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The Cahn-Hilliard equation 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 indicating domains, then the equation is written as
Additional recommended knowledge
where D is a diffusion coefficient with units of Length2 / Time and gives the length of the transition regions between the domains. Here is the partial time derivative and is the Laplacian in n dimensions. Additionally, the quantity is identified as a chemical potential.
Features and applications
Of interest to mathematicians is the existence of a unique solution to the Cahn-Hilliard equation, given smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify
as a free energy functional, then
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.
The Cahn-Hilliard equations finds applications in diverse fields: in interfacial fluid flow, polymer science and in industrial applications. 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.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Cahn-Hilliard_equation". A list of authors is available in Wikipedia.|