Hostname: page-component-7d684dbfc8-hsbzg Total loading time: 0 Render date: 2023-09-26T07:26:23.925Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false
  • English
  • Français

The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids

Published online by Cambridge University Press:  02 January 2013

F. Magaletti ,
F. Picano ,
M. Chinappi ,
L. Marino  and
C. M. Casciola
F. Magaletti
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
F. Picano
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy Linné Flow Center, KTH Mechanics, Osquars Backe 18, SE-100 44 Stockholm, Sweden
M. Chinappi
Affiliation:
Dipartimento di Fisica, Università di Roma La Sapienza, P. le Aldo Moro 5, 00185 Roma, Italy
L. Marino
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
C. M. Casciola*
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy
*
†Email address for correspondence: carlomassimo.casciola@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.

JFM classification

Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Drops and Bubbles Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 95 - 126
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aris, R. 1989 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Badalassi, V. E., Ceniceros, H. D. & Banerjee, S. 2003 Computation of multiphase systems with phase field models. J. Comput. Phys. 190 (2), 371–397.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interface free energy. J. Chem. Phys. 28 (2), 258–267.CrossRefGoogle 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, 688.CrossRefGoogle Scholar
Carlson, A., Bellani, G. & Amberg, G. 2012 Contact line dissipation in short-time dynamic wetting. EPL (Europhys. Lett.) 97, 44004.CrossRefGoogle Scholar
Carlson, A., Do-Quang, M. & Amberg, G. 2009 Modeling of dynamic wetting far from equilibrium. Phys. Fluids 21, 121701.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Ding, H., Gilani, M. N. H. & Spelt, P. D. M. 2010 Sliding, pinch-off and detachment of a droplet on a wall in shear flow. J. Fluid Mech. 644 (217), 56.CrossRefGoogle Scholar
Fisk, S. & Widom, B. 1969 Structure and free energy of the interface between fluid phases in equilibrium near the critical point. J. Chem. Phys. 50, 3219.CrossRefGoogle Scholar
Giacomello, A., Meloni, S., Chinappi, M. & Casciola, C. M. 2012 Cassie–Baxter and Wenzel states on a nanostructured surface: phase diagram, metastabilities, and transition mechanism by atomistic free energy calculations. Langmuir 28, 10764–10772.CrossRefGoogle ScholarPubMed
Gurtin, M. E., Polignone, D. & Vinals, J. 1995 Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 06, 815, doi:10.1142/S0218202596000341.CrossRefGoogle Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries* 1. J. Comput. Phys. 39 (1), 201–225.CrossRefGoogle Scholar
Huang, J. J., Shu, C. & Chew, Y. T. 2009 Mobility-dependent bifurcations in capillarity-driven two-phase fluid systems by using a lattice Boltzmann phase-field model. Intl J. Numer. Meth. Fluids 60 (2), 203–225.CrossRefGoogle Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1), 96–127.CrossRefGoogle Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.CrossRefGoogle Scholar
Jeng, U. et al. 1998 Viscosity effect on capillary waves at liquid interfaces. J. Phys.: Condens. Matter 10, 4955.Google Scholar
Kevorkian, J. & Cole, J. D. 1996 Multiple Scale and Singular Perturbation Methods, vol. 114. Springer.CrossRefGoogle Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2006 On scaling of diffuse-interface models. Chem. Engng Sci. 61 (8), 2364–2378.CrossRefGoogle Scholar
Kim, J. 2005 A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204 (2), 784–804.CrossRefGoogle Scholar
Lighthill, J. 2001 Waves in Rluids. Cambridge University Press.Google Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi–incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. Ser. A: Math., Phys. Engng Sci. 454, 2617.CrossRefGoogle Scholar
Marmottant, P. & Villermaux, E. et al. 2004 On spray formation. J. Fluid Mech. 498 (1), 73–111.CrossRefGoogle Scholar
Menard, T., Tanguy, S. & Berlemont, A. 2007 Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary break-up of a liquid jet. Intl J. Multiphase Flow 33 (5), 510–524.CrossRefGoogle Scholar
Mitrinović, D. M., Tikhonov, A. M., Li, M., Huang, Z. & Schlossman, M. L. 2000 Noncapillary-wave structure at the water–alkane interface. Phys. Rev. Lett. 85, 582–585.CrossRefGoogle ScholarPubMed
Nijmeijer, M. J. P., Bakker, A. F., Bruin, C. & Sikkenk, J. H. 1988 A molecular dynamics simulation of the Lennard–Jones liquid–vapour interface. J. Chem. Phys. 89, 3789.CrossRefGoogle Scholar
Orlandini, S., Meloni, S. & Ciccotti, G. 2011 Hydrodynamics from statistical mechanics: combined dynamical-NEMD and conditional sampling to relax an interface between two immiscible liquids. Phys. Chem. Chem. Phys. 13 (29), 13177–13181.CrossRefGoogle ScholarPubMed
Prosperetti, A. 1981 Motion of two superposed viscous fluids. Phys. Fluids 24, 1217.CrossRefGoogle Scholar
Qian, T., Wang, X. P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360.CrossRefGoogle Scholar
Ren, W. et al. 2011 Derivation of continuum models for the moving contact line problem based on thermodynamic principles. Commun. Math. Sci. 9 (2), 597–606.CrossRefGoogle Scholar
Renardy, Y. & Renardy, M. 2002 Prost: a parabolic reconstruction of surface tension for the volume-of-fluid method. J. Comput. Phys. 183 (2), 400–421.CrossRefGoogle Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34 (9), 977–992.CrossRefGoogle Scholar
Sussman, M. & Puckett, E. G. 2000 A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162 (2), 301–337.CrossRefGoogle Scholar
Thorsen, T., Roberts, R. W., Arnold, F. H. & Quake, S. R. 2001 Dynamic pattern formation in a vesicle-generating microfluidic device. Phys. Rev. Lett. 86 (18), 4163–4166.CrossRefGoogle Scholar
Triezenberg, D. G. & Zwanzig, R. 1972 Fluctuation theory of surface tension. Phys. Rev. Lett. 28 (18), 1183–1185.CrossRefGoogle Scholar
Truesdell, C. & Toupin, R. A. 1960 Principles of classical mechanics and field theory. In Encyclopedia of Physics (ed. Flügge, S.). vol. 3. Springer, Part 1.Google Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y. J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169 (2), 708–759.CrossRefGoogle Scholar
Unverdi, S. O. & Tryggvason, G. 1992 A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100 (1), 25–37.CrossRefGoogle Scholar
Vollmayr-Lee, B. P. & Rutenberg, A. D. 2003 Fast and accurate coarsening simulation with an unconditionally stable time step. Phys. Rev. E 68 (6), 66703.CrossRefGoogle ScholarPubMed
Van der Waals, JD 1979 The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J. Stat. Phys. 20 (2), 200–244.CrossRefGoogle 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, 293–317.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 129 (3), 163–176.CrossRefGoogle Scholar
Yue, P., Zhou, C. & Feng, J. J. 2010 Sharp-interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279–294.CrossRefGoogle Scholar