Unifying binary fluid diffuse-interface models in the sharp-interface limit

2013 ◽  
Vol 736 ◽  
pp. 5-43 ◽  
Author(s):  
David N. Sibley ◽  
Andreas Nold ◽  
Serafim Kalliadasis
Keyword(s):  
Phase Field ◽  
Outer Region ◽  
Mobility Model ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Binary Fluids ◽  
Sharp Interface Limit ◽  
Binary Fluid ◽  
Phase Field Models

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.

The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids

2013 ◽  
Vol 714 ◽  
pp. 95-126 ◽  
Author(s):  
F. Magaletti ◽  
F. Picano ◽  
M. Chinappi ◽  
L. Marino ◽  
C. M. Casciola
Keyword(s):  
Numerical Simulations ◽  
Capillary Waves ◽  
Sharp Interface ◽  
Navier Stokes ◽  
Interface Thickness ◽  
Effective Parameters ◽  
Binary Fluids ◽  
Two Phase ◽  
Navier Stokes Equation ◽  
Sharp Interface Limit

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.

scholarly journals A ternary Cahn–Hilliard–Navier–Stokes model for two-phase flow with precipitation and dissolution

2020 ◽  
pp. 1-35
Author(s):  
Christian Rohde ◽  
Lars von Wolff
Keyword(s):  
Phase Field ◽  
Stokes Equations ◽  
Solid Phase ◽  
Immiscible Fluids ◽  
Sharp Interface ◽  
Ion Concentration ◽  
Navier Stokes ◽  
Interface Model ◽  
Sharp Interface Model ◽  
Two Phase

We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The model is shown to preserve the fundamental conservation constraints and to obey the second law of thermodynamics for a novel free energy formulation. An extended analysis for vanishing interfacial width reveals that in this limit the sharp interface model is recovered, including all relevant transmission conditions. Notably, the new phase-field model is able to realize Navier-slip conditions for solid–fluid interfaces in the limit.

scholarly journals (In)compressibility and parameter identification in phase field models for capillary flows

2017 ◽  
Vol 44 (2) ◽  
pp. 189-214 ◽  
Author(s):  
M. Dehsara ◽  
H. Fu ◽  
S.Dj Mesarovic ◽  
D.P. Sekulic ◽  
M. Krivilyov
Keyword(s):  
Phase Field ◽  
Triple Line ◽  
Slip Boundary Condition ◽  
Diffuse Interface ◽  
Slip Boundary ◽  
Phase Field Models ◽  
Capillary Flows ◽  
Wide Range ◽  
Mobility Parameter ◽  
Wetting Process

Phase field (diffuse interface) models accommodate diffusive triple line motion with variable contact angle, thus allowing for the no-slip boundary condition without the stress singularities. We consider two commonly used classes of phase field models: the compositionally compressible (CC) model with compressibility limited to the fluid mix within the diffuse interface, and the incompressible (IC) model. First, we show that the CC model applied to fluids with dissimilar mass densities exhibits the computational instability leading to the breakup of the triple line. We provide a qualitative physical explanation of this instability and argue that the compositional compressibility within the diffuse interface is inconsistent with the global incompressible flow. Second, we derive the IC model as a systematic approximation to the CC model, based on a suitable choice of continuum velocity field. Third, we benchmark the IC model against sharp interface theory and experimental kinetics. The triple line kinetics is well represented by the triple line mobility parameter. Finally, we investigate the effects of the bulk phase field diffusional mobility parameter on the kinetics of the wetting process and find that within a wide range of magnitudes the bulk mobility does not affect the flow.

Application of Interface-Tracking Method Based on Phase-Field Model to Numerical Analysis of Isothermal and Thermal Two-Phase Flows

2007 ◽  
Author(s):  
Naoki Takada
Keyword(s):  
Phase Field ◽  
Density Ratio ◽  
Phase Field Model ◽  
Field Method ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Navier Stokes ◽  
Interface Tracking ◽  
Two Phase Flows ◽  
Two Phase

For interface-tracking simulation of two-phase flows in various micro-fluidics devices, the applicability of two versions of Navier-Stokes phase-field method (NS-PFM) was examined, combining NS equations for a continuous fluid with a diffuse-interface model based on the van der Waals-Cahn-Hilliard free-energy theory. Through the numerical simulations, the following major findings were obtained: (1) The first version of NS-PFM gives good predictions of interfacial shapes and motions in an incompressible, isothermal two-phase fluid with high density ratio on solid surface with heterogeneous wettability. (2) The second version successfully captures liquid-vapor motions with heat and mass transfer across interfaces in phase change of a non-ideal fluid around the critical point.

scholarly journals A regularized phase-field model for faceting in a kinetically controlled crystal growth

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200227
Author(s):  
T. Philippe ◽  
H. Henry ◽  
M. Plapp
Keyword(s):  
Crystal Growth ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Interface Problem ◽  
Kinetically Controlled ◽  
Interface Theory ◽  
Coarsening Dynamics

At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen–Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t , with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.

On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit

2010 ◽  
Vol 140 (6) ◽  
pp. 1161-1186 ◽  
Author(s):  
Wolfgang Dreyer ◽  
Christiane Kraus
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Van Der Waals ◽  
Phase Field Model ◽  
Entropy Inequality ◽  
Sharp Interface ◽  
Energy Estimates ◽  
Sharp Interface Limit ◽  
Entropy Flux

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

Mass and Volume Conservation in Phase Field Models for Binary Fluids

2013 ◽  
Vol 13 (4) ◽  
pp. 1045-1065 ◽  
Author(s):  
Jie Shen ◽  
Xiaofeng Yang ◽  
Qi Wang
Keyword(s):  
Phase Field ◽  
Numerical Study ◽  
Mixing Layer ◽  
Phase Field Model ◽  
Fluid Phase ◽  
Binary Fluids ◽  
Fluid Mixture ◽  
Volume Fractions ◽  
Phase Field Models ◽  
Constraints Model

AbstractThe commonly used incompressible phase field models for non-reactive, binary fluids, in which the Cahn-Hilliard equation is used for the transport of phase variables (volume fractions), conserve the total volume of each phase as well as the material volume, but do not conserve the mass of the fluid mixture when densities of two components are different. In this paper, we formulate the phase field theory for mixtures of two incompressible fluids, consistent with the quasi-compressible theory [28], to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase. In this formulation, the mass-average velocity is no longer divergence-free (solenoidal) when densities of two components in the mixture are not equal, making it a compressible model subject to an internal con-straint. In one formulation of the compressible models with internal constraints (model 2), energy dissipation can be clearly established. An efficient numerical method is then devised to enforce this compressible internal constraint. Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions. Numerical comparisons show that (a) predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin; and (b) predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone. The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.

Sharp-interface limit of the Cahn–Hilliard model for moving contact lines

2010 ◽  
Vol 645 ◽  
pp. 279-294 ◽  
Author(s):  
PENGTAO YUE ◽  
CHUNFENG ZHOU ◽  
JAMES J. FENG
Keyword(s):  
Contact Line ◽  
Diffusion Length ◽  
Slip Length ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Contact Lines ◽  
Sharp Interface Limit ◽  
Interface Models ◽  
Moving Contact ◽  
Moving Contact Lines

Diffuse-interface models may be used to compute moving contact lines because the Cahn–Hilliard diffusion regularizes the singularity at the contact line. This paper investigates the basic questions underlying this approach. Through scaling arguments and numerical computations, we demonstrate that the Cahn–Hilliard model approaches a sharp-interface limit when the interfacial thickness is reduced below a threshold while other parameters are fixed. In this limit, the contact line has a diffusion length that is related to the slip length in sharp-interface models. Based on the numerical results, we propose a criterion for attaining the sharp-interface limit in computing moving contact lines.

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