scholarly journals Sharp interface limit of a Stokes/Cahn–Hilliard system. Part I: Convergence result

2021 ◽  
Vol 23 (3) ◽  
pp. 353-402
Author(s):  
Helmut Abels ◽  
Andreas Marquardt
Keyword(s):  
Convergence Result ◽  
Sharp Interface ◽  
Sharp Interface Limit ◽  
System Part

scholarly journals Sharp Interface Limit of a Stokes/Cahn–Hilliard System, Part II: Approximate Solutions

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Helmut Abels ◽  
Andreas Marquardt
Keyword(s):  
Fractional Order ◽  
Approximate Solutions ◽  
Sharp Interface ◽  
Main Step ◽  
Two Dimensional ◽  
Matched Asymptotics ◽  
Sharp Interface Limit ◽  
System Part ◽  
Proof Of Convergence ◽  
Problematic Feature

AbstractWe construct rigorously suitable approximate solutions to the Stokes/Cahn–Hilliard system by using the method of matched asymptotics expansions. This is a main step in the proof of convergence given in the first part of this contribution, [3], where the rigorous sharp interface limit of a coupled Stokes/Cahn–Hilliard system in a two dimensional, bounded and smooth domain is shown. As a novelty compared to earlier works, we introduce fractional order terms, which are of significant importance, but share the problematic feature that they may not be uniformly estimated in $$\epsilon $$ ϵ in arbitrarily strong norms. As a consequence, gaining necessary estimates for the error, which occurs when considering the approximations in the Stokes/Cahn–Hilliard system, is rather involved.

scholarly journals (Non-)convergence of solutions of the convective Allen–Cahn equation

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Helmut Abels
Keyword(s):  
Transport Equation ◽  
Mean Curvature ◽  
Convergence Result ◽  
Normal Direction ◽  
Sharp Interface ◽  
Navier Stokes ◽  
Matched Asymptotics ◽  
Sharp Interface Limit ◽  
Convergence Of Solutions ◽  
Cahn Equation

AbstractWe consider the sharp interface limit of a convective Allen–Cahn equation, which can be part of a Navier–Stokes/Allen–Cahn system, for different scalings of the mobility $$m_\varepsilon =m_0\varepsilon ^\theta $$ m ε = m 0 ε θ as $$\varepsilon \rightarrow 0$$ ε → 0 . In the case $$\theta >2$$ θ > 2 we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $$\theta =0$$ θ = 0 . Moreover, we show that an associated mean curvature functional does not converge to the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $$\theta =0,1$$ θ = 0 , 1 by the method of formally matched asymptotics.

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.

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