January 2019, 18(1): 195-225. doi: 10.3934/cpaa.2019011

Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

* Corresponding author

Received  October 2017 Revised  April 2018 Published  August 2018

Fund Project: The authors acknowledge support by the SPP 1506 "Transport Processes at Fluidic Interfaces" of the German Science Foundation (DFG) through grant GA695/6-1 and GA695/6-2. The results are part of the third author's PhD-thesis [33]

Two-phase flow of two Newtonian incompressible viscous fluids with a soluble surfactant and different densities of the fluids can be modeled within the diffuse interface approach. We consider a Navier-Stokes/Cahn-Hilliard type system coupled to non-linear diffusion equations that describe the diffusion of the surfactant in the bulk phases as well as along the diffuse interface. Moreover, the surfactant concentration influences the free energy and therefore the surface tension of the diffuse interface. For this system existence of weak solutions globally in time for general initial data is proved. To this end a two-step approximation is used that consists of a regularization of the time continuous system in the first and a time-discretization in the second step.

Citation: Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011
References:
[1]

H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289 (2009), 45-73. doi: 10.1007/s00220-009-0806-4.

[2]

H. Abels and D. Breit, Weak solutions for a non-Newtonian diffuse interface model with different densities, Nonlinearity, 29 (2016), 3426-3453. doi: 10.1088/0951-7715/29/11/3426.

[3]

H. AbelsD. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480. doi: 10.1007/s00021-012-0118-x.

[4]

H. AbelsD. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1175-1190. doi: 10.1016/j.anihpc.2013.01.002.

[5]

H. Abels and H. Garcke, Weak solutions and diffuse interface models for incompressible two-phase flows, In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, pages 1-60, 2018.

[6]

H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp. doi: 10.1142/S0218202511500138.

[7]

H. Abels, H. Garcke, K. F. Lam and W. Josef, Two-phase flow with surfactants: Diffuse interface models and their analysis. In Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics (D. Bothe and A. Reusken eds), Birkhäuser, Cham, pages 255-270, 2017.

[8]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.

[9]

S. Aland, A. Hahn, C. Kahle and R. Nürnberg, Comparative Simulations of Taylor Flow with Surfactants Based on Sharp- and Diffuse-Interface Models, In Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics (D. Bothe and A. Reusken eds), Birkhäuser, Cham, pages 639-661, 2017.

[10]

H. W. Alt, An abstract existence theorem for parabolic systems, Commun. Pure Appl. Anal., 11 (2012), 2079-2123. doi: 10.3934/cpaa.2012.11.2079.

[11]

J. W. BarrettH. Garcke and R. Nürnberg, On the stable numerical approximation of two-phase flow with insoluble surfactant, ESAIM Math. Model. Numer. Anal., 49 (2015), 421-458.

[12]

J. W. BarrettH. Garcke and R. Nürnberg, Stable finite element approximations of two-phase flow with soluble surfactant, J. Comput. Phys., 297 (2015), 530-564. doi: 10.1016/j.jcp.2015.05.029.

[13]

D. Bothe and J. Prüss, Stability of equilibria for two-phase flows with soluble surfactant, The Quarterly Journal of Mechanics and Applied Mathematics, 63 (2010), 177-199. doi: 10.1093/qjmam/hbq003.

[14]

D. Bothe, J. Prüss and G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, In H. Brezis, M. Chipot, and J. Escher, editors, Nonlinear Elliptic and Parabolic problems, Progress in Nonlinear Differential Equations and Their Applications, volume 64, pages 37-61. Springer, New York, 2005. doi: 10.1007/3-7643-7385-7_3.

[15]

R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.

[16]

G. Dore, Lp regularity for abstract differential equations, In Functional analysis and related topics, 1991 (Kyoto), volume 1540 of Lecture Notes in Math., pages 25-38. Springer, Berlin, 1993. doi: 10.1007/BFb0085472.

[17]

S. EngbolmM. Do-QuangG. Amberg and A.-K. Tornberg, On modeling and simulation of surfactants in diffuse interface flow, Communications in Computational Physics, 14 (2013), 875-915. doi: 10.4208/cicp.120712.281212a.

[18]

S. Ganesan and L. Tobiska, Arbitrary Lagrangian-Eulerian finite-element method for computation of two-phase flows with soluble surfactants, J. Comput. Phys., 231 (2012), 3685-3702. doi: 10.1016/j.jcp.2012.01.018.

[19]

H. GarckeK. F. Lam and B. Stinner, Diffuse interface modelling of soluble surfactants in two-phase flow, Commun. Math. Sci., 12 (2014), 1475-1522. doi: 10.4310/CMS.2014.v12.n8.a6.

[20]

H. Garcke and S. Wieland, Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system, SIAM J. Math. Anal., 37 (2006), 2025-2048. doi: 10.1137/040617017.

[21]

A. James and J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, Journal of Computational Physics, 201 (2004), 685-722. doi: 10.1016/j.jcp.2004.06.013.

[22]

S. Khatri and A.-K. Tornberg, A numerical method for two phase flows with insoluble surfactants, Computers and Fluids, 49 (2011), 150-165. doi: 10.1016/j.compfluid.2011.05.008.

[23]

M.-C. LaiY.-H. Tseng and H. Huang, An immersed boundary method for interfacial flows with insoluble surfactant, Journal of Computational Physics, 227 (2008), 7279-7293. doi: 10.1016/j.jcp.2008.04.014.

[24]

Y. Li and J. Kim, A comparison study of phase-field models for an immiscible binary mixture with surfactant, The European Physical Journal B-Condensed Matter and Complex Systems, 85 (2012), 1-9.

[25]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1 volume 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications.

[26]

H. Liu and Y. Zhang, Phase-field modeling droplet dynamics with soluble surfactants, Journal of Computational Physics, 229 (2010), 9166-9187.

[27]

M. Muradoglu and G. Tryggvason, A front-tracking method for computation of interfacial flows with soluble surfactants, Journal of Computational Physics, 227 (2008), 2238-2262.

[28]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, second edition, 2004.

[29]

J. Simon, Compact sets in the space Lp(0; T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[30]

K. TeigenX. LiJ. LowengrubF. Wang and A. Voigt, A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Communications in Mathematical Sciences, 7 (2009), 1009-1037.

[31]

K. TeigenP. SongJ. Lowengrub and A. Voigt, A diffuse-interface method for two-phase flows with soluble surfactants, Journal of Computational Physics, 230 (2011), 375-393. doi: 10.1016/j.jcp.2010.09.020.

[32]

R. van der Sman and S. van der Graaf, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces, Rheology Acta, 46 (2006), 3-11.

[33]

J. Weber, Analysis of diffuse interface models for two-phase flows with and without surfactants, 2016. PhD thesis, University of Regensburg, urn:nbn:de:bvb:355-epub-342471.

[34]

J. XuZ. LiJ. Lowengrub and H. Zhao, A level-set method for interfacial flows with surfactant, Journal of Computational Physics, 212 (2006), 590-616. doi: 10.1016/j.jcp.2005.07.016.

[35]

E. Zeidler, Applied Functional Analysis, volume 108 of Applied Mathematical Sciences, Springer-Verlag, New York, 1995.

show all references

References:
[1]

H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys., 289 (2009), 45-73. doi: 10.1007/s00220-009-0806-4.

[2]

H. Abels and D. Breit, Weak solutions for a non-Newtonian diffuse interface model with different densities, Nonlinearity, 29 (2016), 3426-3453. doi: 10.1088/0951-7715/29/11/3426.

[3]

H. AbelsD. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480. doi: 10.1007/s00021-012-0118-x.

[4]

H. AbelsD. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1175-1190. doi: 10.1016/j.anihpc.2013.01.002.

[5]

H. Abels and H. Garcke, Weak solutions and diffuse interface models for incompressible two-phase flows, In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, pages 1-60, 2018.

[6]

H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp. doi: 10.1142/S0218202511500138.

[7]

H. Abels, H. Garcke, K. F. Lam and W. Josef, Two-phase flow with surfactants: Diffuse interface models and their analysis. In Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics (D. Bothe and A. Reusken eds), Birkhäuser, Cham, pages 255-270, 2017.

[8]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.

[9]

S. Aland, A. Hahn, C. Kahle and R. Nürnberg, Comparative Simulations of Taylor Flow with Surfactants Based on Sharp- and Diffuse-Interface Models, In Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics (D. Bothe and A. Reusken eds), Birkhäuser, Cham, pages 639-661, 2017.

[10]

H. W. Alt, An abstract existence theorem for parabolic systems, Commun. Pure Appl. Anal., 11 (2012), 2079-2123. doi: 10.3934/cpaa.2012.11.2079.

[11]

J. W. BarrettH. Garcke and R. Nürnberg, On the stable numerical approximation of two-phase flow with insoluble surfactant, ESAIM Math. Model. Numer. Anal., 49 (2015), 421-458.

[12]

J. W. BarrettH. Garcke and R. Nürnberg, Stable finite element approximations of two-phase flow with soluble surfactant, J. Comput. Phys., 297 (2015), 530-564. doi: 10.1016/j.jcp.2015.05.029.

[13]

D. Bothe and J. Prüss, Stability of equilibria for two-phase flows with soluble surfactant, The Quarterly Journal of Mechanics and Applied Mathematics, 63 (2010), 177-199. doi: 10.1093/qjmam/hbq003.

[14]

D. Bothe, J. Prüss and G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, In H. Brezis, M. Chipot, and J. Escher, editors, Nonlinear Elliptic and Parabolic problems, Progress in Nonlinear Differential Equations and Their Applications, volume 64, pages 37-61. Springer, New York, 2005. doi: 10.1007/3-7643-7385-7_3.

[15]

R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.

[16]

G. Dore, Lp regularity for abstract differential equations, In Functional analysis and related topics, 1991 (Kyoto), volume 1540 of Lecture Notes in Math., pages 25-38. Springer, Berlin, 1993. doi: 10.1007/BFb0085472.

[17]

S. EngbolmM. Do-QuangG. Amberg and A.-K. Tornberg, On modeling and simulation of surfactants in diffuse interface flow, Communications in Computational Physics, 14 (2013), 875-915. doi: 10.4208/cicp.120712.281212a.

[18]

S. Ganesan and L. Tobiska, Arbitrary Lagrangian-Eulerian finite-element method for computation of two-phase flows with soluble surfactants, J. Comput. Phys., 231 (2012), 3685-3702. doi: 10.1016/j.jcp.2012.01.018.

[19]

H. GarckeK. F. Lam and B. Stinner, Diffuse interface modelling of soluble surfactants in two-phase flow, Commun. Math. Sci., 12 (2014), 1475-1522. doi: 10.4310/CMS.2014.v12.n8.a6.

[20]

H. Garcke and S. Wieland, Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system, SIAM J. Math. Anal., 37 (2006), 2025-2048. doi: 10.1137/040617017.

[21]

A. James and J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, Journal of Computational Physics, 201 (2004), 685-722. doi: 10.1016/j.jcp.2004.06.013.

[22]

S. Khatri and A.-K. Tornberg, A numerical method for two phase flows with insoluble surfactants, Computers and Fluids, 49 (2011), 150-165. doi: 10.1016/j.compfluid.2011.05.008.

[23]

M.-C. LaiY.-H. Tseng and H. Huang, An immersed boundary method for interfacial flows with insoluble surfactant, Journal of Computational Physics, 227 (2008), 7279-7293. doi: 10.1016/j.jcp.2008.04.014.

[24]

Y. Li and J. Kim, A comparison study of phase-field models for an immiscible binary mixture with surfactant, The European Physical Journal B-Condensed Matter and Complex Systems, 85 (2012), 1-9.

[25]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1 volume 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications.

[26]

H. Liu and Y. Zhang, Phase-field modeling droplet dynamics with soluble surfactants, Journal of Computational Physics, 229 (2010), 9166-9187.

[27]

M. Muradoglu and G. Tryggvason, A front-tracking method for computation of interfacial flows with soluble surfactants, Journal of Computational Physics, 227 (2008), 2238-2262.

[28]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, second edition, 2004.

[29]

J. Simon, Compact sets in the space Lp(0; T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[30]

K. TeigenX. LiJ. LowengrubF. Wang and A. Voigt, A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Communications in Mathematical Sciences, 7 (2009), 1009-1037.

[31]

K. TeigenP. SongJ. Lowengrub and A. Voigt, A diffuse-interface method for two-phase flows with soluble surfactants, Journal of Computational Physics, 230 (2011), 375-393. doi: 10.1016/j.jcp.2010.09.020.

[32]

R. van der Sman and S. van der Graaf, Diffuse interface model of surfactant adsorption onto flat and droplet interfaces, Rheology Acta, 46 (2006), 3-11.

[33]

J. Weber, Analysis of diffuse interface models for two-phase flows with and without surfactants, 2016. PhD thesis, University of Regensburg, urn:nbn:de:bvb:355-epub-342471.

[34]

J. XuZ. LiJ. Lowengrub and H. Zhao, A level-set method for interfacial flows with surfactant, Journal of Computational Physics, 212 (2006), 590-616. doi: 10.1016/j.jcp.2005.07.016.

[35]

E. Zeidler, Applied Functional Analysis, volume 108 of Applied Mathematical Sciences, Springer-Verlag, New York, 1995.

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