Hostname: page-component-5f745c7db-rgzdr Total loading time: 0 Render date: 2025-01-06T08:16:57.775Z Has data issue: true hasContentIssue false
  • English
  • Français

Thermodynamically consistent phase-field modelling of activated solute transport in binary solvent fluids

Published online by Cambridge University Press:  24 January 2023

Jisheng Kou Open the ORCID record for Jisheng Kou [Opens in a new window] ,
Amgad Salama Open the ORCID record for Amgad Salama [Opens in a new window]  and
Xiuhua Wang Open the ORCID record for Xiuhua Wang [Opens in a new window]
Jisheng Kou*
Affiliation:
Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province, Shaoxing University, Shaoxing, 312000 Zhejiang, PR China School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000 Hubei, PR China
Amgad Salama
Affiliation:
Mechanical Engineering Department, Faculty of Engineering, University of Saskatchewan, Saskatoon, SK, S7N 5A9, Canada
Xiuhua Wang*
Affiliation:
School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000 Hubei, PR China
*
†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

Active diffusion of substances in binary immiscible and incompressible fluids with different densities occurs universally in nature and industry, but relevant mathematical models and numerical simulation have been studied scarcely so far. In this paper, a thermodynamically consistent phase-field model is established to describe the activated solute transport in binary fluids with different densities. A mixed free-energy function for multiple solutes is proposed, which leads to different solute chemical potentials in binary solvent fluids, thus it has the ability to characterize the solubility difference of the solutes in two solvents. The two-phase flow is governed by a general hydrodynamic phase-field model that can account for general average velocity and different densities. The proposed model is derived rigorously using the second law of thermodynamics. Moreover, a general multi-component solute diffusion model is established using the Maxwell–Stefan approach, which involves the crossing influences between different solutes. To solve the model effectively, an efficient, linearized and decoupled numerical method is proposed for the model as well. The proposed numerical method can preserve the thermodynamical consistency, i.e. obeying an energy dissipation law at the discrete level, as well as guarantee the mass conservation law for the solutes and solvent fluids. Numerical experiments are carried out to show that the proposed model and numerical method can simulate various processes of the solute active diffusion in two-phase solvent fluids.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 955 , 25 January 2023 , A41
Check for updates
Copyright
© The Author(s), 2023. Published by 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

REFERENCES

Abels, H., Garcke, H. & Grün, G. 2012 Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math Models Methods Appl. Sci. 22, 1150013.CrossRefGoogle Scholar
Aki, G.L., Dreyer, W., Giesselmann, J. & Kraus, C. 2014 A quasi-incompressible diffuse interface model with phase transition. Math. Models Methods Appl. Sci. 24, 827–861.CrossRefGoogle Scholar
Aland, S. & Voigt, A. 2012 Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. Intl J. Numer. Meth. Fluids 69, 747–761.CrossRefGoogle Scholar
Anderson, D.M., McFadden, G.B. & Wheeler, A.A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.CrossRefGoogle Scholar
Boyer, F. 2002 A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31, 41–68.CrossRefGoogle Scholar
Brenner, H. 2009 a Bi-velocity hydrodynamics. Physica A 388, 3391–3398.CrossRefGoogle Scholar
Brenner, H. 2009 b Bi-velocity hydrodynamics. Multicomponent fluids. Intl J. Engng Sci. 47, 902–929.CrossRefGoogle Scholar
Brenner, H. 2013 Bivelocity hydrodynamics. Diffuse mass flux vs. diffuse volume flux. Physica A 392, 558–566.CrossRefGoogle Scholar
Cahn, J.W. & Allen, S.M. 1977 A microscopic theory for domain wall motion and its experimental verification in Fe–Al alloy domain growth kinetics. J. Phys. Colloq. C7, C7–51.Google Scholar
Cahn, J.W. & Hilliard, J.E. 1958 Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Chen, Y. & Shen, J. 2016 Efficient, adaptive energy stable schemes for the incompressible Cahn–Hilliard Navier–Stokes phase-field models. J. Comput. Phys. 308, 40–56.CrossRefGoogle Scholar
Ding, H., Spelt, P.D.M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 2078–2095.CrossRefGoogle Scholar
Ding, H. & Yuan, C. 2014 On the diffuse interface method using a dual-resolution Cartesian grid. J. Comput. Phys. 273, 243–254.CrossRefGoogle Scholar
Drechsler, M., Giavazzi, F., Cerbino, R. & Palacios, I.M. 2017 Active diffusion and advection in Drosophila oocytes result from the interplay of actin and microtubules. Nat. Commun. 8, 1520.CrossRefGoogle ScholarPubMed
von den Driesch, N., Wirths, S., Troitsch, R., Mussler, G., Breuer, U., Moutanabbir, O., Grützmacher, D. & Buca, D. 2020 Thermally activated diffusion and lattice relaxation in (Si)GeSn materials. Phys. Rev. Mater. 4, 033604.CrossRefGoogle 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, 879–915.CrossRefGoogle Scholar
Garcke, H., Lam, K.F. & Stinner, B. 2014 Diffuse interface modelling of soluble surfactants in two-phase flow. Commun. Math. Sci. 12, 1475–1522.CrossRefGoogle Scholar
Gomez, H. & Hughes, T. 2011 Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230, 5310–5327.CrossRefGoogle Scholar
Gong, X., Gong, Z. & Huang, H. 2014 An immersed boundary method for mass transfer across permeable moving interfaces. J. Comput. Phys. 278, 148–168.CrossRefGoogle Scholar
Gong, Y., Zhao, J., Yang, X. & Wang, Q. 2018 Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities. SIAM J. Sci. Comput. 40, B138–B167.CrossRefGoogle Scholar
Grün, G. & Klingbeil, F. 2014 Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257, 708–725.CrossRefGoogle Scholar
Guo, Z., Lin, P., Lowengrub, J. & Wise, S.M. 2017 Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier–Stokes–Cahn–Hilliard system: primitive variable and projection-type schemes. Comput. Meth. Appl. Mech. Engng 326, 144–174.CrossRefGoogle Scholar
Guzmán-Lastra, F., Löwen, H. & Mathijssen, A.J.T.M. 2021 Active carpets drive non-equilibrium diffusion and enhanced molecular fluxes. Nat. Commun. 12, 1906.CrossRefGoogle ScholarPubMed
Haddada, M.E. & Tierra, G. 2022 A thermodynamically consistent model for two-phase incompressible flows with different densities. Derivation and efficient energy-stable numerical schemes. Comput. Meth. Appl. Mech. Engng 389, 114328.CrossRefGoogle Scholar
Hass, A. & Fine, P. 2010 Sequential selective extraction procedures for the study of heavy metals in soils, sediments, and waste materials – a critical review. Crit. Rev. Environ. Sci. Technol. 40, 365–399.CrossRefGoogle Scholar
Johansson, J., Svensson, C.P.T., Martensson, T., Samuelson, L. & Seifert, W. 2005 Mass transport model for semiconductor nanowire growth. J. Phys. Chem. B 109, 13567–13571.CrossRefGoogle ScholarPubMed
Kim, J. 2012 Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12, 613–661.CrossRefGoogle Scholar
Kou, J. & Sun, S. 2018 a Thermodynamically consistent modeling and simulation of multi-component two-phase flow with partial miscibility. Comput. Meth. Appl. Mech. Engng 331, 623–649.CrossRefGoogle Scholar
Kou, J. & Sun, S. 2018 b Entropy stable modeling of non-isothermal multi-component diffuse-interface two-phase flows with realistic equations of state. Comput. Meth. Appl. Mech. Engng 341, 221–248.CrossRefGoogle 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, 3219–3248.CrossRefGoogle Scholar
Kou, J., Sun, S. & Wang, X. 2020 a A novel energy factorization approach for the diffuse-interface model with Peng–Robinson equation of state. SIAM J. Sci. Comput. 42, B30–B56.CrossRefGoogle Scholar
Kou, J., Wang, X., Zeng, M. & Cai, J. 2020 b Energy stable and mass conservative numerical method for a generalized hydrodynamic phase-field model with different densities. Phys. Fluids 32, 117103.CrossRefGoogle Scholar
Li, J. & Wang, Q. 2014 A class of conservative phase field models for multiphase fluid flows, J. Appl. Mech., 81, 021004.CrossRefGoogle Scholar
Liu, C., Shen, J. & Yang, X. 2015 Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. J. Sci. Comput. 62, 601–622.CrossRefGoogle Scholar
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454, 2617–2654.CrossRefGoogle Scholar
Patankar, N.A. 2016 Thermodynamics of trapping gases for underwater superhydrophobicity. Langmuir 32, 7023–7028.CrossRefGoogle ScholarPubMed
Qiao, Z., Zhang, Z. & Tang, T. 2011 An adaptive time-stepping strategy for the molecular beam epitaxy models. SIAM J. Sci. Comput. 33, 1395–1414.CrossRefGoogle Scholar
Qin, Y., Huang, H., Zhu, Y., Liu, C. & Xu, S. 2022 A phase field model for mass transport with semi-permeable interfaces. J. Comput. Phys. 464, 111334.CrossRefGoogle Scholar
Roudbari, M.S., Simsek, G., van Brummelen, E.H. & van der Zee, K.G. 2018 Diffuse-interface two-phase flow models with different densities: a new quasi-incompressible form and a linear energy-stable method. Math. Models Meth. Appl. Sci. 28, 733–770.CrossRefGoogle Scholar
Shen, J. & Yang, X. 2010 A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput. 32, 1159–1179.CrossRefGoogle Scholar
Shen, J., Yang, X. & Wang, Q. 2013 Mass and volume conservation in phase field models for binary fluids. Commun. Comput. Phys. 13, 1045–1065.CrossRefGoogle Scholar
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press.Google Scholar
van der Sman, R.G.M. & van der Graaf, S. 2006 Diffuse interface model of surfactant adsorption onto flat and droplet interfaces. Rheol Acta 46, 3–11.CrossRefGoogle Scholar
Wang, X., Gong, X., Sugiyama, K., Takagi, S. & Huang, H. 2020 a An immersed boundary method for mass transfer through porous biomembranes under large deformations. J. Comput. Phys. 413, 109444.CrossRefGoogle Scholar
Wang, X., Kou, J. & Cai, J. 2020 b Stabilized energy factorization approach for Allen–Cahn equation with logarithmic Flory–Huggins potential. J. Sci. Comput. 82, 25.CrossRefGoogle Scholar
Wang, X., Kou, J. & Gao, H. 2021 Linear energy stable and maximum principle preserving semi-implicit scheme for Allen–Cahn equation with double well potential. Commun. Nonlinear Sci. Numer. Simul. 98, 105766.CrossRefGoogle Scholar
Wang, Y., Shu, C., Shao, J., Wu, J. & Niu, X. 2015 A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio. J. Comput. Phys. 290, 336–351.CrossRefGoogle Scholar
Watson, A.J., Schuster, U., Shutler, J.D., Holding, T., Ashton, I.G.C., Landschützer, P., Woolf, D.K. & Goddijn-Murphy, L. 2020 Revised estimates of ocean-atmosphere CO$_2$ flux are consistent with ocean carbon inventory. Nat. Commun. 11, 4422.CrossRefGoogle ScholarPubMed
Wells, G.N., Kuhl, E. & Garikipati, K. 2006 A discontinuous Galerkin method for the Cahn–Hilliard equation. J. Comput. Phys. 218, 860–877.CrossRefGoogle Scholar
Wesselingh, J.A. & Krishna, R. 2000 Mass Transfer in Multicomponent Mixtures. Delft University Press.Google Scholar
Yang, X. & Zhao, J. 2019 On linear and unconditionally energy stable algorithms for variable mobility Cahn–Hilliard type equation with logarithmic Flory–Huggins potential. Commun. Comput. Phys. 25, 703–728.CrossRefGoogle 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.CrossRefGoogle Scholar
Yue, P., Zhou, C. & Feng, J.J. 2007 Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 223, 1–9.CrossRefGoogle Scholar
Zhu, G., Kou, J., Yao, B., Wu, Y.-S., Yao, J. & Sun, S. 2019 Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants. J. Fluid Mech. 879, 327–359.CrossRefGoogle Scholar