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Journal articles on the topic 'Navier-Stokes-Cahn-Hilliard model'

Author: Grafiati

Published: 30 November 2024

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1

Li, Xiaoli, and Jie Shen. "On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case." Mathematical Models and Methods in Applied Sciences 30, no. 12 (October 19, 2020): 2263–97. http://dx.doi.org/10.1142/s0218202520500438.

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We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme.

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2

Medjo, T. Tachim. "A Cahn-Hilliard-Navier-Stokes model with delays." Discrete and Continuous Dynamical Systems - Series B 21, no. 8 (September 2016): 2663–85. http://dx.doi.org/10.3934/dcdsb.2016067.

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3

Medjo, T. "Robust control of a Cahn-Hilliard-Navier-Stokes model." Communications on Pure and Applied Analysis 15, no. 6 (September 2016): 2075–101. http://dx.doi.org/10.3934/cpaa.2016028.

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4

Kotschote, Matthias, and Rico Zacher. "Strong solutions in the dynamical theory of compressible fluid mixtures." Mathematical Models and Methods in Applied Sciences 25, no. 07 (April 14, 2015): 1217–56. http://dx.doi.org/10.1142/s0218202515500311.

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In this paper we investigate the compressible Navier–Stokes–Cahn–Hilliard equations (the so-called NSCH model) derived by Lowengrub and Truskinovsky. This model describes the flow of a binary compressible mixture; the fluids are supposed to be macroscopically immiscible, but partial mixing is permitted leading to narrow transition layers. The internal structure and macroscopic dynamics of these layers are induced by a Cahn–Hilliard law that the mixing ratio satisfies. The PDE constitute a strongly coupled hyperbolic–parabolic system. We establish a local existence and uniqueness result for strong solutions.

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5

Boyer, Franck, and Sebastian Minjeaud. "Hierarchy of consistent n-component Cahn–Hilliard systems." Mathematical Models and Methods in Applied Sciences 24, no. 14 (October 16, 2014): 2885–928. http://dx.doi.org/10.1142/s0218202514500407.

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In this paper, we propose a new generalization of the well-known Cahn–Hilliard two-phase model for the modeling of n-phase mixtures. The model is derived using the consistency principle: we require that our n-phase model exactly coincides with the classical two-phase model when only two phases are present in the system. We give conditions for the model to be well-posed. We also present numerical results (including simulations obtained when coupling the Cahn–Hilliard system with the Navier–Stokes so as to obtain a phase-field model for multiphase flows) to illustrate the capability of such modeling.

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6

LAM, KEI FONG, and HAO WU. "Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis." European Journal of Applied Mathematics 29, no. 4 (October 9, 2017): 595–644. http://dx.doi.org/10.1017/s0956792517000298.

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We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

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7

Li, Xiaoli, and Jie Shen. "On fully decoupled MSAV schemes for the Cahn–Hilliard–Navier–Stokes model of two-phase incompressible flows." Mathematical Models and Methods in Applied Sciences 32, no. 03 (January 31, 2022): 457–95. http://dx.doi.org/10.1142/s0218202522500117.

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We construct first- and second-order time discretization schemes for the Cahn–Hilliard–Navier–Stokes system based on the multiple scalar auxiliary variables (MSAV) approach for gradient systems and (rotational) pressure-correction for Navier–Stokes equations. These schemes are linear, fully decoupled, unconditionally energy stable, and only require solving a sequence of elliptic equations with constant coefficients at each time step. We carry out a rigorous error analysis for the first-order scheme, establishing optimal convergence rate for all relevant functions in different norms. We also provide numerical experiments to verify our theoretical results.

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8

Lasarzik, Robert. "Analysis of a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model." Nonlinear Analysis 213 (December 2021): 112526. http://dx.doi.org/10.1016/j.na.2021.112526.

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9

Deugoue, Gabriel, Boris Jidjou Moghomye, and Theodore Tachim Medjo. "Splitting-up scheme for the stochastic Cahn–Hilliard Navier–Stokes model." Stochastics and Dynamics 21, no. 01 (March 18, 2020): 2150005. http://dx.doi.org/10.1142/s0219493721500052.

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In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

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10

Chen, Jie, Shuyu Sun, and Zhangxin Chen. "Coupling Two-Phase Fluid Flow with Two-Phase Darcy Flow in Anisotropic Porous Media." Advances in Mechanical Engineering 6 (January 1, 2014): 871021. http://dx.doi.org/10.1155/2014/871021.

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This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.

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11

Metzger, Stefan. "On convergent schemes for two-phase flow of dilute polymeric solutions." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (November 2018): 2357–408. http://dx.doi.org/10.1051/m2an/2018042.

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We construct a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS 26 (2016) 823–866]). The model consists of Cahn-Hilliard type equations describing the evolution of the fluids and the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three spatial dimensions for the velocity and the pressure of the fluids with an elastic extra-stress tensor on the right-hand side in the momentum equation which originates from the presence of dissolved polymer chains. The polymers are modeled by dumbbells subjected to a finitely extensible, nonlinear elastic (FENE) spring-force potential. Their density and orientation are described by a Fokker-Planck type parabolic equation with a center-of-mass diffusion term. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of these finite element approximations converges towards a weak solution of the coupled Cahn-Hilliard-Navier-Stokes-Fokker-Planck system. To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.

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12

Anders, Denis, and Kerstin Weinberg. "A Thermodynamically Consistent Approach to Phase-Separating Viscous Fluids." Journal of Non-Equilibrium Thermodynamics 43, no. 2 (April 25, 2018): 185–91. http://dx.doi.org/10.1515/jnet-2017-0052.

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AbstractThe de-mixing properties of heterogeneous viscous fluids are determined by an interplay of diffusion, surface tension and a superposed velocity field. In this contribution a variational model of the decomposition, based on the Navier–Stokes equations for incompressible laminar flow and the extended Korteweg–Cahn–Hilliard equations, is formulated. An exemplary numerical simulation using {C}^{1}-continuous finite elements demonstrates the capability of this model to compute phase decomposition and coarsening of the moving fluid.

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13

Boyer, F., C. Lapuerta, S. Minjeaud, B. Piar, and M. Quintard. "Cahn–Hilliard/Navier–Stokes Model for the Simulation of Three-Phase Flows." Transport in Porous Media 82, no. 3 (May 7, 2009): 463–83. http://dx.doi.org/10.1007/s11242-009-9408-z.

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14

Sibley, David N., Andreas Nold, and Serafim Kalliadasis. "Unifying binary fluid diffuse-interface models in the sharp-interface limit." Journal of Fluid Mechanics 736 (November 1, 2013): 5–43. http://dx.doi.org/10.1017/jfm.2013.521.

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AbstractRecent results published by Gugenberger et al. on surface diffusion (Phys. Rev. E, vol. 78, 2008, 016703), show that the sharp-interface limit of the phase field models often adopted in the literature fails to produce the appropriate boundary conditions. With this knowledge, we consider the sharp-interface limit of phase field models for binary fluids, obtained carefully, where hydrodynamic equations are coupled to phase field evolution based on Cahn–Hilliard or Allen–Cahn theories, in a variety of guises, and unify and contrast their forms and behaviours in the sharp-interface limit. In particular, a tensorial mobility model is analysed, which allows the bulk fluids in the outer region to satisfy classical Navier–Stokes type equations to all orders in the Cahn number.

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15

Huang, Qiming, and Junxiang Yang. "Linear and Energy-Stable Method with Enhanced Consistency for the Incompressible Cahn–Hilliard–Navier–Stokes Two-Phase Flow Model." Mathematics 10, no. 24 (December 12, 2022): 4711. http://dx.doi.org/10.3390/math10244711.

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The Cahn–Hilliard–Navier–Stokes model is extensively used for simulating two-phase incompressible fluid flows. With the absence of exterior force, this model satisfies the energy dissipation law. The present work focuses on developing a linear, decoupled, and energy dissipation-preserving time-marching scheme for the hydrodynamics coupled Cahn–Hilliard model. An efficient time-dependent auxiliary variable approach is first introduced to design equivalent equations. Based on equivalent forms, a BDF2-type linear scheme is constructed. In each time step, the unique solvability and the energy dissipation law can be analytically estimated. To enhance the energy stability and the consistency, we correct the modified energy by a practical relaxation technique. Using the finite difference method in space, the fully discrete scheme is described, and the numerical solutions can be separately implemented. Numerical results indicate that the proposed scheme has desired accuracy, consistency, and energy stability. Moreover, the flow-coupled phase separation, the falling droplet, and the dripping droplet are well simulated.

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16

Brunk, Aaron, and Mária Lukáčová-Medvid’ová. "Global existence of weak solutions to viscoelastic phase separation part: I. Regular case." Nonlinearity 35, no. 7 (June 16, 2022): 3417–58. http://dx.doi.org/10.1088/1361-6544/ac5920.

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Abstract We prove the existence of weak solutions to a viscoelastic phase separation problem in two space dimensions. The mathematical model consists of a Cahn–Hilliard-type equation for two-phase flows and the Peterlin–Navier–Stokes equations for viscoelastic fluids. We focus on the case of a polynomial-like potential and suitably bounded coefficient functions. Using the Lagrange–Galerkin finite element method complex behavior of solution for spinodal decomposition including transient polymeric network structures is demonstrated.

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17

Liang, Zhilei, and Dehua Wang. "Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows." Mathematical Models and Methods in Applied Sciences 30, no. 12 (October 23, 2020): 2445–86. http://dx.doi.org/10.1142/s0218202520500475.

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A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

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18

Tachim Medjo, T. "Pullback attractor of a three dimensional globally modified Cahn–Hilliard-Navier–Stokes model." Applicable Analysis 97, no. 6 (March 13, 2017): 1016–41. http://dx.doi.org/10.1080/00036811.2017.1296952.

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19

Espath, L. F. R., A. F. Sarmiento, P. Vignal, B. O. N. Varga, A. M. A. Cortes, L. Dalcin, and V. M. Calo. "Energy exchange analysis in droplet dynamics via the Navier–Stokes–Cahn–Hilliard model." Journal of Fluid Mechanics 797 (May 23, 2016): 389–430. http://dx.doi.org/10.1017/jfm.2016.277.

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We develop the energy budget equation of the coupled Navier–Stokes–Cahn–Hilliard (NSCH) system. We use the NSCH equations to model the dynamics of liquid droplets in a liquid continuum. Buoyancy effects are accounted for through the Boussinesq assumption. We physically interpret each quantity involved in the energy exchange to gain further insight into the model. Highly resolved simulations involving density-driven flows and the merging of droplets allow us to analyse these energy budgets. In particular, we focus on the energy exchanges when droplets merge, and describe flow features relevant to this phenomenon. By comparing our numerical simulations to analytical predictions and experimental results available in the literature, we conclude that modelling droplet dynamics within the framework of NSCH equations is a sensible approach worthy of further research.

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20

Shin, Jaemin, Junxiang Yang, Chaeyoung Lee, and Junseok Kim. "The Navier–Stokes–Cahn–Hilliard model with a high-order polynomial free energy." Acta Mechanica 231, no. 6 (April 2, 2020): 2425–37. http://dx.doi.org/10.1007/s00707-020-02666-y.

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21

Minjeaud, Sebastian. "An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model." Numerical Methods for Partial Differential Equations 29, no. 2 (April 27, 2012): 584–618. http://dx.doi.org/10.1002/num.21721.

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22

Deugoue, G., and T. Tachim Medjo. "Large deviation for a 2D Cahn-Hilliard-Navier-Stokes model under random influences." Journal of Mathematical Analysis and Applications 486, no. 1 (June 2020): 123863. http://dx.doi.org/10.1016/j.jmaa.2020.123863.

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23

Shen, Mingguang, and Ben Q. Li. "A Phase Field Approach to Modeling Heavy Metal Impact in Plasma Spraying." Coatings 12, no. 10 (September 22, 2022): 1383. http://dx.doi.org/10.3390/coatings12101383.

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A phase field model enhanced with the shared memory parallelism OpenMP was proposed, capable of modeling the impact of a heavy metal droplet under practical plasma spraying conditions. The finite difference solution of the Navier-Stokes equations, coupled with the Cahn-Hilliard equation, tracks the gas-liquid interface. The liquid fraction, defined over the computational domain, distinguishes fluid from solid. The model is employed for Ni and YSZ drop impacts after ruling out the effect of mesh size. The model exhibits a reasonable parallel-computing efficiency, and the predicted maximum spread factors agree well with analytical models.

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24

Zhao, Xiaopeng. "On the strong solution of 3D non-isothermal Navier–Stokes–Cahn–Hilliard equations." Journal of Mathematical Physics 64, no. 3 (March 1, 2023): 031506. http://dx.doi.org/10.1063/5.0099260.

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In this paper, we consider the global existence of strong solutions of a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. In the diffuse interface model, the evolution of the velocity u is ruled by the Navier–Stokes system, while the order parameter φ representing the difference of the fluid concentration of the two fluids is assumed to satisfy a convective Cahn–Hilliard equation. The effects of the temperature are prescribed by a suitable form of heat equation. By using a refined pure energy method, we prove the existence of the global strong solution by assuming that [Formula: see text] is sufficiently small, and higher order derivatives can be arbitrarily large.

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25

Gondyul, Е. А., and V. V. Lisitsa. "Modeling of unsteady flows of multiphase viscous fluid in a pore space." Interexpo GEO-Siberia 2, no. 2 (May 18, 2022): 32–37. http://dx.doi.org/10.33764/2618-981x-2022-2-2-32-37.

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The authors have developed and implemented a numerical algorithm to model unsteady flows of a viscous multiphase isothermal fluid by finite difference method using the projection method for the numerical solution of the Navier-Stokes equation. The projection method implies splitting the initial system of equations by physical processes, in which convective transport and the effect of the pressure gradient are separately taken into account. As a result, at each step, it is necessary to solve the Poisson equation to find the pressure field. The solution of SLAE is performed by a parallel direct solver based on LU decomposition. An explicit scheme is used to solve the Cahn-Hilliard equation to update the phase field, the parameter of which is taken into account when adding surface forces to the Navier-Stokes equation. Computational experiments showing qualitative and quantitative agreement with experimental and numerical data from the literature are presented.

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26

Balashov, Vladislav, Evgenii Savenkov, and Alexander Zlotnik. "Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 1 (February 25, 2019): 1–13. http://dx.doi.org/10.1515/rnam-2019-0001.

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Abstract We propose a numerical algorithm for simulations of two-component viscous compressible isothermal flows with surface effects in 3D domains of complex shape with voxel representation of geometry. The basic mathematical model is the regularized system of Navier–Stokes–Cahn–Hilliard equations. Simulations of droplet spreading over a flat base and displacement of one liquid by another one in a pore space of real rock sample are carried out. The simulation results demonstrate the applicability and good efficiency of the used system of equations, the corresponding difference scheme, and its implementation algorithms for numerical solution of the considered class of problems.

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27

Tachim Medjo, T. "The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise." Communications on Pure & Applied Analysis 18, no. 3 (2019): 1117–38. http://dx.doi.org/10.3934/cpaa.2019054.

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28

Tachim Medjo, T. "The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise." Communications on Pure & Applied Analysis 18, no. 6 (2019): 2961–82. http://dx.doi.org/10.3934/cpaa.2019132.

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29

Deugoué, G., A. Ndongmo Ngana, and T. Tachim Medjo. "On the strong solutions for a stochastic 2D nonlocal Cahn–Hilliard–Navier–Stokes model." Dynamics of Partial Differential Equations 17, no. 1 (2020): 19–60. http://dx.doi.org/10.4310/dpde.2020.v17.n1.a2.

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30

Naso, Aurore, and Lennon Ó. Náraigh. "A flow-pattern map for phase separation using the Navier–Stokes–Cahn–Hilliard model." European Journal of Mechanics - B/Fluids 72 (November 2018): 576–85. http://dx.doi.org/10.1016/j.euromechflu.2018.08.002.

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31

Magaletti, F., F. Picano, M. Chinappi, L. Marino, and C. M. Casciola. "The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids." Journal of Fluid Mechanics 714 (January 2, 2013): 95–126. http://dx.doi.org/10.1017/jfm.2012.461.

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AbstractThe Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.

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32

NOCHETTO, RICARDO H., ABNER J. SALGADO, and SHAWN W. WALKER. "A DIFFUSE INTERFACE MODEL FOR ELECTROWETTING WITH MOVING CONTACT LINES." Mathematical Models and Methods in Applied Sciences 24, no. 01 (October 31, 2013): 67–111. http://dx.doi.org/10.1142/s0218202513500474.

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We introduce a diffuse interface model for the phenomenon of electrowetting on dielectric and present an analysis of the arising system of equations. Moreover, we study discretization techniques for the problem. The model takes into account different material parameters on each phase and incorporates the most important physical processes, such as incompressibility, electrostatics and dynamic contact lines; necessary to properly reflect the relevant phenomena. The arising nonlinear system couples the variable density incompressible Navier–Stokes equations for velocity and pressure with a Cahn–Hilliard type equation for the phase variable and chemical potential, a convection diffusion equation for the electric charges and a Poisson equation for the electric potential. Numerical experiments are presented, which illustrate the wide range of effects the model is able to capture, such as splitting and coalescence of droplets.

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Tachim Medjo, Theodore, Cristina Tone, and Florentina Tone. "Maximum principle of optimal control of a Cahn–Hilliard–Navier–Stokes model with state constraints." Optimal Control Applications and Methods 42, no. 3 (January 24, 2021): 807–32. http://dx.doi.org/10.1002/oca.2701.

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Tierra, Francisco Guillén-González and Giordano. "Splitting Schemes for a Navier-Stokes-Cahn-Hilliard Model for Two Fluids with Different Densities." Journal of Computational Mathematics 32, no. 6 (June 2014): 643–64. http://dx.doi.org/10.4208/jcm.1405-m4410.

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35

Tachim Medjo, T. "Unique strong and attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model." Applicable Analysis 96, no. 16 (September 21, 2016): 2695–716. http://dx.doi.org/10.1080/00036811.2016.1236924.

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36

He, Yinnian, and Xinlong Feng. "Uniform H2-regularity of solution for the 2D Navier–Stokes/Cahn–Hilliard phase field model." Journal of Mathematical Analysis and Applications 441, no. 2 (September 2016): 815–29. http://dx.doi.org/10.1016/j.jmaa.2016.04.040.

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37

Tachim Medjo, T. "Weak solution of a stochastic 3D Cahn-Hilliard-Navier-Stokes model driven by jump noise." Journal of Mathematical Analysis and Applications 484, no. 1 (April 2020): 123680. http://dx.doi.org/10.1016/j.jmaa.2019.123680.

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38

Budiana, Eko Prasetya, Pranowo Pranowo, Catur Harsito, Dominicus Danardono Dwi Prija Tjahjana, and Syamsul Hadi. "Numerical Simulation of Droplet Coalescence Using Meshless Radial Basis Function and Domain Decomposition Method." CFD Letters 17, no. 4 (October 31, 2024): 1–17. http://dx.doi.org/10.37934/cfdl.17.4.117.

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The present investigation of the dynamic two-binary droplet interactions has gained attention since its use to expand and improve several numerical methods. Generally, its interactions are classified into coalescence, bouncing, reflective, and stretching separation. This study simulated droplet coalescence using the meshless radial basis function (RBF) method. These methods are used to solve the Navier-Stokes equations combined with the Cahn-Hilliard equations to track the interface between two fluids. This work uses the fractional step method to calculate the pressure-velocity coupling in the Navier-Stokes equations. The numerical results were compared with the available data in the literature to validate the proposed method. Based on the validation, the proposed method conforms well with the literature. To identify further coalescence characteristics, the model considered different values in viscosity (2, 4, and 8 cP), collision velocity (1.5 m/s and 3 m/s), and surface tension (0.014, 0.028, and 0.056 N/m) parameters. The increasing viscosity was linearly proportional to the collision time, whereas increased surface tension and collision velocity shortened the collision time.

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39

Frigeri, Sergio. "Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities." Mathematical Models and Methods in Applied Sciences 26, no. 10 (August 25, 2016): 1955–93. http://dx.doi.org/10.1142/s0218202516500494.

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We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists in a Navier–Stokes type system coupled with a convective nonlocal Cahn–Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non-degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

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40

Tachim Medjo, T. "On the existence and uniqueness of solution to a stochastic 2D Cahn–Hilliard–Navier–Stokes model." Journal of Differential Equations 263, no. 2 (July 2017): 1028–54. http://dx.doi.org/10.1016/j.jde.2017.03.008.

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41

Deugoué, G., and T. Tachim Medjo. "The exponential behavior of a stochastic globally modified Cahn–Hilliard–Navier–Stokes model with multiplicative noise." Journal of Mathematical Analysis and Applications 460, no. 1 (April 2018): 140–63. http://dx.doi.org/10.1016/j.jmaa.2017.11.050.

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42

Guo, Z., and P. Lin. "A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects." Journal of Fluid Mechanics 766 (February 4, 2015): 226–71. http://dx.doi.org/10.1017/jfm.2014.696.

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AbstractIn this paper, we develop a phase-field model for binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. The governing equations of the model including the Navier–Stokes equations with additional stress term, Cahn–Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, some numerical examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared with the corresponding analytical solutions and the existing numerical results as validations for our model.

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Zanella, R., G. Tegze, R. Le Tellier, and H. Henry. "Two- and three-dimensional simulations of Rayleigh–Taylor instabilities using a coupled Cahn–Hilliard/Navier–Stokes model." Physics of Fluids 32, no. 12 (December 1, 2020): 124115. http://dx.doi.org/10.1063/5.0031179.

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Jia, Hongen, Xue Wang, and Kaitai Li. "A novel linear, unconditional energy stable scheme for the incompressible Cahn–Hilliard–Navier–Stokes phase-field model." Computers & Mathematics with Applications 80, no. 12 (December 2020): 2948–71. http://dx.doi.org/10.1016/j.camwa.2020.10.006.

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PERNA, TOMAS. "ROLE OF SYMMETRY IN OPTIMIZATION OF FEM SIMULATION CALCULATIONS." MM Science Journal 2022, no. 2 (June 1, 2022): 5670–74. http://dx.doi.org/10.17973/mmsj.2022_06_2022059.

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At the studying the Mannesmann piercing process, we unify two approaches to the problem solving. Namely, the commercial FEM software procedure and the mathematical model of the process via the mathematical model of the considered FEM-simulation bound by certain unifying symmetries. Such phenomenon seemingly exists only if the FE-mesh is initiated to be physically interpreted. We shortly outline, how to come to the slightly modified Cahn-Hilliard equation as to the mathematical model of the FE-simulation possessing quasi-symmetry given by a lattice of colloidal assembly formed by the chosen FE-mesh. Separation of two cylindrical surfaces of the pierced product together with the inpainting role of the piercing plug are described with respect to the background given by the Navier-Stokes equations related to the flow between the both surfaces. Influence of the involved groups related to the considered quasi-symmetry is illustrated by the convergence/divergence of the Newton-Raphson number in the CPU-time.

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Gao, Yali, Xiaoming He, Liquan Mei, and Xiaofeng Yang. "Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn--Hilliard--Navier--Stokes--Darcy Phase Field Model." SIAM Journal on Scientific Computing 40, no. 1 (January 2018): B110—B137. http://dx.doi.org/10.1137/16m1100885.

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Deugoué, G., B. Jidjou Moghomye, and T. Tachim Medjo. "Existence of a solution to the stochastic nonlocal Cahn–Hilliard Navier–Stokes model via a splitting-up method." Nonlinearity 33, no. 7 (May 29, 2020): 3424–69. http://dx.doi.org/10.1088/1361-6544/ab8020.

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Yang, Junxiang, and Junseok Kim. "A novel Cahn–Hilliard–Navier–Stokes model with a nonstandard variable mobility for two-phase incompressible fluid flow." Computers & Fluids 213 (December 2020): 104755. http://dx.doi.org/10.1016/j.compfluid.2020.104755.

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Tian, Ben, Bing Zhang, Junkai Deng, Dong Wang, Houjun Gong, Yang Li, Kerong Guo, Sen Yang, and Xiaoqin Ke. "Morphological evolution during liquid-liquid phase separation governed by composition change pathways." Journal of Applied Physics 132, no. 6 (August 14, 2022): 064702. http://dx.doi.org/10.1063/5.0089516.

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Liquid-liquid phase separation (LLPS) phenomenon are widely recognized to be of vital importance for physics, materials science, and biology. It is highly desired to develop powerful tools to study the LLPS behavior and related physical mechanisms. For this purpose, a phase-field method was developed here which combines the Cahn-Hilliard diffusion equation and the Navier-Stokes equation. The morphological evolution of LLPS behavior with the change in composition was comprehensively investigated under a prototypical ternary theoretical phase diagram. The phase-field simulation results indicated that the microstructural evolution was controlled by the phase diagram and driven by the coupling of diffusion and gravity effect. Moreover, the intermediate morphological microstructures and corresponding interfacial properties during LLPS could be tuned by selecting different composition change pathways. Furthermore, gravity-dependent density overturning and consequent Rayleigh-Taylor instability were observed in a unique LLPS process, demonstrating that the proposed model can capture the critical features of LLPS phenomenon.

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Feng, Xiaobing. "Fully Discrete Finite Element Approximations of the Navier--Stokes--Cahn-Hilliard Diffuse Interface Model for Two-Phase Fluid Flows." SIAM Journal on Numerical Analysis 44, no. 3 (January 2006): 1049–72. http://dx.doi.org/10.1137/050638333.

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