Cahn-Hilliard#

In this subsection, the parameters for multiphase flow simulation using the Cahn-Hilliard equations are specified.

In this method, the two fluids considered are given an index of \(0\) and \(1\) respectively. The amount of fluid at any given quadrature point is represented by a phase order parameter \(\phi\) between \(-1\) and \(1\). The interface between the two fluids is naturally diffuse and the motion of the interface are driven by differences in the local chemical potential \(\eta\).

Note

A Canh-Hilliard theory documentation explaining the origins of the diffuse interface will be added in an upcoming update.

The equations solved are as follows:

\[ \begin{align}\begin{aligned}\begin{split}\partial_t\phi + (\mathbf{a} \cdot \nabla) \phi - \nabla \cdot (M(\phi)\nabla \eta) = 0 \\\end{split}\\ \eta - \frac{\lambda}{\epsilon^2}(\phi^3 - \phi) + \lambda \nabla^2 \phi + \xi h^2 \nabla^2 \eta = 0\end{aligned}\end{align} \]

where:

  • \(\mathbf{a}\) corresponds to the velocity field; this vector field is used when the problem is driven by convection.

  • \(M(\phi)\) is the mobility function; two cases are considered: a constant mobility model, i.e, \(M = D\) with \(D\) the mobility constant, or a quartic mobility model with \(M(\phi) = D(1-\phi^2)^2\).

  • \(\xi\) is a smoothing coefficient for the chemical potential.

  • \(h\) is a local measure of the cell size. There may be different h in an adapted mesh.

  • \(\epsilon\) is a parameter linked to the thickness of the interface.

Note

At the moment, a maximum of two fluids is supported. By convention, air is usually the fluid 0 and the other fluid of interest is the fluid 1. The fluid 1 (0) corresponds to a value of \(\phi = -1\) (\(\phi = 1\)). See Initial Conditions for the definition of the Cahn-Hilliard initial conditions and Physical properties - Two phase simulations for the definition of the physical properties of both fluids. Do not forget to set cahn hilliard = true in the Multiphysics subsection of the .prm file.

The default values of the Cahn-Hilliard parameters are given in the text box below:

subsection cahn hilliard

  set potential smoothing coefficient = 1

  subsection epsilon
    set verbosity = quiet
    set method    = automatic
    set value     = 1
  end
end

  • potential smoothing coefficient: defines the \(\xi\) parameter in the equations above. Its value should be increased if the potential presents excessive oscillations (in advective problems for instance).

  • epsilon: defines the \(\epsilon\) parameter. The reader is refered to the Cahn-Hilliard Method section for additional details about this parameter.

    • verbosity: enables the display of the interface thickness values in the terminal. This does not affect the printing of output fils. Choices are quiet (default, no output) or verbose (output at every iteration).

    • method: sets the method for the computation of epsilon. The first choice is automatic: the interface thickness is determined automatically using the minimum cell diameter obtained by the deal.ii minimum_cell_diameter method. The other choice is manual: allows the user to define the interface thickness by hand by setting the desired value with value.

Attention

The mobility model and mobility constant must be set in the Physical Properties section.