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Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect

Published online by Cambridge University Press:  25 July 2016

HAO WU*
Affiliation:
School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, 200433 Shanghai, China email: [email protected]

Abstract

We investigate a non-isothermal diffuse-interface model that describes the dynamics of two-phase incompressible flows with thermo-induced Marangoni effect. The governing PDE system consists of the Navier--Stokes equations coupled with convective phase-field and energy transport equations, in which the surface tension, fluid viscosity and thermal diffusivity are allowed to be temperature dependent functions. First, we establish the existence and uniqueness of local strong solutions when the spatial dimension is two and three. Then, in the two-dimensional case, assuming that the L-norm of the initial temperature is suitably bounded with respect to the coefficients of the system, we prove the existence of global weak solutions as well as the existence and uniqueness of global strong solutions.

Type
Papers
Copyright
Copyright © Cambridge University Press 2016 

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References

[1] Abels, H. (2009) On a diffuse interface model for two-phase flows of viscous incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194 (2), 463506.CrossRefGoogle Scholar
[2] Anderson, D.-M., McFadden, G.-B. & Wheeler, A.-A. (1998) Diffuse-interface methods in fluid mechanics. Annu. Review of Fluid Mech. 30, 139165.CrossRefGoogle Scholar
[3] Borcia, R. & Bestehorn, M. (2003) Phase-field model for Marangoni convection in liquid-gas systems with a deformable interface. Phys. Rev. E 67, 066307.CrossRefGoogle ScholarPubMed
[4] Borcia, R. & Bestehorn, M. (2006) Phase-field models for Marangoni convection in planar layers. J. Optoelectron. Adv. Mater. 8, 10371039.Google Scholar
[5] Boyer, F. (1999) Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20 (2), 175212.Google Scholar
[6] Cahn, J.-W. & Hillard, J.-E. (1958) Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258267.CrossRefGoogle Scholar
[7] Climent-Ezquerra, B., Guillén-González, F. & Jesus Moreno-Iraberte, M. (2009) Regularity and time-periodicity for a nematic liquid crystal model. Nonlinear Anal. 71 (1 & 2), 539549.CrossRefGoogle Scholar
[8] Eleuteri, M., Rocca, E. & Schimperna, G. (2014) Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids. arXiv:1406.1635.Google Scholar
[9] Eleuteri, M., Rocca, E. & Schimperna, G. (2015) On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete Contin. Dyn. Syst. 35 (6), 24972522.CrossRefGoogle Scholar
[10] Feng, J., Liu, C., Shen, J. & Yue, P.-T. (2005) An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: Advantages and challenges. In: Calderer, M.-C. T. & Terentjev, E. (editors), Modeling of Soft Matter, IMA Volumes in Mathematics and its Applications, Vol. 141, Springer, New York, pp. 126.Google Scholar
[11] Feng, X.-B., He, Y.-N. & Liu, C. (2007) Analysis of finite element approximations of a phase field model for two phase fluids. Math. Comp. 76 (258), 539571.CrossRefGoogle Scholar
[12] Gal, C. & Grasselli, M. (2010) Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete Conti. Dyna. Sys. 28 (1), 139.CrossRefGoogle Scholar
[13] Gal, C. & Grasselli, M. (2010) Asymptotic behavior of a Cahn–Hilliard–navier–stokes system in 2D. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (1), 401436.CrossRefGoogle Scholar
[14] Gal, C. & Grasselli, M. (2010) Trajectory attractors for binary fluid mixtures in 3D. Chinese Ann. Math. Ser. B 31 (5), 655678.CrossRefGoogle Scholar
[15] Guo, Z., Lin, P. & Wang, Y. (2014) Continuous finite element schemes for a phase field model in two-layer fluid Bénard–Marangoni convection computations. Comp. Phys. Commun. 185 (1), 6378.CrossRefGoogle Scholar
[16] Hou, T. & Li, C. (2005) Global well-posedness of the viscous Boussinesq equations. Discrete Conti. Dynam. Sys. 12 (1), 112.CrossRefGoogle Scholar
[17] Hua, J.-S., Lin, P., Liu, C. & Wang, Q. (2011) Energy law preserving C 0 finite element schemes for phase field models in two-phase flow computations. J. Comp. Phys. 230 (19), 71157131.CrossRefGoogle Scholar
[18] Huang, A.-M. (2015) The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with varibale viscosity and the thermal diffusivity. Nonlinear Anal. 113, 401429.CrossRefGoogle Scholar
[19] Lai, M.-J., Pan, R.-H. & Zhao, K. (2011) Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch. Ration. Mech. Anal. 199 (3), 739760.CrossRefGoogle Scholar
[20] Li, H.-P., Pan, R.-H. & Zhang, W.-Z. (2015) Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion. J. Hyper. Differ. Equ. 12 (3), 469488.CrossRefGoogle Scholar
[21] Lieberman, G.-M. (1996) Second Order Parabolic Differential Equations, World Scientific, Singapore.CrossRefGoogle Scholar
[22] Lin, F.-H. & Liu, C. (1995) Nonparabolic dissipative system modeling the flow of liquid crystals. Comm. Pure Appl. Math. XLVIII, 501537.CrossRefGoogle Scholar
[23] Liu, C. & Shen, J. (2003) A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179 (3 & 4), 211228.CrossRefGoogle Scholar
[24] Liu, C., Shen, J., Feng, J. & Yue, P.-T. (2005) Variational approach in two-phase flows of complex fluids: transport and induced elastic stress. In: Miranville, A. (editor), Mathematical Models and Methods in Phase Transitions, Nova Publishers, New York, pp. 259278.Google Scholar
[25] Lorca, S.-A. & Boldrini, J.-L. (1996) Stationary solutions for generalized Boussinesq models. J. Diff. Equ. 124 (2), 389406.CrossRefGoogle Scholar
[26] Lorca, S.-A. & Boldrini, J.-L. (1996) The initial value problem for a generalized Boussinesq model: Regularity and global existence of strong solutions. Mat. Contemp. 11, 7194.Google Scholar
[27] Lorca, S.-A. & Boldrini, J.-L. (1999) The initial value problem for a generalized Boussinesq model. Nonlinear Anal. 36 (4), 457480.CrossRefGoogle Scholar
[28] Lowengrub, J. & Truskinovsky, L. (1998) Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, 26172654.CrossRefGoogle Scholar
[29] Marangoni, C. (1871) Ueber die Ausbreitung der Tropfen einer Flussigkeit auf der Oberflache einer anderen. Ann. Phys. Chem. (Poggendorff) 143 (7), 337354.CrossRefGoogle Scholar
[30] Mendes-Tatsis, M.-A. & Agble, D. (2000) The effect of surfactants on Marangoni convection in the isobutanol/water system. J. Non-Equilib. Thermodyn. 25 (3 & 4), 239249.Google Scholar
[31] Pata, V. & Zelik, S. (2007) A result on the existence of global attractors for semigroups of closed operators. Commun. Pure Appl. Anal. 6 (2), 481486.CrossRefGoogle Scholar
[32] Simon, J. (1987) Compact sets in the space Lp (0,T;B). Ann. Mat. Pura Appl. 146 (1), 6596.CrossRefGoogle Scholar
[33] Sternling, C.-V. & Scriven, E. (1959) Interfacial turbulence: Hydrodynamic instability and the marangoni effect. A. I. Ch. E. J. 5, 514523.CrossRefGoogle Scholar
[34] Sun, P.-T., Liu, C. & Xu, J.-C. (2009) Phase-field model of thermo-induced Marangoni effects in the mixtures and its numerical similations with mixed finite element method. Commun. Comput. Phys. 6 (5), 10951117.CrossRefGoogle Scholar
[35] Sun, Y.-Z. & Zhang, Z.-F. (2013) Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity. J. Differ. Equ. 255 (6), 10691085.CrossRefGoogle Scholar
[36] Temam, R. (1977) Navier–Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and its Applications, Vol. 2, Oxford, North-Holland, Amsterdam, New York.Google Scholar
[37] Thompson, J. (1855) On certain curious motions observable at the surfaces of wine and other alcoholic liquors. Phil. Mag. 10 (67), 330333.CrossRefGoogle Scholar
[38] Wang, C. & Zhang, Z.-F. (2011) Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. Adv. Math. 228 (1), 4362.CrossRefGoogle Scholar
[39] Wu, H. & Xu, X. (2013) Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects. Commun. Math. Sci. 11 (2), 603633.CrossRefGoogle Scholar
[40] Yue, P.-T., Feng, J., Liu, C. & Shen, J. (2004) A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293317.CrossRefGoogle Scholar
[41] Yue, P.-T., Feng, J., Liu, C. & Shen, J. (2005) Interfacial forces and Marangoni flow on a nematic drop retracting in an isotropic fluid. J. Colloid. Intert. Sci. 290 (1), 281288.CrossRefGoogle Scholar
[42] Zhao, K. (2011) Global regularity for a coupled Cahn–Hilliard–Boussinesq system on bounded domains. Quart. Appl. Math. 69 (2), 331356.CrossRefGoogle Scholar
[43] Zheng, S. (2004) Nonlinear Evolution Equations. Pitman series Monographs and Survey in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, Florida.CrossRefGoogle Scholar