2.1.2 Navier-Stokes equation
The incompressible Navier-Stokes equations were applied to describe the transport of momentum and mass of both solvent liquid and biofilm liquid:
\(\rho\frac{\partial\mathbf{u}}{\partial t}+\rho\left(\mathbf{u}\bullet\nabla\right)\mathbf{u}=\nabla\bullet\mathbf{\sigma}+\frac{\varphi}{\varepsilon^{2}}\psi\bullet\nabla\phi\), \(\mathbf{\ }\mathbf{\sigma}=-p\mathbf{1}+\mathbf{T}\mathbf{(}\phi)\), \(\ \nabla\bullet\mathbf{u}=0\),
where σ is the Cauchy stress tensor, T is the deviatoric part of the stress tensor, \(p\) is the pressure [Pa], and 1 is the second-order identity tensor.