Interdisciplinary Applied Mathematics

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The governing equation for this case is obtained in terms of the order parameter Ф, which is defined as


Ф


?? i + n2


where щ denotes the number density of each species. This order parameter changes very    fast    very    close to    the    interface,    but    it    is    almost    constant    in


the bulk. The viscosity of the binary mixture can then be expressed as a


linear combination of its two components, i.e.,


Pm(z)



1+Ф(»


2



+ Рь



1-Ф(г)


2


The thermodynamics of the binary mixture are described via a free-energy functional plus other contributions to account for surface effects. The corresponding semigrand potential proposed in (Andrienko et al., 2003), is given


by


U{Ф) = ^ I dV2(УФ)2 + /(Ф) — рф) + *s,


where a is a length scale characteristic of the molecule size, f (Ф) is the Helmholtz free-energy density, Ts is the surface energy, and p is the chemical potential. The governing equation is then obtained by minimizing the above functional to obtain


(10.8)


д2Ф Ф 1    1 + Ф


к^+2~2ТЫ1^Ф


where T is the temperature. This is a boundary value problem, which was solved in (Andrienko et al., 2003), for a channel with identical walls located sufficiently far from each other so that the film layers do not overlap. The solution of the above equation reveals a prewetting transition that depends on the temperature; it is sudden, and it jumps from a thin film to a thick film (first-order transition). For thin films a small slip length is obtained, but for    thick    films a    large    slip    length    is obtained    that    also    depends    on

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