2.1.3 Constitutive equation of viscoelasticity
To study the viscoelastic fluids, we applied the Oldroyd-B model by considering the water as a Newtonian fluid and the biofilm as a non-Newtonian fluid. The constitutive Oldroyd-B equation is given by
\(\mathbf{\tau}+\lambda\mathbf{\tau}^{\nabla}=2\mu_{b}\mathbf{d}\), \(\ \mathbf{\tau}^{\nabla}=\frac{\partial\mathbf{\tau}}{\partial t}+\mathbf{u}\bullet\nabla\mathbf{\tau-}\left(\nabla\mathbf{u}\right)^{T}\mathbf{\bullet\tau}\mathbf{-}\mathbf{\tau}\mathbf{\bullet}\mathbf{(}\nabla\mathbf{u}\mathbf{)}\),
\(\mathbf{T}\left(\phi\right)=V_{b}\mathbf{(\tau+}2\mu_{s}\mathbf{d)}+V_{s}\left(2\mu_{s}\mathbf{d}\right)\),
where τ is the extra (i.e., viscous) stress tensor,\(\ \mathbf{\tau}^{\nabla}\) denotes the upper-convected time derivative (Oldroyd derivative) of the stress tensor, λ=\(\mu_{b}/G_{b}\) is the biofilm elastic relaxation time [s].\(G_{b}\) is the biofilm shear modulus [Pa]. \(\mu_{b}\) and\(\mu_{s}\) are dynamic viscosities of the biofilm and solvent [Pa∙s], respectively. Here we define\(\mathbf{\sigma}_{e}=2\mu_{s}\mathbf{d}\) as the elastic deviatoric component of the stress tensor. Finally, d is the symmetric part of the velocity gradient,\(\mathbf{d}=\frac{1}{2}(\nabla\mathbf{u}+\left(\nabla\mathbf{u}\right)^{T})\).