December  2020, 7(2): 291-312. doi: 10.3934/jcd.2020012

An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations

1. 

Keldysh Institute of Applied Mathematics, Miusskaya sqr., 4, 125047 Moscow, Russia

2. 

National Research University Higher School of Economics, Pokrovskii bd. 11, 109028 Moscow, Russia, Keldysh Institute of Applied Mathematics, Miusskaya sqr., 4, 125047 Moscow, Russia

* Corresponding author

Received  September 2019 Published  July 2020

Fund Project: The study was supported by the Russian Science Foundation, project no. 19-11-00169

We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range $ (0,1) $. This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.

Citation: Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
References:
[1]

M. O. Abu-Al-SaudS. Popinet and H. A. Tchelepi, A conservative and well-balanced surface tension model, J. Comput. Phys., 371 (2018), 896-913.  doi: 10.1016/j.jcp.2018.02.022.  Google Scholar

[2]

D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar

[3]

A. Arakawa and V. R. Lamb, Computational design of the basic dynamical processes of the UCLA general circulation model, Methods Comput. Phys., 17 (1977), 173-265.   Google Scholar

[4]

V. A. Balashov and E. B. Savenkov, Quasi-hydrodynamic model of multiphase fluid flows taking into account phase interaction, J. Appl. Mech. Tech. Phys., 59 (2018), 434-444.  doi: 10.1134/S0021894418030069.  Google Scholar

[5]

V. A. Balashov and E. B. Savenkov, Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations, J. Comput. Appl. Math., 372 (2020), 112743, 16 pp. doi: 10.1016/j.cam.2020.112743.  Google Scholar

[6]

V. BalashovE. Savenkov and A. Zlotnik, Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics, Russian J. Numer. Anal. Math. Modelling, 34 (2019), 1-13.  doi: 10.1515/rnam-2019-0001.  Google Scholar

[7]

V. Balashov and A. Zlotnik, An energy dissipative spatial discretization for the regularized compressible Navier–Stokes–Cahn–Hilliard system of equations, Math. Model. Anal., 25 (2020), 110-129.  doi: 10.3846/mma.2020.10577.  Google Scholar

[8]

V. BalashovA. Zlotnik and E. Savenkov, Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface, Russian J. Numer. Anal. Math. Modelling, 32 (2017), 347-358.  doi: 10.1515/rnam-2017-0033.  Google Scholar

[9] M. J. Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, Cambridge University Press, 2017.  doi: 10.1017/9781316145098.  Google Scholar
[10]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[11]

B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations, CIMNE, Barcelona, 2008. Google Scholar

[12]

K. Connington and T. Lee, A review of spurious currents in the lattice Boltzmann method for multiphase flows, J. Mech. Sci. Technology, 26 (2012), 3857-3863.  doi: 10.1007/s12206-012-1011-5.  Google Scholar

[13]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.  Google Scholar

[14]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.  doi: 10.1016/0362-546X(94)00205-V.  Google Scholar

[15]

A. Yu. Demianov, O. Yu. Dinariev and N. V. Evseev, Introduction to the Density Functional Method in Hydrodynamics, Fizmatlit, Moscow, 2014. Google Scholar

[16]

T. G. Elizarova, Quasi-Gas Dynamic Equations, Computational Fluid and Solid Mechanics. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.  Google Scholar

[17]

F. FrankC. LiuF. O. Alpak and B. Riviere, A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging, Comput. Geosci., 22 (2018), 543-563.  doi: 10.1007/s10596-017-9709-1.  Google Scholar

[18]

Y. GongJ. Zhao and Q. Wang, An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities, Comput. Phys. Comm., 219 (2017), 20-34.  doi: 10.1016/j.cpc.2017.05.002.  Google Scholar

[19]

L. GoudenègeD. Martin and G. Vial, High order finite element calculations for the Cahn-Hilliard equation, J. Sci. Comput., 52 (2012), 294-321.  doi: 10.1007/s10915-011-9546-7.  Google Scholar

[20]

J.-L. Guermond and B. Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74 (2014), 284-305.  doi: 10.1137/120903312.  Google Scholar

[21]

Z. GuoP. LinJ. Lowengrub and S. M. Wise, 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. Eng., 326 (2017), 144-174.  doi: 10.1016/j.cma.2017.08.011.  Google Scholar

[22]

D. J. E. HarvieM. R. Davidson and M. Rudman, An analysis of parasitic current generation in volume of fluid simulations, Appl. Math. Model., 30 (2006), 1056-1066.  doi: 10.1016/j.apm.2005.08.015.  Google Scholar

[23] P. C. Hiemenz and T. P. Lodge, Polymer Chemistry, 2$^{nd}$ edition, CRC Press, 2007.   Google Scholar
[24]

D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.  Google Scholar

[25]

D. JametD. Torres and J. U. Brackbill, On the theory and computation of surface tension: The elimination of parasitic currents through energy conservation in the second-gradient method, J. Comput. Phys., 182 (2002), 262-276.  doi: 10.1006/jcph.2002.7165.  Google Scholar

[26]

J. Liu, Thermodynamically Consistent Modeling and Simulation of Multiphase Flows, PhD dissertation, the University of Texas at Austin, 2014. Google Scholar

[27]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1978), 2617-2654.  doi: 10.1098/rspa.1998.0273.  Google Scholar

[28]

S. Minjeaud, An adaptive pressure correction method without spurious velocities for diffuse-interface models of incompressible flows, J. Comput. Phys., 236 (2013), 143-156.  doi: 10.1016/j.jcp.2012.11.022.  Google Scholar

[29]

N. Provatas and K. Elde, Phase-Field Methods in Material Science and Engineering, Willey-VCH, Weinheim, 2010. doi: 10.1002/9783527631520.  Google Scholar

[30]

Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging, RKhD, Moscow-Izhevsk, 2009 [in Russian]. Google Scholar

[31]

M. Svärd, A new Eulerian model for viscous and heat conducting compressible flows, Phys. A, 506 (2018), 350-375.  doi: 10.1016/j.physa.2018.03.097.  Google Scholar

[32]

G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models, Arch. Comput. Meth. Eng., 22 (2015), 269-289.  doi: 10.1007/s11831-014-9112-1.  Google Scholar

[33]

P. YueC. Zhou and J. J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations, J. Comput. Phys., 223 (2007), 1-9.  doi: 10.1016/j.jcp.2006.11.020.  Google Scholar

[34]

I. Zacharov et al., "Zhores" – Petaflops supercomputer for data-driven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology, preprint, arXiv: 1902.07490. Google Scholar

[35]

A. Zlotnik, On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes systems of equations in polar coordinates, Russ. J. Numer. Anal. Math. Model., 33 (2018), 199-210.  doi: 10.1515/rnam-2018-0017.  Google Scholar

[36]

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), 303-319.  doi: 10.1134/S0965542516020160.  Google Scholar

[37]

A. A. Zlotnik and T. A. Lomonosov, Conditions for $L^2$-dissipativity of linearized explicit difference schemes with regularization for 1D barotropic gas dynamics equations, Comput. Math. Math. Phys., 59 (2019), 452-464.  doi: 10.1134/S0965542519030151.  Google Scholar

[38]

A. A. Zlotnik and T. A. Lomonosov, On $L^2$-dissipativity of a linearized explicit finite-difference scheme with quasi-gasdynamic regularization for the barotropic gas dynamics system of equations, Dokl. Math., 101 (2020) (in press). Google Scholar

show all references

References:
[1]

M. O. Abu-Al-SaudS. Popinet and H. A. Tchelepi, A conservative and well-balanced surface tension model, J. Comput. Phys., 371 (2018), 896-913.  doi: 10.1016/j.jcp.2018.02.022.  Google Scholar

[2]

D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar

[3]

A. Arakawa and V. R. Lamb, Computational design of the basic dynamical processes of the UCLA general circulation model, Methods Comput. Phys., 17 (1977), 173-265.   Google Scholar

[4]

V. A. Balashov and E. B. Savenkov, Quasi-hydrodynamic model of multiphase fluid flows taking into account phase interaction, J. Appl. Mech. Tech. Phys., 59 (2018), 434-444.  doi: 10.1134/S0021894418030069.  Google Scholar

[5]

V. A. Balashov and E. B. Savenkov, Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations, J. Comput. Appl. Math., 372 (2020), 112743, 16 pp. doi: 10.1016/j.cam.2020.112743.  Google Scholar

[6]

V. BalashovE. Savenkov and A. Zlotnik, Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics, Russian J. Numer. Anal. Math. Modelling, 34 (2019), 1-13.  doi: 10.1515/rnam-2019-0001.  Google Scholar

[7]

V. Balashov and A. Zlotnik, An energy dissipative spatial discretization for the regularized compressible Navier–Stokes–Cahn–Hilliard system of equations, Math. Model. Anal., 25 (2020), 110-129.  doi: 10.3846/mma.2020.10577.  Google Scholar

[8]

V. BalashovA. Zlotnik and E. Savenkov, Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface, Russian J. Numer. Anal. Math. Modelling, 32 (2017), 347-358.  doi: 10.1515/rnam-2017-0033.  Google Scholar

[9] M. J. Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, Cambridge University Press, 2017.  doi: 10.1017/9781316145098.  Google Scholar
[10]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[11]

B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations, CIMNE, Barcelona, 2008. Google Scholar

[12]

K. Connington and T. Lee, A review of spurious currents in the lattice Boltzmann method for multiphase flows, J. Mech. Sci. Technology, 26 (2012), 3857-3863.  doi: 10.1007/s12206-012-1011-5.  Google Scholar

[13]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.  Google Scholar

[14]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.  doi: 10.1016/0362-546X(94)00205-V.  Google Scholar

[15]

A. Yu. Demianov, O. Yu. Dinariev and N. V. Evseev, Introduction to the Density Functional Method in Hydrodynamics, Fizmatlit, Moscow, 2014. Google Scholar

[16]

T. G. Elizarova, Quasi-Gas Dynamic Equations, Computational Fluid and Solid Mechanics. Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-00292-2.  Google Scholar

[17]

F. FrankC. LiuF. O. Alpak and B. Riviere, A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging, Comput. Geosci., 22 (2018), 543-563.  doi: 10.1007/s10596-017-9709-1.  Google Scholar

[18]

Y. GongJ. Zhao and Q. Wang, An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities, Comput. Phys. Comm., 219 (2017), 20-34.  doi: 10.1016/j.cpc.2017.05.002.  Google Scholar

[19]

L. GoudenègeD. Martin and G. Vial, High order finite element calculations for the Cahn-Hilliard equation, J. Sci. Comput., 52 (2012), 294-321.  doi: 10.1007/s10915-011-9546-7.  Google Scholar

[20]

J.-L. Guermond and B. Popov, Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74 (2014), 284-305.  doi: 10.1137/120903312.  Google Scholar

[21]

Z. GuoP. LinJ. Lowengrub and S. M. Wise, 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. Eng., 326 (2017), 144-174.  doi: 10.1016/j.cma.2017.08.011.  Google Scholar

[22]

D. J. E. HarvieM. R. Davidson and M. Rudman, An analysis of parasitic current generation in volume of fluid simulations, Appl. Math. Model., 30 (2006), 1056-1066.  doi: 10.1016/j.apm.2005.08.015.  Google Scholar

[23] P. C. Hiemenz and T. P. Lodge, Polymer Chemistry, 2$^{nd}$ edition, CRC Press, 2007.   Google Scholar
[24]

D. Jacqmin, Calculation of two-phase Navier–Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.  Google Scholar

[25]

D. JametD. Torres and J. U. Brackbill, On the theory and computation of surface tension: The elimination of parasitic currents through energy conservation in the second-gradient method, J. Comput. Phys., 182 (2002), 262-276.  doi: 10.1006/jcph.2002.7165.  Google Scholar

[26]

J. Liu, Thermodynamically Consistent Modeling and Simulation of Multiphase Flows, PhD dissertation, the University of Texas at Austin, 2014. Google Scholar

[27]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1978), 2617-2654.  doi: 10.1098/rspa.1998.0273.  Google Scholar

[28]

S. Minjeaud, An adaptive pressure correction method without spurious velocities for diffuse-interface models of incompressible flows, J. Comput. Phys., 236 (2013), 143-156.  doi: 10.1016/j.jcp.2012.11.022.  Google Scholar

[29]

N. Provatas and K. Elde, Phase-Field Methods in Material Science and Engineering, Willey-VCH, Weinheim, 2010. doi: 10.1002/9783527631520.  Google Scholar

[30]

Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging, RKhD, Moscow-Izhevsk, 2009 [in Russian]. Google Scholar

[31]

M. Svärd, A new Eulerian model for viscous and heat conducting compressible flows, Phys. A, 506 (2018), 350-375.  doi: 10.1016/j.physa.2018.03.097.  Google Scholar

[32]

G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models, Arch. Comput. Meth. Eng., 22 (2015), 269-289.  doi: 10.1007/s11831-014-9112-1.  Google Scholar

[33]

P. YueC. Zhou and J. J. Feng, Spontaneous shrinkage of drops and mass conservation in phase-field simulations, J. Comput. Phys., 223 (2007), 1-9.  doi: 10.1016/j.jcp.2006.11.020.  Google Scholar

[34]

I. Zacharov et al., "Zhores" – Petaflops supercomputer for data-driven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology, preprint, arXiv: 1902.07490. Google Scholar

[35]

A. Zlotnik, On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes systems of equations in polar coordinates, Russ. J. Numer. Anal. Math. Model., 33 (2018), 199-210.  doi: 10.1515/rnam-2018-0017.  Google Scholar

[36]

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), 303-319.  doi: 10.1134/S0965542516020160.  Google Scholar

[37]

A. A. Zlotnik and T. A. Lomonosov, Conditions for $L^2$-dissipativity of linearized explicit difference schemes with regularization for 1D barotropic gas dynamics equations, Comput. Math. Math. Phys., 59 (2019), 452-464.  doi: 10.1134/S0965542519030151.  Google Scholar

[38]

A. A. Zlotnik and T. A. Lomonosov, On $L^2$-dissipativity of a linearized explicit finite-difference scheme with quasi-gasdynamic regularization for the barotropic gas dynamics system of equations, Dokl. Math., 101 (2020) (in press). Google Scholar

Figure 1.  $ \tilde\Psi_0(C)\equiv\Psi_0(\tilde\rho, C) $ for $ \omega_2 > \omega_1 $ and some fixed $ \tilde\rho>0 $
Figure 4.  Distributions of $ C $ and $ \rho $ along the segment $ x_1\in[0.5X, X] $ and $ x_2 = x_3 = 0.5X $, at the vicinity of the interface at $ t = 20\cdot10^3\Delta t $
Figure 2.  Location of nodes of $ \omega_{h\bar{k},\bar{l}^*,\bar{m}^*} $ (thick dots) and $ \omega_{h\bar{k}^*,\bar{l},\bar{m}} $ (red crosses), where $ u_k $ and $ \Pi_{lk} $, $ l\neq k $, are respectively defined
Figure 3.  Droplet interface evolution in the case (Ⅰ) for $ R = 0.25X $. Distribution of $ C(x) $ in the section $ x_1,x_2\in[0.5X,0.87X] $, $ x_3 = 0.5X $, is represented
Figure 5.  Evolution of $ \sigma_L(t) $: for $ R = 0.18X $ in the cases (Ⅰ)-(Ⅲ) (left) and for several $ R $ in the case (Ⅲ) (right)
Figure 6.  Observable dependence of $ \Delta p $ on $ 1/R_a $ for different $ R $ in (46)
Figure 7.  Evolution of $ {\bar{E}}_{\text{kin}} $ and $ \mathcal{E}_h-\tilde{E} $ for the droplet with $ R = 0.25X $ and $ \tilde{E} = 9.16034\cdot 10^{-5}\, \text{J} $
Figure 8.  $ {\bar{E}}_{\text{kin}} (t) $ computed by schemes $ A $ (from this paper) and $ B $ (from [6])
Figure 9.  Evolution of $ C_{\min} $ and $ C_{\max} $ (for spinodal decomposition)
Figure 10.  $ \bar{E}_\text{kin}(t) $ for $ \alpha^\ast = 0 $ and $ 0.5 $ and some $ \Delta t $ (the break in the graph line means that computations collapse due to instability)
Figure 12.  $ \bar{E}_{\text{kin}}(t) $ for some $ \alpha^\ast\geq0.5 $ and $ \Delta t $ (the break in the graph line means that computations collapse due to instability)
Figure 13.  Evolution of $ {\bar{E}}_{\text{kin}} $ and $ \mathcal{E}_h $ (for spinodal decomposition)
Figure 14.  Isosurfaces $ C = 0.5 $ at different time moments (for spinodal decomposition)
[1]

Dieter Bothe, Jan Prüss. Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 673-696. doi: 10.3934/dcdss.2017034

[2]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401

[3]

Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325

[4]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1

[5]

Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138

[6]

Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419

[7]

Feng Ma, Mingfang Ni. A two-phase method for multidimensional number partitioning problem. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 203-206. doi: 10.3934/naco.2013.3.203

[8]

Juan Vicente Gutiérrez-Santacreu. Two scenarios on a potential smoothness breakdown for the three-dimensional Navier–Stokes equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2593-2613. doi: 10.3934/dcds.2020142

[9]

Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405

[10]

T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067

[11]

Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evolution Equations & Control Theory, 2012, 1 (1) : 171-194. doi: 10.3934/eect.2012.1.171

[12]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[13]

Massimo Lanza de Cristoforis, aolo Musolino. A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2509-2542. doi: 10.3934/cpaa.2014.13.2509

[14]

Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159

[15]

Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna. On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2497-2522. doi: 10.3934/dcds.2015.35.2497

[16]

Clément Cancès. On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Networks & Heterogeneous Media, 2010, 5 (3) : 635-647. doi: 10.3934/nhm.2010.5.635

[17]

Piotr Bogusław Mucha, Milan Pokorný, Ewelina Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1079-1099. doi: 10.3934/dcdss.2014.7.1079

[18]

Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041

[19]

Christian Klingenberg, Marlies Pirner, Gabriella Puppo. A consistent kinetic model for a two-component mixture with an application to plasma. Kinetic & Related Models, 2017, 10 (2) : 445-465. doi: 10.3934/krm.2017017

[20]

Mike Hay, Andrew N. W. Hone, Vladimir S. Novikov, Jing Ping Wang. Remarks on certain two-component systems with peakon solutions. Journal of Geometric Mechanics, 2019, 11 (4) : 561-573. doi: 10.3934/jgm.2019028

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]