2.10. Viscosity Models

A number of different viscosity models are available.

2.10.1. Newtonian Fluid

The …

2.10.2. Arrhenius

For free energies involing a compositional order parameter \(\phi\) one can define a local viscosity following a relation owing to Arrhenius:

\[\eta(\mathbf{r}) = \eta_{-}^{(1-\phi/\phi^\star)/2} \eta_{+}^{(1+\phi/\phi^\star)/2}.\]

For example for a symmetric binary fluid with \(\phi^\star = \pm (-A/B)^{1/2}\), \(\eta_+\) is the viscosity when the composition is \(\phi = +\phi^\star\) and \(\eta_-\) is the viscosity when \(\phi = -\phi^\star\).

This may be specified in the input file via

viscosity_model                  arrhenius
viscosity_arrhenius_eta_plus     0.1
viscosity_arrhenius_eta_minus    0.5
viscoisty_arrhenius_phistar      1.0

Values for \(\eta_+\) and \(\eta_-\) must be specified, and set the shear viscosity in the corresponding phases. If the value for \(\phi^\star\) is omitted, it will default to that specified by the free energy parameters \((-A/B)^{1/2}\). However, one is free to choose a \(\phi^\star\) for the viscosity model independently of the value that would be computed from the free energy parameters.

The values of \(\eta_+\) and \(\eta_-\) override any Newtonian viscosity that may also be specified in the input. The local bulk viscosity is computed via

\[\eta_\nu (\mathbf{r}) = (\eta_\nu/\eta) \eta (\mathbf{r})\]

where \(\eta\) and \(\eta_\nu\) are the fixed Newtonian shear and bulk viscosities, respectively.