Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T02:35:18.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

Published online by Cambridge University Press:  27 September 2019

Guangpu Zhu Open the ORCID record for Guangpu Zhu [Opens in a new window] ,
Jisheng Kou ,
Bowen Yao ,
Yu-shu Wu ,
Jun Yao  and
Shuyu Sun
Guangpu Zhu
Affiliation:
Research Center of Multiphase Flow in Porous Media, School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China
Jisheng Kou
Affiliation:
School of Civil Engineering, Shaoxing University, Shaoxing 312000, Zhejiang, China School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, Hubei, China
Bowen Yao
Affiliation:
Department of Petroleum Engineering, Colorado School of Mines, 1600 Arapahoe Street, Golden, CO 80401, USA
Yu-shu Wu
Affiliation:
Department of Petroleum Engineering, Colorado School of Mines, 1600 Arapahoe Street, Golden, CO 80401, USA
Jun Yao*
Affiliation:
Research Center of Multiphase Flow in Porous Media, School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China
Shuyu Sun*
Affiliation:
Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Droplet dynamics on a solid substrate is significantly influenced by surfactants. It remains a challenging task to model and simulate the moving contact line dynamics with soluble surfactants. In this work, we present a derivation of the phase-field moving contact line model with soluble surfactants through the first law of thermodynamics, associated thermodynamic relations and the Onsager variational principle. The derived thermodynamically consistent model consists of two Cahn–Hilliard type of equations governing the evolution of interface and surfactant concentration, the incompressible Navier–Stokes equations and the generalized Navier boundary condition for the moving contact line. With chemical potentials derived from the free energy functional, we analytically obtain certain equilibrium properties of surfactant adsorption, including equilibrium profiles for phase-field variables, the Langmuir isotherm and the equilibrium equation of state. A classical droplet spread case is used to numerically validate the moving contact line model and equilibrium properties of surfactant adsorption. The influence of surfactants on the contact line dynamics observed in our simulations is consistent with the results obtained using sharp interface models. Using the proposed model, we investigate the droplet dynamics with soluble surfactants on a chemically patterned surface. It is observed that droplets will form three typical flow states as a result of different surfactant bulk concentrations and defect strengths, specifically the coalescence mode, the non-coalescence mode and the detachment mode. In addition, a phase diagram for the three flow states is presented. Finally, we study the unbalanced Young stress acting on triple-phase contact points. The unbalanced Young stress could be a driving or resistance force, which is determined by the critical defect strength.

JFM classification

Interfacial Flows (free surface): Contact lines Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 327 - 359
Check for updates
Copyright
© 2019 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

Alke, A. & Bothe, D. 2009 3D numerical modeling of soluble surfactant at fluidic interfaces based on the volume-of-fluid method. Fluid Dyn. Mater. Process. 5 (4), 345–372.Google Scholar
Alpak, F., Samardžić, A. & Frank, F. 2018 A distributed parallel direct simulator for pore-scale two-phase flow on digital rock images using a finite difference implementation of the phase-field method. J. Petrol. Sci. Engng 166, 806–824.10.1016/j.petrol.2017.11.022Google Scholar
Alpak, F. O., Riviere, B. & Frank, F. 2016 A phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition. Comput. Geosci. 20 (5), 881–908.10.1007/s10596-015-9551-2Google Scholar
Aniszewski, W., Arrufat, T., Crialesi-Esposito, M., Dabiri, S., Fuster, D., Ling, Y., Lu, J., Malan, L., Pal, S. & Scardovelli, R. 2019 Parallel, Robust, Interface Simulator (PARIS).Google Scholar
Blake, T. D., Fernandez-Toledano, J.-C., Doyen, G. & De Coninck, J. 2015 Forced wetting and hydrodynamic assist. Phys. Fluids 27 (11), 112101.10.1063/1.4934703Google Scholar
Booty, M. & Siegel, M. 2010 A hybrid numerical method for interfacial fluid flow with soluble surfactant. J. Comput. Phys. 229 (10), 3864–3883.10.1016/j.jcp.2010.01.032Google Scholar
Cai, X., Marschall, H., Wörner, M. & Deutschmann, O. 2014 A phase field method with adaptive mesh refinement for numerical simulation of 3D wetting processes with OpenFOAM® . In 2nd International Symposium on Multiscale Multiphase Process Engineering (MMPE), Hamburg, Germany. DECHEMA.Google Scholar
Chen, H., Sun, S. & Zhang, T. 2018 Energy stability analysis of some fully discrete numerical schemes for incompressible Navier–Stokes equations on staggered grids. J. Sci. Comput. 75 (1), 427–456.10.1007/s10915-017-0543-3Google Scholar
Chen, J., Sun, S. & Wang, X.-P. 2014 A numerical method for a model of two-phase flow in a coupled free flow and porous media system. J. Comput. Phys. 268, 1–16.10.1016/j.jcp.2014.02.043Google Scholar
Copetti, M. & Elliott, C. M. 1992 Numerical analysis of the Cahn–Hilliard equation with a logarithmic free energy. Numer. Math. 63 (1), 39–65.10.1007/BF01385847Google Scholar
Cuenot, B., Magnaudet, J. & Spennato, B. 1997 The effects of slightly soluble surfactants on the flow around a spherical bubble. J. Fluid Mech. 339, 25–53.10.1017/S0022112097005053Google Scholar
Ding, H. & Spelt, P. D. 2007 Wetting condition in diffuse interface simulations of contact line motion. Phys. Rev. E 75 (4), 046708.Google Scholar
Dupuis, A. & Yeomans, J. 2005 Modeling droplets on superhydrophobic surfaces: equilibrium states and transitions. Langmuir 21 (6), 2624–2629.10.1021/la047348iGoogle Scholar
Engblom, S., Do-Quang, M., Amberg, G. & Tornberg, A.-K. 2013 On diffuse interface modeling and simulation of surfactants in two-phase fluid flow. Commun. Comput. Phys. 14 (4), 879–915.10.4208/cicp.120712.281212aGoogle Scholar
Fan, W., Sun, H., Yao, J., Fan, D. & Yang, Y. 2019 Homogenization approach for liquid flow within shale system considering slip effect. J. Cleaner Prod. 235, 146–157.10.1016/j.jclepro.2019.06.290Google Scholar
Frank, F., Liu, C., Alpak, F. O., Berg, S. & Riviere, B. 2018 Direct numerical simulation of flow on pore-scale images using the phase-field method. SPE J. 23 (5), 1833–1850.10.2118/182607-PAGoogle Scholar
Gao, M. & Wang, X.-P. 2012 A gradient stable scheme for a phase field model for the moving contact line problem. J. Comput. Phys. 231 (4), 1372–1386.10.1016/j.jcp.2011.10.015Google Scholar
Gao, M. & Wang, X.-P. 2014 An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity. J. Comput. Phys. 272, 704–718.10.1016/j.jcp.2014.04.054Google Scholar
Garcke, H., Lam, K. F. & Stinner, B. 2014 Diffuse interface modelling of soluble surfactants in two-phase flow. Commun. Math. Sci. 12 (8), 1475–1522.10.4310/CMS.2014.v12.n8.a6Google Scholar
Guermond, J.-L. & Salgado, A. 2009 A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228 (8), 2834–2846.10.1016/j.jcp.2008.12.036Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.10.1017/S0022112099006874Google Scholar
James, A. J. & Lowengrub, J. 2004 A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 201 (2), 685–722.10.1016/j.jcp.2004.06.013Google Scholar
Khatri, S. & Tornberg, A.-K. 2014 An embedded boundary method for soluble surfactants with interface tracking for two-phase flows. J. Comput. Phys. 256, 768–790.10.1016/j.jcp.2013.09.019Google Scholar
Komura, S. & Kodama, H. 1997 Two-order-parameter model for an oil–water-surfactant system. Phys. Rev. E 55 (2), 1722–1727.Google Scholar
Kou, J. & Sun, S. 2018a Thermodynamically consistent modeling and simulation of multi-component two-phase flow with partial miscibility. Comput. Meth. Appl. Mech. Engng 331, 623–649.10.1016/j.cma.2017.11.023Google Scholar
Kou, J. & Sun, S. 2018b Thermodynamically consistent simulation of nonisothermal diffuse-interface two-phase flow with Peng–Robinson equation of state. J. Comput. Phys. 371, 581–605.10.1016/j.jcp.2018.05.047Google Scholar
Kou, J., Sun, S. & Wang, X. 2018 Linearly decoupled energy-stable numerical methods for multicomponent two-phase compressible flow. SIAM J. Numer. Anal. 56 (6), 3219–3248.10.1137/17M1162287Google Scholar
Lai, M.-C., Tseng, Y.-H. & Huang, H. 2010 Numerical simulation of moving contact lines with surfactant by immersed boundary method. Commun. Comput. Phys. 8 (4), 735.10.4208/cicp.281009.120210aGoogle Scholar
Laradji, M., Guo, H., Grant, M. & Zuckermann, M. J. 1992 The effect of surfactants on the dynamics of phase separation. J. Phys.: Condens. Matter 4 (32), 6715.Google Scholar
Li, J., Yu, B., Wang, Y., Tang, Y. & Wang, H. 2015 Study on computational efficiency of composite schemes for convection-diffusion equations using single-grid and multigrid methods. J. Therm. Sci. Technol. 10 (1), JTST0009-JTST.Google Scholar
Li, Y. & Kim, J. 2012 A comparison study of phase-field models for an immiscible binary mixture with surfactant. Eur. Phys. J. B 85 (10), 340.10.1140/epjb/e2012-30184-1Google Scholar
Liu, H., Ba, Y., Wu, L., Li, Z., Xi, G. & Zhang, Y. 2018 A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants. J. Fluid Mech. 837, 381–412.10.1017/jfm.2017.859Google Scholar
Liu, H. & Zhang, Y. 2010 Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 229 (24), 9166–9187.10.1016/j.jcp.2010.08.031Google Scholar
Liu, P., Yan, X., Yao, J. & Sun, S. 2019 Modeling and analysis of the acidizing process in carbonate rocks using a two-phase thermal-hydrologic-chemical coupled model. Chem. Engng Sci. 207, 215–234.10.1016/j.ces.2019.06.017Google Scholar
Malan, L., Ling, Y., Scardovelli, R., Llor, A. & Zaleski, S. 2019 Detailed numerical simulations of pore competition in idealized micro-spall using the VOF method. Comput. Fluids. 189, 60–72.10.1016/j.compfluid.2019.05.011Google Scholar
Moukalled, F., Mangani, L. & Darwish, M. 2016 The finite volume method in computational fluid dynamics. In An Advanced Introduction with OpenFOAM and Matlab, pp. 436–439. Springer.Google Scholar
Muradoglu, M. & Tryggvason, G. 2008 A front-tracking method for computation of interfacial flows with soluble surfactants. J. Comput. Phys. 227 (4), 2238–2262.10.1016/j.jcp.2007.10.003Google Scholar
Onsager, L. 1931a Reciprocal relations in irreversible processes. I. Phys. Rev. 37 (4), 405.10.1103/PhysRev.37.405Google Scholar
Onsager, L. 1931b Reciprocal relations in irreversible processes. II. Phys. Rev. 38 (12), 2265.10.1103/PhysRev.38.2265Google Scholar
Patil, N. D., Gada, V. H., Sharma, A. & Bhardwaj, R. 2016 On dual-grid level-set method for contact line modeling during impact of a droplet on hydrophobic and superhydrophobic surfaces. Intl J. Multiphase Flow 81, 54–66.10.1016/j.ijmultiphaseflow.2016.01.005Google Scholar
Pätzold, G. & Dawson, K. 1995 Numerical simulation of phase separation in the presence of surfactants and hydrodynamics. Phys. Rev. E 52 (6), 6908.Google Scholar
Qian, T., Wang, X.-P. & Sheng, P. 2003 Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68 (1), 016306.Google Scholar
Qian, T., Wang, X.-P. & Sheng, P. 2004 Power-law slip profile of the moving contact line in two-phase immiscible flows. Phys. Rev. Lett. 93 (9), 094501.10.1103/PhysRevLett.93.094501Google Scholar
Qian, T., Wang, X.-P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360.10.1017/S0022112006001935Google Scholar
Raffa, P., Broekhuis, A. A. & Picchioni, F. 2016 Polymeric surfactants for enhanced oil recovery: a review. J. Petrol. Sci. Engng 145, 723–733.10.1016/j.petrol.2016.07.007Google Scholar
Shang, X., Luo, Z. & Bai, B. 2019 Droplets trapped by a wetting surface with chemical defects in shear flows. Chem. Engng Sci. 195, 433–441.10.1016/j.ces.2018.09.041Google Scholar
Shen, J., Xu, J. & Yang, J. 2018 The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416.10.1016/j.jcp.2017.10.021Google Scholar
Shen, J. & Yang, X. 2015 Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53 (1), 279–296.10.1137/140971154Google Scholar
Shen, J., Yang, X. & Yu, H. 2015 Efficient energy stable numerical schemes for a phase field moving contact line model. J. Comput. Phys. 284, 617–630.10.1016/j.jcp.2014.12.046Google Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45 (1), 269–292.10.1146/annurev-fluid-011212-140734Google Scholar
Sui, Y., Ding, H. & Spelt, P. D. 2014 Numerical simulations of flows with moving contact lines. Annu. Rev. Fluid Mech. 46 (1), 97–119.10.1146/annurev-fluid-010313-141338Google Scholar
Teigen, K. E., Song, P., Lowengrub, J. & Voigt, A. 2011 A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230 (2), 375–393.10.1016/j.jcp.2010.09.020Google Scholar
Titta, A., Le Merrer, M., Detcheverry, F., Spelt, P. & Biance, A.-L. 2018 Level-set simulations of a 2D topological rearrangement in a bubble assembly: effects of surfactant properties. J. Fluid Mech. 838, 222–247.10.1017/jfm.2017.887Google Scholar
Van der Sman, R. & Van der Graaf, S. 2006 Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol. Acta 46 (1), 3–11.10.1007/s00397-005-0081-zGoogle Scholar
Wang, X.-P., Qian, T. & Sheng, P. 2008 Moving contact line on chemically patterned surfaces. J. Fluid Mech. 605, 59–78.10.1017/S0022112008001456Google Scholar
Wei, B., Hou, J., Sukop, M. C. & Liu, H. 2019 Pore scale study of amphiphilic fluids flow using the Lattice Boltzmann model. Intl J. Heat Mass Transfer 139, 725–735.10.1016/j.ijheatmasstransfer.2019.05.056Google Scholar
Wodlei, F., Sebilleau, J., Magnaudet, J. & Pimienta, V. 2018 Marangoni-driven flower-like patterning of an evaporating drop spreading on a liquid substrate. Nature Commun. 9 (1), 820.10.1038/s41467-018-03201-3Google Scholar
Xu, J.-J. & Ren, W. 2014 A level-set method for two-phase flows with moving contact line and insoluble surfactant. J. Comput. Phys. 263, 71–90.10.1016/j.jcp.2014.01.012Google Scholar
Xu, J.-J., Yang, Y. & Lowengrub, J. 2012 A level-set continuum method for two-phase flows with insoluble surfactant. J. Comput. Phys. 231 (17), 5897–5909.10.1016/j.jcp.2012.05.014Google Scholar
Xu, X., Di, Y. & Yu, H. 2018 Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines. J. Fluid Mech. 849, 805–833.10.1017/jfm.2018.428Google Scholar
Yan, X., Huang, Z., Yao, J., Zhang, Z., Liu, P., Li, Y. & Fan, D. 2019 Numerical simulation of hydro-mechanical coupling in fractured vuggy porous media using the equivalent continuum model and embedded discrete fracture model. Adv. Water Resour. 126, 137–154.10.1016/j.advwatres.2019.02.013Google Scholar
Yang, X. & Ju, L. 2017 Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model. Comput. Meth. Appl. Mech. Engng 315, 691–712.10.1016/j.cma.2016.10.041Google Scholar
Yang, X., Zhao, J., Wang, Q. & Shen, J. 2017 Numerical approximations for a three-component Cahn–Hilliard phase-field model based on the invariant energy quadratization method. Math. Models Meth. Appl. Sci. 27 (11), 1993–2030.10.1142/S0218202517500373Google Scholar
Yu, H. & Yang, X. 2017 Numerical approximations for a phase-field moving contact line model with variable densities and viscosities. J. Comput. Phys. 334, 665–686.10.1016/j.jcp.2017.01.026Google 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, 293–317.10.1017/S0022112004000370Google Scholar
Yun, A., Li, Y. & Kim, J. 2014 A new phase-field model for a water–oil-surfactant system. Appl. Math. Comput. 229, 422–432.Google Scholar
Zampogna, G. A., Magnaudet, J. & Bottaro, A. 2019 Generalized slip condition over rough surfaces. J. Fluid Mech. 858, 407–436.10.1017/jfm.2018.780Google Scholar
Zeng, Q.-D., Yao, J. & Shao, J. 2019 Study of hydraulic fracturing in an anisotropic poroelastic medium via a hybrid EDFM-XFEM approach. Comput. Geotech. 105, 51–68.10.1016/j.compgeo.2018.09.010Google Scholar
Zhang, J., Eckmann, D. M. & Ayyaswamy, P. S. 2006 A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport. J. Comput. Phys. 214 (1), 366–396.10.1016/j.jcp.2005.09.016Google Scholar
Zhang, L., Kang, Q., Yao, J., Gao, Y., Sun, Z., Liu, H. & Valocchi, A. J. 2015 Pore scale simulation of liquid and gas two-phase flow based on digital core technology. Sci. Chin. Technol. Sci. 58 (8), 1375–1384.10.1007/s11431-015-5842-zGoogle Scholar
Zhang, Z., Xu, S. & Ren, W. 2014 Derivation of a continuum model and the energy law for moving contact lines with insoluble surfactants. Phys. Fluids 26 (6), 062103.10.1063/1.4881195Google Scholar
Zhao, J., Kang, Q., Yao, J., Viswanathan, H., Pawar, R., Zhang, L. & Sun, H. 2018 The effect of wettability heterogeneity on relative permeability of two – phase flow in porous media: a lattice Boltzmann study. Water Resour. Res. 54 (2), 1295–1311.10.1002/2017WR021443Google Scholar
Zhu, G., Chen, H., Yao, J. & Sun, S. 2019a Efficient energy-stable schemes for the hydrodynamics coupled phase-field model. Appl. Math. Model. 70, 82–108.10.1016/j.apm.2018.12.017Google Scholar
Zhu, G., Kou, J., Sun, S., Yao, J. & Li, A. 2018 Decoupled, energy stable schemes for a phase-field surfactant model. Comput. Phys. Commun. 233, 67–77.10.1016/j.cpc.2018.07.003Google Scholar
Zhu, G., Kou, J., Sun, S., Yao, J. & Li, A. 2019b Numerical approximation of a phase-field surfactant model with fluid flow. J. Sci. Comput. 80, 223–247.10.1007/s10915-019-00934-1Google Scholar
Zhu, G.-P., Yao, J., Sun, H., Zhang, M., Xie, M.-J., Sun, Z.-X. & Lu, T. 2016 The numerical simulation of thermal recovery based on hydraulic fracture heating technology in shale gas reservoir. J. Nat. Gas Sci. Engng 28, 305–316.10.1016/j.jngse.2015.11.051Google Scholar