Numerical approximations for a phase-field moving contact line model with variable densities and viscosities

We consider the numerical approximations of a two-phase hydrodynamics coupled phase-field model that incorporates the variable densities, viscosities and moving contact line boundary conditions. The model is a nonlinear, coupled system that consists of incompressible NavierStokes equations with the generalized Navier boundary condition, and the CahnHilliard equations with moving contact line boundary conditions. By some subtle explicitimplicit treatments to nonlinear terms, we develop two efficient, unconditionally energy stable numerical schemes, in particular, a linear decoupled energy stable scheme for the system with static contact line condition, and a nonlinear energy stable scheme for the system with dynamic contact line condition. An efficient spectral-Galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes. Various numerical results show that the proposed schemes are efficient and accurate.

[1]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[2]  Chunfeng Zhou,et al.  Sharp-interface limit of the Cahn–Hilliard model for moving contact lines , 2010, Journal of Fluid Mechanics.

[3]  S. H. Davis,et al.  On the motion of a fluid-fluid interface along a solid surface , 1974, Journal of Fluid Mechanics.

[4]  Jun Li,et al.  Energy Stable Numerical Schemes for a Hydrodynamic Model of Nematic Liquid Crystals , 2016, SIAM J. Sci. Comput..

[5]  P. Lin,et al.  A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects , 2014, Journal of Fluid Mechanics.

[6]  J. S. Rowlinson,et al.  Translation of J. D. van der Waals' “The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density” , 1979 .

[7]  Jie Shen,et al.  Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immersed in viscous fluids , 2013, J. Comput. Phys..

[8]  Daozhi Han,et al.  Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows , 2017, J. Sci. Comput..

[9]  Feng Chen,et al.  An efficient and energy stable scheme for a phase‐field model for the moving contact line problem , 2016 .

[10]  N. Hadjiconstantinou.  Regular Article: Hybrid Atomistic–Continuum Formulations and the Moving Contact-Line Problem , 1999 .

[11]  Ping Sheng,et al.  Molecular scale contact line hydrodynamics of immiscible flows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Abner J. Salgado,et al.  A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines , 2013 .

[13]  Lili Ju,et al.  Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model , 2017 .

[14]  Jie Shen,et al.  A Phase-Field Model and Its Numerical Approximation for Two-Phase Incompressible Flows with Different Densities and Viscosities , 2010, SIAM J. Sci. Comput..

[15]  Xiaofeng Yang,et al.  Numerical Approximations for Allen-Cahn Type Phase Field Model of Two-Phase Incompressible Fluids with Moving Contact Lines , 2017 .

[16]  Jie Shen,et al.  A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios , 2012, J. Comput. Phys..

[17]  ShenJie.  Efficient spectral-Galerkin method I , 1994 .

[18]  Xiaofeng Yang,et al.  Error analysis of stabilized semi-implicit method of Allen-Cahnequation , 2009 .

[19]  Min Gao,et al.  An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity , 2014, J. Comput. Phys..

[20]  Tao Tang,et al.  Stability Analysis of Large Time-Stepping Methods for Epitaxial Growth Models , 2006, SIAM J. Numer. Anal..

[21]  Jie Shen,et al.  Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach , 2011 .

[22]  Jie Shen,et al.  Decoupled Energy Stable Schemes for a Phase-Field Model of Two-Phase Incompressible Flows with Variable Density , 2014, Journal of Scientific Computing.

[23]  Jie Shen,et al.  Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids , 2014, SIAM J. Sci. Comput..

[24]  James J. Feng,et al.  A diffuse-interface method for simulating two-phase flows of complex fluids , 2004, Journal of Fluid Mechanics.

[25]  Robbins,et al.  Simulations of contact-line motion: Slip and the dynamic contact angle. , 1989, Physical review letters.

[26]  Abner J. Salgado,et al.  A splitting method for incompressible flows with variable density based on a pressure Poisson equation , 2009, J. Comput. Phys..

[27]  Jie Shen,et al.  Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method , 2006, J. Comput. Phys..

[28]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[29]  Zhilin Li,et al.  A level-set method for interfacial flows with surfactant , 2006, J. Comput. Phys..

[30]  P. Sheng,et al.  A variational approach to moving contact line hydrodynamics , 2006, Journal of Fluid Mechanics.

[31]  Jie Shen,et al.  A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids , 2016, J. Comput. Phys..

[32]  H. Abels,et al.  Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities , 2011, 1104.1336.

[33]  L. Quartapelle,et al.  A projection FEM for variable density incompressible flows , 2000 .

[34]  Sebastian Aland,et al.  Time integration for diffuse interface models for two-phase flow , 2013, J. Comput. Phys..

[35]  Banavar,et al.  Molecular dynamics of Poiseuille flow and moving contact lines. , 1988, Physical review letters.

[36]  J. Lendvai,et al.  Correlation between resistivity increment and volume fraction of G.P. zones in an Al-3·2 wt % Zn-2·2 wt % Mg alloy , 1976 .

[37]  Jie Shen,et al.  Efficient energy stable numerical schemes for a phase field moving contact line model , 2015, J. Comput. Phys..

[38]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[39]  D. J. Eyre.  Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation , 1998 .

[40]  Ping Lin,et al.  Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes , 2015 .

[41]  M. Gurtin,et al.  TWO-PHASE BINARY FLUIDS AND IMMISCIBLE FLUIDS DESCRIBED BY AN ORDER PARAMETER , 1995, patt-sol/9506001.

[42]  John William Strutt,et al.  Scientific Papers: On the Theory of Surface Forces. II. Compressible Fluids , 2009 .

[43]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[44]  Xiaofeng Yang,et al.  Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , 2010 .

[45]  J. Lowengrub,et al.  A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant , 2004 .

[46]  Xiaofeng Yang,et al.  Decoupled energy stable schemes for phase-field vesicle membrane model , 2015, J. Comput. Phys..

[47]  Jie Shen,et al.  Error analysis of fully discrete velocity-correction methods for incompressible flows , 2008, Math. Comput..

[48]  Daozhi Han,et al.  Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model , 2017, J. Comput. Phys..

[49]  Xiaofeng Yang,et al.  Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends , 2016, J. Comput. Phys..

[50]  H. K. Moffatt.  Viscous and resistive eddies near a sharp corner , 1964, Journal of Fluid Mechanics.

[51]  Suchuan Dong,et al.  On imposing dynamic contact-angle boundary conditions for wall-bounded liquid–gas flows , 2012 .

[52]  E. B. Dussan,et al.  LIQUIDS ON SOLID SURFACES: STATIC AND DYNAMIC CONTACT LINES , 1979 .

[53]  Xiaofeng Yang,et al.  Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows , 2010 .

[54]  Ping Sheng,et al.  Power-law slip profile of the moving contact line in two-phase immiscible flows. , 2004, Physical review letters.

[55]  Xiaofeng Yang,et al.  Error estimates for finite element approximations of consistent splitting schemes for incompressible flows , 2007 .

[56]  Junseok Kim.  Phase-Field Models for Multi-Component Fluid Flows , 2012 .

[57]  J. Waals.  The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density , 1979 .

[58]  Joel Koplik,et al.  Molecular dynamics of fluid flow at solid surfaces , 1989 .

[59]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[60]  R. Sec.,et al.  XX. On the theory of surface forces.—II. Compressible fluids , 1892 .

[61]  Jie Shen,et al.  Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows , 2015, SIAM J. Numer. Anal..

[62]  Sébastian Minjeaud.  An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model , 2013 .

[63]  Jia Zhao,et al.  Decoupled Energy Stable Schemes for a Phase Field Model of Three-Phase Incompressible Viscous Fluid Flow , 2016, Journal of Scientific Computing.

[64]  Jie Shen,et al.  An efficient moving mesh spectral method for the phase-field model of two-phase flows , 2009, J. Comput. Phys..

[65]  Xiaofeng Yang,et al.  Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach , 2017 .

[66]  P. Spelt.  A level-set approach for simulations of flows with multiple moving contact lines with hysteresis , 2005 .

[67]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[68]  Endre Süli,et al.  Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth , 2010, Math. Comput..

[69]  Xiaofeng Yang,et al.  Mass and Volume Conservation in Phase Field Models for Binary Fluids , 2013 .

[70]  Min Gao,et al.  A gradient stable scheme for a phase field model for the moving contact line problem , 2012, J. Comput. Phys..

[71]  E. Weinan,et al.  Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics , 2005 .

[72]  Jian-Jun Xu,et al.  A level-set method for two-phase flows with moving contact line and insoluble surfactant , 2014, J. Comput. Phys..

[73]  Jie Shen,et al.  Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[74]  Luis A. Caffarelli,et al.  An L∞ bound for solutions of the Cahn-Hilliard equation , 1995 .

[75]  Brian Wetton,et al.  Computation of a Moving Drop/Bubble on a Solid Surface using a Front-Tracking Method , 2004 .

[76]  J. Li,et al.  Numerical simulation of moving contact line problems using a volume-of-fluid method , 2001 .

[77]  Roland Glowinski,et al.  A least-squares/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line , 2011, J. Comput. Phys..