The Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations

2018 ◽  
pp. 1-13
Author(s):  
Helmut Abels ◽  
Johannes Daube ◽  
Christiane Kraus ◽  
Dietmar Kröner
Keyword(s):  
Sharp Interface ◽  
Navier Stokes ◽  
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.

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.

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.

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.

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