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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cahn-Hilliard equation ： ウィキペディア英語版 Cahn–Hilliard equation The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If $c$ is the concentration of the fluid, with $c=\backslash pm1$ indicating domains, then the equation is written as :$\backslash frac\; =\; D\backslash nabla^2\backslash left(c^3-c-\backslash gamma\backslash nabla^2\; c\backslash right),$ where $D$ is a diffusion coefficient with units of $\backslash text^2/\backslash text$ and $\backslash sqrt$ gives the length of the transition regions between the domains. Here $\backslash partial/$ is the partial time derivative and $\backslash nabla^2$ is the Laplacian in $n$ dimensions. Additionally, the quantity $\backslash mu\; =\; c^3-c-\backslash gamma\backslash nabla^2\; c$ is identified as a chemical potential. Related to it is the Allen–Cahn equation, as well as the Stochastic Cahn–Hilliard Equation and the Stochastic Allen–Cahn equation. ==Features and applications== Of interest to mathematicians is the existence of a unique solution of the Cahn–Hilliard equation, given by smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify :$F()=\backslash int\; d^n\; x\; \backslash left(c\backslash right|^2\backslash right),$ as a free energy functional, then :$\backslash frac\; =\; -\backslash int\; d^n\; x\; \backslash left|\backslash nabla\backslash mu\backslash right|^2,$ so that the free energy decays to zero. This also indicates segregation into domains is the asymptotic outcome of the evolution of this equation. In real experiments, the segregation of an initially mixed binary fluid into domains is observed. The segregation is characterized by the following facts. * There is a transition layer between the segregated domains, with a profile given by the function $c(x)\; =\; \backslash tanh\backslash left(\backslash frac$ because this function is an equilibrium solution of the Cahn–Hilliard equation. * Of interest also is the fact that the segregated domains grow in time as a power law. That is, if $L(t)$ is a typical domain size, then $L(t)\backslash propto\; t^$. This is the Lifshitz–Slyozov law, and has been proved rigorously for the Cahn–Hilliard equation and observed in numerical simulations and real experiments on binary fluids. * The Cahn–Hilliard equation has the form of a conservation law, $\backslash frac\; =\; \backslash nabla\; \backslash cdot\; \backslash bold(x),$ with $\backslash bold(x)=D\backslash nabla\; \backslash mu$. Thus the phase separation process conserves the total concentration $C=\backslash int\; d^n\; x\; c\backslash left(x,t\backslash right)$, so that $\backslash frac=0$. * When one phase is significantly more abundant, the Cahn–Hilliard equation can show the phenomenon known as Ostwald ripening, where the minority phase forms spherical droplets, and the smaller droplets are absorbed through diffusion into the larger ones. The Cahn–Hilliard equations finds applications in diverse fields: in complex fluids and soft matter (interfacial fluid flow, polymer science and in industrial applications). The solution of the Cahn–Hilliard equation for a binary mixture demonstrated to coincide well with the solution of a Stefan problem and the model of Thomas and Windle.〔F. J. Vermolen, M.G. Gharasoo, P. L. J. Zitha, J. Bruining. (2009). Numerical Solutions of Some Diffuse Interface Problems: The Cahn-Hilliard Equation and the Model of Thomas and Windle.'' IntJMultCompEng'',7(6):523–543.〕 Of interest to researchers at present is the coupling of the phase separation of the Cahn–Hilliard equation to the Navier–Stokes equations of fluid flow. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cahn–Hilliard equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.