2.1.1 Phase field model
The phase-field model was adapted from Yue et al. (2006) and Zhang et al. (2010), and applied to biofilm-fluid interactions. Two types of incompressible, immiscible fluids were studied as two components of a single fluid. The fluid interface was assumed to be a thin, nonzero thickness transition region with distributed interfacial forces (Kim, 2012).
In our model, we consider the bulk water phase (solvent) and biofilm as two immiscible fluids. The PF variable ϕ \(\in[-1,1]\) is defined as the difference between volume fractions of each component.\(\ V_{b}+V_{s}=1,\ V_{b}=\frac{1+\phi}{2},\ V_{s}=\frac{1-\phi}{2}\),\(\text{\ ρ}\left(\phi\right)=V_{b}\rho_{b}+V_{s}\rho_{s}\), where \(V_{b}\) and \(V_{s}\) are the volume fractions of biofilm and solvent. \(\rho\) is the density of fluids [kg/m3], \(\rho_{b}\) represents the density of a biofilm and \(\rho_{s}\) represents the density of a solvent. The governing equation for the PF model is the advective Cahn-Hilliard equation, which uses a chemical potential representing the diffuse interface of two fluids:
\(\frac{\partial\phi}{\partial t}+\mathbf{u}\bullet\nabla\phi=\nabla\bullet\frac{\gamma\varphi}{\varepsilon^{2}}\nabla\psi\),\(\text{\ \ ψ}=-\nabla\bullet\varepsilon^{2}\nabla\phi+\left(\phi^{2}-1\right)\phi\),
where \(\mathbf{u}\) is the fluid velocity [m/s], γ is the mobility [m3⋅s/kg], \(\varphi\) is the mixing energy density [N], and \(\varepsilon\) is the interface thickness parameter [m]. Another dependent variable of the PF model is phase field variable \(\psi\). The mixing energy density is defined as:\(\ \phi=\frac{3\vartheta\varepsilon}{\sqrt{8}}\), where\(\vartheta\) is surface tension coefficient [N/m]. The mobility\(\gamma\) is defined as:\(\ \gamma=\chi\varepsilon^{2}\), where χ is the mobility coefficient [m⋅s/kg].