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Energy exchange analysis in droplet dynamics via the Navier–Stokes–Cahn–Hilliard model

Published online by Cambridge University Press:  23 May 2016

L. F. R. Espath*
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
A. F. Sarmiento
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia Applied Mathematics and Computational Science, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
P. Vignal
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia Material Science and Engineering, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
B. O. N. Varga
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
A. M. A. Cortes
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
L. Dalcin
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia Extreme Computing Research Center, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia National Scientific and Technical Research Council (CONICET), Santa Fe, Argentina
V. M. Calo
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia Applied Geology Department, Western Australian School of Mines, Faculty of Science and Engineering, Curtin University, Perth, Western Australia, 6845, Australia
*
Email address for correspondence: espath@gmail.com

Abstract

We develop the energy budget equation of the coupled Navier–Stokes–Cahn–Hilliard (NSCH) system. We use the NSCH equations to model the dynamics of liquid droplets in a liquid continuum. Buoyancy effects are accounted for through the Boussinesq assumption. We physically interpret each quantity involved in the energy exchange to gain further insight into the model. Highly resolved simulations involving density-driven flows and the merging of droplets allow us to analyse these energy budgets. In particular, we focus on the energy exchanges when droplets merge, and describe flow features relevant to this phenomenon. By comparing our numerical simulations to analytical predictions and experimental results available in the literature, we conclude that modelling droplet dynamics within the framework of NSCH equations is a sensible approach worthy of further research.

Type
Papers
Copyright
© 2016 Cambridge University Press 

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References

Aarts, D. G. A. L., Lekkerkerker, H. N. W., Guo, H., Wegdam, G. H. & Bonn, D. 2005 Hydrodynamics of droplet coalescence. Phys. Rev. Lett. 95 (16), 164503.Google Scholar
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1), 139165.CrossRefGoogle Scholar
Buffa, A., De Falco, C. & Sangalli, G. 2011a Isogeometric analysis: stable elements for the 2D Stokes equation. Intl J. Numer. Meth. Fluids 65 (11–12), 14071422.Google Scholar
Buffa, A., Rivas, J., Sangalli, G. & Vázquez, R. 2011b Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49 (2), 818844.CrossRefGoogle Scholar
Buffa, A., Sangalli, G. & Vázquez, R. 2010 Isogeometric analysis in electromagnetics: B-splines approximation. Comput. Meth. Appl. Mech. Engng 199 (17), 11431152.Google Scholar
Cahn, J. W. 1959 Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys. 30 (5), 11211124.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (2), 258267.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1959 Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31 (3), 688699.CrossRefGoogle Scholar
Chung, J. & Hulbert, G. M. 1993 A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-𝛼 method. Trans ASME J. Appl. Mech. 60 (2), 371375.Google Scholar
Clift, R., Grace, J. R & Weber, M. E. 2005 Bubbles, Drops, and Particles. Courier Corporation.Google Scholar
Collier, N., Dalcin, L. & Calo, V. M. 2014 On the computational efficiency of isogeometric methods for smooth elliptic problems using direct solvers. Intl J. Numer. Meth. Engng 100 (8), 620632.CrossRefGoogle Scholar
Collier, N., Dalcin, L., Pardo, D. & Calo, V. M. 2013 The cost of continuity: performance of iterative solvers on isogeometric finite elements. SIAM J. Sci. Comput. 35 (2), A767A784.CrossRefGoogle Scholar
Collier, N., Pardo, D., Dalcin, L., Paszynski, M. & Calo, V. M. 2012 The cost of continuity: a study of the performance of isogeometric finite elements using direct solvers. Comput. Meth. Appl. Mech. Engng 213, 353361.Google Scholar
Côrtes, A. M. A., Coutinho, A. L. G. A., Dalcin, L. & Calo, V. M. 2015 Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system. J. Comput. Sci 11, 123136.Google Scholar
Côrtes, A. M. A., Vignal, P., Sarmiento, A., García, D., Collier, N., Dalcin, L. & Calo, V. M. 2014 Solving nonlinear, high-order partial differential equations using a high-performance isogeometric analysis framework. In High Performance Computing, Communications in Computer and Information Science, vol. 485, pp. 236247. Springer.Google Scholar
Dalcin, L., Collier, N., Vignal, P., Côrtes, A. M. A. & Calo, V. M.2015 PetIGA: a framework for high-performance isogeometric analysis. Preprint arXiv:1305.4452.CrossRefGoogle Scholar
Eggers, J., Lister, J. R. & Stone, A. H. 1999 Coalescence of liquid drops. J. Fluid Mech. 401, 293310.CrossRefGoogle Scholar
Emmerich, H. 2003 The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts and Applications of Phase-Field Models, Lecture Notes in Physics, vol. 73. Springer.Google Scholar
Espath, L. F. R., Braun, A. L., Awruch, A. M. & Dalcin, L. D. 2015a A nurbs-based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach. Intl J. Numer. Meth. Engng 102 (13), 18391868.Google Scholar
Espath, L. F. R., Pinto, L. C., Laizet, S. & Silvestrini, J. H. 2015b High-fidelity simulations of the lobe-and-cleft structures and the deposition map in particle-driven gravity currents. Phys. Fluids 27 (5), 056604.Google Scholar
Evans, J. A. & Hughes, T. J. R. 2013a Isogeometric divergence-conforming B-splines for the Darcy–Stokes–Brinkman equations. Math. Models Meth. Appl. Sci. 23 (04), 671741.Google Scholar
Evans, J. A. & Hughes, T. J. R. 2013b Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math. Models Meth. Appl. Sci. 23 (08), 14211478.CrossRefGoogle Scholar
Evans, J. A. & Hughes, T. J. R. 2013c Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. J. Comput. Phys. 241, 141167.CrossRefGoogle Scholar
Gomez, H., Cueto-Felgueroso, L. & Juanes, R. 2013 Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium. J. Comput. Phys. 238, 217239.Google Scholar
Gomez, H., Hughes, T. J. R., Nogueira, X. & Calo, V. M. 2010 Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Comput. Meth. Appl. Mech. Engng 199 (25), 18281840.Google Scholar
Gómez, H., Calo, V. M., Bazilevs, Y. & Hughes, T. J. R. 2008 Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput. Meth. Appl. Mech. Engng 197 (49), 43334352.CrossRefGoogle Scholar
Gomez, H. & Nogueira, X. 2012a A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional. Commun. Nonlinear Sci. Numer. Simul. 17 (12), 49304946.CrossRefGoogle Scholar
Gomez, H. & Nogueira, X. 2012b An unconditionally energy-stable method for the phase field crystal equation. Comput. Meth. Appl. Mech. Engng 249, 5261.Google Scholar
Guo, Z. & Lin, P. 2015 A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects. J. Fluid Mech. 766, 226271.CrossRefGoogle Scholar
Guo, Z., Lin, P. & Lowengrub, J. S. 2014 A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486507.CrossRefGoogle Scholar
Gurtin, M. E. 1996 Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D 92 (3), 178192.Google Scholar
Gurtin, M. E., Fried, E. & Anand, L. 2010 The Mechanics and Thermodynamics of Continua. Cambridge University Press.Google Scholar
Gurtin, M. E., Polignone, D. & Viñals, J. 1996 Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Meth. Appl. Sci. 6 (06), 815831.CrossRefGoogle Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1), 96127.CrossRefGoogle Scholar
Jamet, D., Lebaigue, O., Coutris, N. & Delhaye, J. M. 2001 The second gradient method for the direct numerical simulation of liquid–vapor flows with phase change. J. Comput. Phys. 169 (2), 624651.Google Scholar
Jansen, K. E., Whiting, C. H. & Hulbert, M. G. 2000 A generalized-𝛼 method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Comput. Meth. Appl. Mech. Engng 190 (3), 305319.CrossRefGoogle Scholar
Josserand, C. & Thoroddsen, S. T. 2016 Drop impact on solid surface. Annu. Rev. Fluid Mech. 48 (1), 365391.Google Scholar
Kavehpour, H. P. 2015 Coalescence of drops. Annu. Rev. Fluid Mech. 47, 245268.Google Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2006 On scaling of diffuse–interface models. Chem. Engng Sci. 61 (8), 23642378.CrossRefGoogle Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2007 Capillary spreading of a droplet in the partially wetting regime using a diffuse–interface model. J. Fluid Mech. 572, 367387.CrossRefGoogle Scholar
Kühnel, W. 2006 Differential Geometry: Curves-Surfaces-Manifolds, Student Mathematical Library, vol. 16. American Mathematical Society.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Cambridge University Press.Google Scholar
marquis de Laplace, P. S. 1805 Traité de mécanique céleste. vol. 4. Courcier.Google Scholar
Liu, C. & Walkington, N. J. 2000 Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37 (3), 725741.Google Scholar
Liu, J.2014 Thermodynamically consistent modeling and simulation of multiphase flows. PhD thesis, The University of Texas at Austin.Google Scholar
Loginova, I., Amberg, G. & Ågren, J. 2001 Phase-field simulations of non-isothermal binary alloy solidification. Acta Materialia 49 (4), 573581.Google Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454 (1978), 26172654.Google Scholar
Meiburg, E. & Kneller, B. 2010 Turbidity currents and their deposits. Annu. Rev. Fluid Mech. 42, 135156.Google Scholar
Myers, D. 1990 Surfaces, Interfaces and Colloids. Wiley-VCH.Google Scholar
Rider, W. J. & Kothe, D. B. 1998 Reconstructing volume tracking. J. Comput. Phys. 141 (2), 112152.Google Scholar
Sarmiento, A., Cortes, A. M. A., Garcia, D., Dalcin, L., Collier, N. & Calo, V. M. 2015 PetIGA-MF: a multi-field high-performance toolbox for structure-preserving B-splines spaces. J. Comput. Sci. (submitted) arXiv:1602.08727.Google Scholar
Sethian, J. A. 1999 Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 3rd edn. Cambridge University Press.Google Scholar
Spatschek, R., Müller-Gugenberger, C., Brener, E. & Nestler, B. 2007 Phase field modeling of fracture and stress-induced phase transitions. Phys. Rev. E 75 (6), 066111.Google Scholar
Thoroddsen, S. T., Takehara, K. & Etoh, T. G. 2005 The coalescence speed of a pendent and a sessile drop. J. Fluid Mech. 527, 85114.Google Scholar
Vignal, P., Dalcin, L., Brown, D. L., Collier, N. & Calo, V. M. 2015a An energy-stable convex splitting for the phase-field crystal equation. Comput. Struct. 158, 355368.Google Scholar
Vignal, P., Sarmiento, A., Côrtes, A. M. A., Dalcin, L. & Calo, V. M. 2015b Coupling Navier–Stokes and Cahn–Hilliard equations in a two-dimensional annular flow configuration. Proc. Comput. Sci. 51, 934943.Google Scholar
Vignal, P. A., Collier, N. & Calo, V. M. 2013 Phase field modeling using petiga. Proc. Comput. Sci. 18, 16141623.Google Scholar
Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 6587.Google Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293317.Google Scholar

Espath et al. supplementary movie

Phase-field evolution, velocities, and energies: simulation case #2.

Download Espath et al. supplementary movie(Video)
Video 1.9 MB

Espath et al. supplementary movie

Phase-field evolution, velocities, and energies: simulation case #3.

Download Espath et al. supplementary movie(Video)
Video 2 MB