Interfacial dynamics with soluble surfactants: A phase-field two-phase flow model with variable densities

Abstract

In this work, we present a hydrodynamics coupled phase-field surfactant model with variable densities. Two scalar auxiliary variables are introduced to transform the original free energy functional into an equivalent form, and then a new thermodynamically consistent model can be obtained. In this model, evolutions of two phase-field variables are described by two Cahn-Hilliard-type equations, and the fluid flow is dominated by incompressible Navier-Stokes equation. The finite difference method on staggered grid is used to solve the above model. Then a classical droplet rising case and a droplet merging case are used to validate our model. Finally, we study the effect of surfactants on droplet deformation and merging. A more prolate profile of droplet is observed under a higher surfactant bulk concentration, which verifies the effect of surfactant in reducing the interfacial tension. Increases in surface Peclet number and initial surfactant bulk concentration can enhance the non-uniformity of surfactant distribution around the interface, which will arise the Marangoni force. The Marangoni force acts as an additional repulsive force to delay the droplet merging.

Cited as: Zhu, G., Li, A. Interfacial dynamics with soluble surfactants: A phase-field two-phase flow model with variable densities. Advances in Geo-Energy Research, 2020, 4(1): 86-98, doi: 10.26804/ager.2020.01.08

References

Alke, A., Bothe, D. 3D numerical modeling of soluble surfactant at fluidic interfaces based on the volume-of-fluid method. Fluid Dyn. Mater. Process. 2009, 5(4): 345-372.

Chen, C., Tsai, Y.L., Lan, C. Adaptive phase field simulation of dendritic crystal growth in a forced flow: 2D vs 3D morphologies. Int. J. Heat Mass Transf. 2009, 52(5-6): 1158-1166.

Comsol, A.B. COMSOL Multiphysics User’s Guide (version 4.3). 2012.

Engblom, S., Do-Quang, M., Amberg, G., et al. On diffuse interface modeling and simulation of surfactants in two-phase fluid flow. Commun. Comput. Phys. 2013, 14(4): 879-915.

Espath, L.F.R., Sarmiento, A.F., Vignal, P., et al. Energy exchange analysis in droplet dynamics via the Navier-Stokes-Cahn-Hilliard model. J. Fluid Mech. 2016, 797: 389-430.

Fonseca, I., Morini, M., Slastikov, V. Surfactants in foam stability: A phase-field model. Arch. Ration. Mech. An. 2007, 183(3): 411-456.

Hysing, S., Turek, S., Kuzmin, D., et al. Quantitative benchmark computations of twodimensional bubble dynamics. Int. J. Numer. Meth. Fl. 2009, 60(11): 1259-1288.

James, A.J., Lowengrub, J. A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 2004, 201(2): 685-722.

Khatri, S., Tornberg, A.K. An embedded boundary method for soluble surfactants with interface tracking for two-phase flows. J. Comput. Phys. 2014, 256: 768-790.

Komura, S., Kodama, H. Two-order-parameter model for an oil-water-surfactant system. Phys. Rev. E 1997, 55(2): 1722-1727.

Kou, J., Sun, S. Thermodynamically consistent modeling and simulation of multi-component two-phase flow with partial miscibility. Comput. Methods Appl. Mech. Eng. 2018a, 331: 623-649.

Kou, J., Sun, S. Thermodynamically consistent simulation of nonisothermal diffuse-interface two-phase flow with Peng-Robinson equation of state. J. Comput. Phys. 2018b, 371: 581-605.

Kou, J., Sun, S., Wang, X. Linearly decoupled energy-stable numerical methods for multicomponent two-phase compressible flow. SIAM J. Numer. Anal. 2018, 56(6): 3219-3248.

Laradji, M., Guo, H., Grant, M., et al. The effect of surfactants on the dynamics of phase separation. J. Phys. Condens. Matter 1992, 4(32): 6715-6728.

Li, J., Yu, B., Wang, Y., et al. Study on computational efficiency of composite schemes for convection-diffusion equations using single-grid and multigrid methods. J. Therm. Sci. Technol. 2015, 10(1): JTST0009.

Li, Y., Kim, J. A comparison study of phase-field models for an immiscible binary mixture with surfactant. Eur. Phys. J. B 2012, 85(10): 340.

Liu, H., Ba, Y., Wu, L., et al. A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants. J. Fluid Mech. 2018, 837: 381-412.

Liu, H., Zhang, Y. Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 2010, 229(24): 9166-9187.

Moukalled, F., Mangani, L., Darwish, M. The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Heidelberg, German, Springer, 2016.

Muradoglu, M., Tryggvason, G. A front-tracking method for computation of interfacial flows with soluble surfactants. J. Comput. Phys. 2008, 227(4): 2238-2262.

Shen, J., Xu, J., Yang, J. The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 2018, 353: 407-416.

Shen, J., Xu, J., Yang, J. A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 2019, 61(3): 474-506.

Shen, J., Yang, X. Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 2015, 53(1): 279-296.

Sheng, G., Zhao, H., Su, Y., et al. An analytical model to couple gas storage and transport capacity in organic matter with noncircular pores. Fuel 2020, 268: 117288.

Van der Sman, R.G.M. Phase field simulations of ice crystal growth in sugar solutions. Int. J. Heat Mass Transf. 2016, 95: 153-161.

Van der Sman, R.G.M., Meinders, M.B.J. Analysis of improved Lattice Boltzmann phase field method for soluble surfactants. Comput. Phys. Commun. 2016, 199: 12-21.

Van der Sman, R.G.M., Van der Graaf, S. Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol. Acta 2006, 46(1): 3-11.

Wang, H., Yuan, X., Liang, H., et al. A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. Capillarity 2019, 2(3): 33-52.

Wang, X., Kou, J., Cai, J. Stabilized energy factorization Approach for Allen-Cahn Equation with Logarithmic Flory-Huggins potential. J. Sci. Comput. 2020, 82(2): 25.

Xu, J., Li, Z., Lowengrub, J., et al. A level-set method for interfacial flows with surfactant. J. Comput. Phys. 2006, 212(2): 590-616.

Xu, J., Yang, Y., Lowengrub, J. A level-set continuum method for two-phase flows with insoluble surfactant. J. Comput. Phys. 2012, 231(17): 5897-5909.

Xu, X., Di, Y., Yu, H. Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines. J. Fluid Mech. 2018, 849: 805-833.

Yan, X., Huang, Z., Yao, J., et al. Numerical simulation of hydro-mechanical coupling in fractured vuggy porous media using the equivalent continuum model and embedded discrete fracture model. Adv. Water Resour. 2019, 126: 137-154.

Yang, Q., Yao, J., Huang, Z., et al. A comprehensive SPH model for three-dimensional multiphase interface simulation. Comput. Fluids 2019, 187: 98-106.

Yang, X. Numerical approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system. J. Sci. Comput. 2018, 74(3): 1533-1553.

Yang, X., Ju, L. Linear and unconditionally energy stable schemes for the binary fluidsurfactant phase field model. Comput. Methods Appl. Mech. Eng. 2017, 318: 1005-1029.

Yuan, Z., Wu, R., Wu, X. Numerical simulations of multi-hop jumping on superhydrophobic surfaces. Int. J. Heat Mass Transf. 2019, 135: 345-353.

Yue, P., Feng, J.J., Liu, C., et al. A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 2004, 515: 293-317.

Zeng, Q., Yao, J., Shao, J. Study of hydraulic fracturing in an anisotropic poroelastic medium via a hybrid EDFM-XFEM approach. Comput. Geotech. 2019, 105: 51-68.

Zhang, L., Jing, W., Yang, Y., et al. The investigation of permeability calculation using digital core simulation technology. Energies 2019, 12(17): 3273.

Zhu, G., Chen, H., Yao, J., et al. Efficient energy-stable schemes for the hydrodynamics coupled phase-field model. Appl. Math. Model. 2019a, 70: 82-108.

Zhu, G., Kou, J., Sun, S., et al. Decoupled, energy stable schemes for a phase-field surfactant model. Comput. Phys. Commun. 2018, 233: 67-77.

Zhu, G., Kou, J., Sun, S., et al. Numerical approximation of a phase-field surfactant model with fluid flow. J. Sci. Comput. 2019b, 80(1): 223-247.

Zhu, G., Kou, J., Yao, B., et al. Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants. J. Fluid Mech. 2019c, 879: 327-359.

Zhu, G., Kou, J., Yao, J., et al. A phase-field moving contact line model with soluble surfactants, J. Comput. Phys. 2020, 405: 109170.

Zhu, G., Yao, J., Li, A., et al. Pore-scale investigation of carbon dioxide-enhanced oil recovery. Energy Fuels 2017, 31(5): 5324-5332.