Efficient numerical methods for simulating surface tension of multi-component mixtures with the gradient theory of fluid interfaces☆

Abstract Surface tension significantly impacts subsurface flow and transport, and it is the main cause of capillary effect, a major immiscible two-phase flow mechanism for systems with a strong wettability preference. In this paper, we consider the numerical simulation of the surface tension of multi-component mixtures with the gradient theory of fluid interfaces. Major numerical challenges include that the system of the Euler–Lagrange equations is solved on the infinite interval and the coefficient matrix is not positive definite. We construct a linear transformation to reduce the Euler–Lagrange equations, and naturally introduce a path function, which is proven to be a monotonic function of the spatial coordinate variable. By using the linear transformation and the path function, we overcome the above difficulties and develop the efficient methods for calculating the interface and its interior compositions. Moreover, the computation of the surface tension is also simplified. The proposed methods do not need to solve the differential equation system, and they are easy to be implemented in practical applications. Numerical examples are tested to verify the efficiency of the proposed methods.

[1]  P.M.W. Cornelisse,et al.  The Square Gradient Theory Applied - Simultaneous Modelling of Interfacial Tension and Phase Behaviour , 1997 .

[2]  Mary F. Wheeler,et al.  Discontinuous Galerkin methods for simulating bioreactive transport of viruses in porous media , 2007 .

[3]  Shuyu Sun,et al.  Compositional modeling of three‐phase flow with gravity using higher‐order finite element methods , 2011 .

[4]  Abbas Firoozabadi,et al.  Thermodynamics of Hydrocarbon Reservoirs , 1999 .

[5]  M. Sahimi,et al.  Thermodynamic Modeling of Phase and Tension Behavior of CO2/Hydrocarbon Systems , 1985 .

[6]  Alain Graciaa,et al.  Modelling of the surface tension of pure components with the gradient theory of fluid interfaces: a simple and accurate expression for the influence parameters , 2003 .

[7]  Zhibao Li,et al.  On the prediction of surface tension for multicomponent mixtures , 2001 .

[8]  C. Miehe,et al.  A geometrically consistent incremental variational formulation for phase field models in micromagnetics , 2012 .

[9]  Maya Neytcheva,et al.  Efficient numerical solution of discrete multi-component Cahn-Hilliard systems , 2014, Comput. Math. Appl..

[10]  Junseok Kim,et al.  A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows , 2009 .

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

[12]  Thomas J. R. Hughes,et al.  A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework , 2014 .

[13]  Tao Tang,et al.  An Adaptive Time-Stepping Strategy for the Molecular Beam Epitaxy Models , 2011, SIAM J. Sci. Comput..

[14]  L. Lake,et al.  Enhanced Oil Recovery , 2017 .

[15]  Jie Shen,et al.  Efficient energy stable schemes with spectral discretization in space for anisotropic , 2013 .

[16]  L. E. Scriven,et al.  Semiempirical theory of surface tension of binary systems , 1980 .

[17]  D. Peng,et al.  A New Two-Constant Equation of State , 1976 .

[18]  A. Firoozabadi,et al.  Molecular thermodynamic modeling of droplet-type microemulsions. , 2012, Langmuir : the ACS journal of surfaces and colloids.

[19]  Xesús Nogueira,et al.  An unconditionally energy-stable method for the phase field crystal equation , 2012 .

[20]  Shuyu Sun,et al.  A Locally Conservative Finite Element Method Based on Piecewise Constant Enrichment of the Continuous Galerkin Method , 2009, SIAM J. Sci. Comput..

[21]  Mary F. Wheeler,et al.  Analysis of Discontinuous Galerkin Methods for Multicomponent Reactive Transport Problems , 2006, Comput. Math. Appl..

[22]  Peng Song,et al.  A diffuse-interface method for two-phase flows with soluble surfactants , 2011, J. Comput. Phys..

[23]  M. Wheeler,et al.  An augmented-Lagrangian method for the phase-field approach for pressurized fractures , 2014 .

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

[25]  Shuyu Sun,et al.  An adaptive finite element method for simulating surface tension with the gradient theory of fluid interfaces , 2014, J. Comput. Appl. Math..

[26]  Susanne Hertz,et al.  Statistical Mechanics Of Phases Interfaces And Thin Films , 2016 .

[27]  P. Kloucek,et al.  Stability of the fractional step T-scheme for the nonstationary Navier-Stokes equations , 1994 .

[28]  B. Nestler,et al.  Phase-field modeling of multi-component systems , 2011 .

[29]  A. Graciaa,et al.  Modeling of the Surface Tension of Multicomponent Mixtures with the Gradient Theory of Fluid Interfaces , 2005 .

[30]  Mary F. Wheeler,et al.  A DYNAMIC, ADAPTIVE, LOCALLY CONSERVATIVE, AND NONCONFORMING SOLUTION STRATEGY FOR TRANSPORT PHENOMENA IN CHEMICAL ENGINEERING , 2004 .

[31]  C. E. Rasmussen,et al.  Surface tension of quantum fluids from molecular simulations. , 2004, The Journal of chemical physics.

[32]  Cor J. Peters,et al.  Modeling of the surface tension of pure components and mixtures using the density gradient theory combined with a theoretically derived influence parameter correlation , 2012 .