Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cell

Yang, Xiaofeng

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 651-678

We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the “zero-energy-contribution” feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.

Classification : 65N12, 65M12, 65M70
Keywords: Finite element, fully-discrete, second-order, fluid-surfactant, Cahn-Hilliard, Hele-Shaw cell

[1] S. Ahmadikhamsi, F. Golfier, C. Oltean, E. Lefèvre and S. A. Bahrani, Impact of surfactant addition on non-newtonian fluid behavior during viscous fingering in Hele-Shaw cell. Phys. Fluids 32 (2020) 012103.

[2] E. Álvarez-Lacalle, J. Ortín and J. Casademunt, Low viscosity contrast fingering in a rotating Hele-Shaw cell. Phys. Fluids 16 (2004) 908–924. | MR | Zbl

[3] E. Álvarez-Lacalle, J. Ortn and J. Casademunt, Relevance of dynamic wetting in viscous fingering patterns. Phys. Rev. E 74 (2006) 025302.

[4] J. Bear, Dynamics of fluids in porous media, Courier Dover Publications, New York (1988). | Zbl

[5] I. Bischofberger, R. Ramachandran and S. R. Nagel, An island of stability in a sea of fingers: emergent global features of the viscous-flow instability. Soft Matter 11 (2015) 7428–7432.

[6] L. L. Carrillo, F. X. Magdaleno, J. Casademunt and J. Ortn, Experiments in a rotating Hele-Shaw cell. Phys. Rev. E 54 (1996) 6260–6267.

[7] J.-D. Chen, Growth of radial viscous fingers in a Hele-Shaw cell. J. Fluid Mech. 201 (1989) 223–242.

[8] C. Chen and X. Yang, Efficient Numerical Scheme for a dendritic Solidification Phase Field model with melt convection. J. Comput. Phys. 388 (2019) 41–62. | MR

[9] C. Chen and X. Yang, Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn-Hilliard Model. Comput. Meth. Appl. Mech. Eng. 351 (2019) 35–59. | MR

[10] C.-Y. Chen, Y.-S. Huang and J. A. Miranda, Diffuse-interface approach to rotating Hele-Shaw flows. Phys. Rev. E 84 (2011) 046302.

[11] K. Cheng, C. Wang and S. M. Wise, An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation. Commun. Comput. Phys. 26 (2019) 1335–1364. | MR

[12] M. Chinaud, V. Voulgaropoulos and P. Angeli, Surfactant effects on the coalescence of a drop in a Hele-Shaw cell. Phys. Rev. E. 94 (2016) 033101.

[13] J. Y. Y. Chui, P. De Anna and R. Juanes, Interface evolution during radial miscible viscous fingering. Phys. Rev. E 92 (2015) 041003.

[14] L. Dedè, H. Garcke and K. F. Lam, A hele–shaw–cahn–hilliard model for incompressible two-phase flows with different densities. J. Math. Fluid Mech. 20 (2018) 531–567. | MR

[15] C. Derec, P. Boltenhagen, S. Neveu and J. C. Bacri, Magnetic instability between miscible fluids in a Hele-Shaw cell. Magnetohydrodynamics 44 (2008) 135–142.

[16] K. Erik Teigen, P. Song, J. Lowengrub and A. Voigt, A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230 (2011) 375–393. | MR | Zbl

[17] R. Farajzadeh, A. A. Eftekhari, H. Hajibeygi, S. Kahrobaei, J. M. Van Der Meer, S. Vincent-Bonnieu and W. R. Rossen, Simulation of instabilities and fingering in surfactant alternating gas (sag) foam enhanced oil recovery. J. Nat. Gas Sci. Eng. 34 (2016) 1191–1204.

[18] I. Fonseca, M. Morini and V. Slastikov, Surfactants in foam stability: A phase-field approach. Arch. Ration. Mech. Anal. 183 (2007) 411–456. | MR | Zbl

[19] G. Gompper and M. Schick, Self-assembling amphiphilic systems, in phase trasitions and critical phenomena, edited by C. Domb and J. Lebowitz. Academic Press, London, 16 (1994).

[20] S. Gu, H. Zhang and Z. Zhang, An energy-stable finite-difference scheme for the binary fluid-surfactant system. J. Comput. Phys. 367 (2014) 3–11. | MR

[21] D. Han nd X. Wang, Decoupled energy-law preserving numerical schemes for the cahn–hilliard–darcy system. Numer. Methods Partial Differ. Equ. 32 (2016) 936–954. | MR

[22] D. Han and X. Wang, A Second Order in Time, Decoupled, Unconditionally Stable Numerical Scheme for the Cahn–Hilliard–Darcy System. J. Sci. Comput. 14 (2018) 1210–1233. | MR

[23] F. Hecht, New development in freefem++. J. Numer. Math. 20 (2012) 251–265. | MR | Zbl

[24] Y.-S. Huang and C.-Y. Chen, A numerical study on radial Hele-Shaw flow: Influence of fluid miscibility and injection scheme. Comput. Mech. 55 (2015) 407. | MR

[25] S. Komura and H. Kodama, Two-order-parameter model for an oil-water-surfactant system. Phys. Rev. E. 55 (1997) 1722–1727.

[26] M. Laradji, H. Guo, M. Grant and M. J. Zuckermann, The effect of surfactants on the dynamics of phase separation. J. Phy. Condens. Matter 4 (1992) 6715.

[27] M. Laradji, O. G. Mouristen, S. Toxvaerd and M. J. Zuckermann, Molecular dynamics simulatiens af phase separation in the presence ef surfactants. Phys. Rev. E. 50 (1994) 1722–1727.

[28] H. Liu and Y. Zhang, Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 229 (2010) 9166–9187. | Zbl

[29] S. Mollaei and A. H. Darooneh, Spreading, fingering instability and shrinking of a hydrosoluble surfactant on water. Exp. Therm. Fluid Sci. 86 (2017) 98–101.

[30] D. A. Nield and A. Bejan, Convection in porous media, Springer-Verlag, New York, 2nd ed. (1999). | MR | Zbl

[31] K. Okumura, Viscous dynamics of drops and bubbles in Hele-Shaw cells; Drainage, drag fraction, coalescence, and bursting. Adv. Colloid Inter. Sci. 225 (2018) 64–75.

[32] S. Pramanik and M. Mishra, Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell. Phys. Rev. E 91 (2015) 033006.

[33] S. Pramanik, T. K. Hota and M. Mishra, Influence of viscosity contrast on buoyantly unstable miscible fluids in porous media. J. Fluid Mech. 780 (2015) 388–406. | MR

[34] J.-R. Roan and E. I. Shakhnovich, Phase separation of a binary fluid containing surfactants in a Hele-Shaw cell. Phys. Rev. E 59 (1999) 2109.

[35] P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1958) 312–329. | MR | Zbl

[36] P. Satyajit and M. Manoranjan, Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media. Chem. Eng. Sci. 122 (2015) 523–532.

[37] V. Sharma, S. Nand, S. Pramanik, C.-Y. Chen and M. Mishra, Control of radial miscible viscous fingering. J. Fluid Mech. 884 (2020) A16. | MR

[38] J. Shen and X. Yang, Numerical Approximations of Allen-Cahn and Cahn-Hilliard Equations. Disc. Contin. Dyn. Sys. A 28 (2010) 1669–1691. | MR | Zbl

[39] J. Shen and X. Yang, The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems. Contemp. Math. 754 (2020) 217–245. | MR

[40] M. Sun, X. Feng and K. Wang, Numerical simulation of binary fluid–surfactant phase field model coupled with geometric curvature on the curved surface. Comput. Methods Appl. Mech. Eng. 367 (2020) 113123. | MR

[41] C. H. Teng, I. L. Chern and M. C. Lai, Simulating binary fluid-surfactant dynamics by a phase field model. Dis. Contin. Dyn. Syst.-B 17 (2010) 1289–1307. | MR | Zbl

[42] T. Teramoto and F. Yonezawa, Droplet growth dynamics in a water-oil-surfactant system. J. Colloid Inter. Sci. 235 (2001) 329–333.

[43] R. Tsuzuki, Q. Li, Y. Nagatsu and C.-Y. Chen, Numerical study of immiscible viscous fingering in chemically reactive Hele-Shaw flows: Production of surfactants. Phys. Rev. Fluids 4 (2019) 104003.

[44] R. G. M. Van Der Sman and S. Van Der Graaf, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol. Acta 46 (2006) 3–11.

[45] R. G. M. Van Der Sman and M. B. J. Meinders, Analysis of improved lattice boltzmann phase field method for soluble surfactants. Comput. Phys. Comm. 199 (2016) 12–21. | MR

[46] R. A. Wooding, Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech. 39 (1969) 477–95.

[47] X. Yang, A new efficient Fully-decoupled and Second-order time-accurate scheme for Cahn-Hilliard phase-field model of three-phase incompressible flow. Comput. Methods Appl. Mech. Eng. 376 (2021) 13589. | MR

[48] X. Yang, A novel fully-decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier-Stokes equations coupled with mass-conserved Allen-Cahn phase-field model of two-phase incompressible flow. Int. J. Numer. Methods Eng. 122 (2021) 1283–1306. | MR

[49] X. Yang, A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen-Cahn type flow-coupled binary surfactant model. Comput. Methods Appl. Mech. Eng. 373 (2021) 113502. | MR

[50] X. Yang, A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model. J. Comput. Phys. 432 (2021) 110015. | MR | Zbl

[51] X. Yang, Efficient and Energy Stable scheme for the hydrodynamically coupled three components Cahn-Hilliard phase-field model using the stabilized-Invariant Energy Quadratization (S-IEQ) Approach. J. Comput. Phys. 438 (2021) 110342. | MR | Zbl

[52] X. Yang, On a novel fully-decoupled, second-order accurate energy stable numerical scheme for a binary fluid-surfactant phase-field model. SIAM J. Sci. Comput. 43 (2021) B479–B507. | MR | Zbl

[53] X. Yang and L. Ju, Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Meth. Appl. Mech. Eng. 318 (2017) 1005–1029. | MR | Zbl

[54] X. Yang and H. Yu, Efficient Second Order Unconditionally Stable Schemes for a Phase Field Moving Contact Line Model Using an Invariant Energy Quadratization Approach. SIAM J. Sci. Comput. 40 (2018) B889–B914. | MR | Zbl

[55] J. Zhang and X. Yang, Unconditionally energy stable large time stepping method for the $L^2$-gradient flow based ternary phase-field model with precise nonlocal volume conservation. Comput. Methods Appl. Mech. Eng. 361 (2020) 112743. | MR | Zbl

[56] J. Zhang, C. Chen, J. Wang and X. Yang, Efficient, Second order accurate, and unconditionally energy stable numerical scheme for a new hydrodynamics coupled binary phase-field surfactant system. Comput. Phys. Comm. 251 (2020) 107122. | MR | Zbl

[57] G. Zhu, J. Kou, S. Sun, J. Yao and A. Li, Decoupled, energy stable schemes for a phase-field surfactant model. Comput. Phys. Commun. 233 (2018) 67–77. | MR | Zbl