. Comput Methods Appl Mech Eng 299:316–336
Xu J, Vilanova G, Gomez H (2017) Full-scale, three-dimensional simulation of early-stage tumor growth: the onset of malignancy. Comput Methods Appl Mech Eng 314:126–146
Zhang L, Gerstenberger A, Wang X, Liu WK (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193(21):2051–2067
Acknowledgements
The authors would like to thank Robert Style for the experimental data used in Fig. 4.
Funding
HG and HC were partially supported by the European Research Council through the FP7 Ideas Starting Grant Program (Contract #307201). HG and JB were partially supported by Xunta de Galicia, co-financed with FEDER funds. YB was supported by AFOSR Grant No. FA9550-16-1-0131.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
The original version of this article was revised because due to an unfortunate turn of events during processing of this article essential data was omitted from figure 5. Figure 5 has been updated with the correct version that should be regarded as the final version by the reader.
An erratum to this article is available at https://doi.org/10.1007/s11012-017-0699-9.
Rights and permissions
About this article
Cite this article
Bueno, J., Casquero, H., Bazilevs, Y. et al. Three-dimensional dynamic simulation of elastocapillarity. Meccanica 53, 1221–1237 (2018). https://doi.org/10.1007/s11012-017-0667-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-017-0667-4