Cahn-Hilliard equation

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In this example we will show how to solve the Cahn-Hilliard equation in Medusa. The solution provides an implementation of periodic boundary conditions and shows how to combine implicit and explicit operators. This example uses a custom biharmonic operator, so we recommend firstly reading the tutorial on Customization.

Cahn-Hilliard equation

The Cahn-Hilliard equation describes the separation of binary fluid into pure domains. We define concentration, a scalar field $c(x, y) \in [-1, 1]$ to describe the fluid, with values $c \pm 1$ representing pure phases. In this case the equation can be written as

\[ \frac{\partial c}{\partial t} = D \nabla^2 (c^3 - c - \gamma \nabla^2 c), \] where $D$ is the diffusion constant and $\sqrt{\gamma}$ is the characteristic transition length between domains.

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