Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface.

*(English)*Zbl 1422.65145Summary: A regularized system of equations describing a flow of isothermal two-component mixture with diffuse interface is studied. The equation of energy balance and its corollary, i.e., the law of non-increasing of the total energy are derived under general assumptions on the Helmholtz free energy of the mixture. Necessary and sufficient conditions for linearized stability of constant solutions are obtained (in particular case). A difference approximation of the problem is constructed in the two-dimensional periodic case on a nonuniform rectangular grid. The results of numerical experiments demonstrate a qualitative well-posedness of the problem and the applicability of the criterion of linearized stabilization in the original nonlinear formulation.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35Q35 | PDEs in connection with fluid mechanics |

##### Keywords:

two-component mixture; diffuse interface; regularized Navier-Stokes-Cahn-Hilliard equations; finite-difference scheme
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\textit{V. Balashov} et al., Russ. J. Numer. Anal. Math. Model. 32, No. 6, 347--358 (2017; Zbl 1422.65145)

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[1] | F. O. Alpak, B. Riviere, and F. Frank, A phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition. Comput. Geosci. 20 (2016), 881-908. · Zbl 1391.76797 |

[2] | D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30 (1998), 139-165. |

[3] | V. A. Balashov and E. B. Savenkov, Quasi-hydrodynamic equations for diffuse interface type multiphase flow model with surface effects. Preprint No. 75, Keldysh Inst. Appl. Math., 2015 (in Russian). |

[4] | V. A. Balashov and E. B. Savenkov, Numerical study of two-dimensional quasi-hydrodynamic model of two-phase isothermal fluid flow with surface effects. Preprint No. 13, Keldysh Inst. Appl. Math., 2016 (in Russian). |

[5] | B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations. CIMNE, Barcelona, 2008. |

[6] | B. Chetverushkin, N. D’Ascenzo, S. Ishanov, and V. Savel’ev, Hyperbolic type explicit kinetic scheme of magneto gas dynamics for high performance computing systems. Russ. J. Numer. Anal. Math. Modelling30 (2015), No. 1, 27-36. · Zbl 1310.82003 |

[7] | B. Ducomet and A. Zlotnik, On a regularization of the magnetic gas dynamics system of equations. Kinetic Relat. Models6 (2013), No. 3, 533-543. · Zbl 1264.76117 |

[8] | T. G. Elizarova, Quasi-Gas Dynamic Equations. Springer, Berlin-Heidelberg, 2009. · Zbl 1169.76001 |

[9] | T. G. Elizarova, A. A. Zlotnik, and B. N. Chetverushkin, On quasi-gasdynamic and quasi-hydrodynamic equations for binary mixtures of gases. Doklady Math. 90 (2014), No. 3, 1-5. · Zbl 1315.76028 |

[10] | X. Feng, Fully discrete finite element approximation of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase flows. SIAM J. Numer. Anal. 44 (2006), 1049-1072. · Zbl 1344.76052 |

[11] | O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics. Springer, Berlin-Heidelberg-New York, 1985. · Zbl 0588.35003 |

[12] | J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. Royal Soc. London A454 (1998), 2617-2654. · Zbl 0927.76007 |

[13] | P. Meakin and A. M. Tartakovsky, Modelling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev. Geophys. 47 (2009), RG3002. |

[14] | M. V. Popov and T. G. Elizarova, Smoothed MHD equations for numerical simulations of ideal quasi-neutral gas dynamic flows. Comput. Phys. Commun. 196 (2015), 348-361. · Zbl 1360.76358 |

[15] | Yu. V. Sheretov, Continuum Dynamics Under Spatiotemporal Averaging. RChD, Moscow, Izhevsk, 2009 (in Russian). · Zbl 1357.76005 |

[16] | Yu. V. Sheretov, Regularized Hydrodynamic Equations. Tver’ State Univ., Tver’, 2016 (in Russian). |

[17] | A. A. Zlotnik, Parabolicity of a quasi-hydrodynamic system of equations and the stability of its small perturbations. Math. Notes83 (2008), No. 5, 610-623. · Zbl 1160.35326 |

[18] | A. A. Zlotnik, Energy equalities and estimates for barotropic quasi-gasdynamic and quasi-hydrodynamic systems of equations. Comput. Math. Math. Phys. 50 (2010), No. 2, 310-321. · Zbl 1224.35356 |

[19] | A. A. Zlotnik, On construction of quasi-gasdynamic systems of equations and the barotropic system with a potential body force. Math. Model. 24 (2012), No. 4, 65-79 (in Russian). · Zbl 1289.76068 |

[20] | A. A. Zlotnik, On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force. Comput. Math. Math. Phys. 56 (2016), No. 2, 303-319. · Zbl 1382.76198 |

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