Higher Dimensional Bubble Profiles in a Sharp Interface Limit of the FitzHugh--Nagumo System

2018 ◽  
Vol 50 (5) ◽  
pp. 5072-5095 ◽  
Author(s):  
Chao-Nien Chen ◽  
Yung-Sze Choi ◽  
Yeyao Hu ◽  
Xiaofeng Ren
Keyword(s):  
Sharp Interface ◽  
Sharp Interface Limit ◽  
Higher Dimensional

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.

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.

scholarly journals Phase-field model of cell motility: Traveling waves and sharp interface limit

2016 ◽  
Vol 354 (10) ◽  
pp. 986-992 ◽  
Author(s):  
Leonid Berlyand ◽  
Mykhailo Potomkin ◽  
Volodymyr Rybalko
Keyword(s):  
Cell Motility ◽  
Traveling Waves ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Sharp Interface ◽  
Sharp Interface Limit

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.

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