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Governing equations

$\frac{\partial h_1}{\partial t} \,=\, \nabla \left(Q_{11} \nabla \frac{\delta F}{\delta h_1} + Q_{12} \nabla \frac{\delta F}{\delta h_2} \right)$

$\frac{\partial h_2}{\partial t} \,=\, \nabla \left(Q_{21} \nabla \frac{\delta F}{\delta h_1} + Q_{22} \nabla \frac{\delta F}{\delta h_2} \right)$

with the free energy $F \,=\, \int [ \rho_{s} +\rho_{VW}]\,d{\mathbf{x}}$

and the mobility matrix

$\mathbf{Q} =\frac{1}{3 \mu_1} \begin{pmatrix} h_1^3 & \frac{3 h_1^2}{2} \left(h_2 -\frac{h_1}{3} \right) \\[.3ex] \frac{3 h_1^2}{2} \left(h_2 -\frac{h_1}{3} \right) & \frac{(h_2 -h_1)^3 (\mu_1-\mu_2)}{\mu_2} +h_2^3 \end{pmatrix}$

The terms of the density of the excess surface energy are in the simplest case given by the Laplace (or gradient) term

$\rho_{s} = \frac{1}{2}[\sigma_1 (\nabla h_1)^2 + \sigma_2 (\nabla h_2)^2 ]$

and the density of the energy of the van-der-Waals interaction

$\rho_{VW} = - \frac{A_{g21s}}{12\pi h_2^2} - \frac{A_{21s}}{12\pi h_1^2} - \frac{A_{12g}}{12\pi(h_2-h_1)^2}$

where (Israelachvili 1992)

$A_{ijkl}\,\approx\,\frac{3h\nu_e}{8\sqrt{2}}\, \frac{(n_i^2-n_j^2)(n_l^2-n_k^2)}{(n_i^2+n_j^2)^{1/2}(n_l^2+n_k^2)^{1/2}[(n_i^2+n_j^2)^{1/2}+(n_l^2+n_k^2)^{1/2}]}$

and

$A_{ijk}=A_{ijjk}\,$

literature: BGS05

<usebib>Eleiht</usebib>

Two layer films

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Governing equations

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