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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[35]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[35]

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

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