Convergence of the phase field model to its sharp interface limits

Caginalp, Gunduz; Chen, Xinfu

doi:10.1017/s0956792598003520

Cited by 149 publications

(160 citation statements)

References 35 publications

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“…When CH is used to represent an interface that does not intersect a solid wall, as is the case for drop deformation, the sharp-interface limit is well established. [7][8][9] In such a limit, achieved at finite interfacial thickness ⑀, the results no longer depend on ⑀, and the diffusion across the interface becomes negligible. The results thus become comparable with sharp-interface computations and experiments, in which the interface is exceedingly thin, down to molecular scale.…”

Section: Introductionmentioning

confidence: 99%

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

2011

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The Cahn–Hilliard model uses diffusion between fluid components to regularize the stress singularity at a moving contact line. In addition, it represents the dynamics of the near-wall layer by the relaxation of a wall energy. The first part of the paper elucidates the role of the wall relaxation in a flowing system, with two main results. First, we show that wall energy relaxation produces a dynamic contact angle that deviates from the static one, and derive an analytical formula for the deviation. Second, we demonstrate that wall relaxation competes with Cahn–Hilliard diffusion in defining the apparent contact angle, the former tending to “rotate” the interface at the contact line while the latter to “bend” it in the bulk. Thus, varying the two in coordination may compensate each other to produce the same macroscopic solution that is insensitive to the microscopic dynamics of the contact line. The second part of the paper exploits this competition to develop a computational strategy for simulating realistic flows with microscopic slip length at a reduced cost. This consists in computing a moving contact line with a diffusion length larger than the real slip length, but using the wall relaxation to correct the solution to that corresponding to the small slip length. We derive an analytical criterion for the required amount of wall relaxation, and validate it by numerical results on dynamic wetting in capillary tubes and drop spreading.

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Section: Introductionmentioning

confidence: 99%

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

2011

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“…These computations demonstrated that for the results to be quantitatively accurate, two conditions have to be met. First, the interface should be sufficiently thin so that the theoretical model approaches the so-called sharp-interface limit [19]. Second, this thin region must be adequately resolved by fine mesh; it typically requires some 10 grid points.…”

Section: Introductionmentioning

confidence: 99%

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

Zhou

Yue

Feng

et al. 2010

Journal of Computational Physics

117

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-This work presents a three-dimensional finite-element algorithm, based on the phase-field model, for computing interfacial flows of Newtonian and complex fluids. A 3D adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk. It is key to accurate resolution of the interface at manageable computational costs.The coupled Navier-Stokes and Cahn-Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics.One is the ease in simulating topological changes such as interfacial rupture and coalescence.Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

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“…Ilmanen [17] considered the corresponding L 2 -gradient flows and proved the convergence of the Allen-Cahn equation to the meancurvature flow, in the varifold formulation of Brakke [5]. Convergence of various other phase field problems to the corresponding sharp interface models have been shown either formally or rigorously [1,6,8,9,18,27,38], sometimes in quite involved weak formulations.…”

Section: Related Results and Main Techniquesmentioning

confidence: 99%

Convergence of phase-field approximations to the Gibbs–Thomson law

Röger

Tonegawa

2007

Calc. Var.

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We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van der Waals-CahnHilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.

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Convergence of the phase field model to its sharp interface limits

Cited by 149 publications

References 35 publications

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

Convergence of phase-field approximations to the Gibbs–Thomson law

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