## 5 Mathematical Models
### Models at the Macro-Scale
#### 5.1.1 The Navier-Stokes Equations for Viscoelastic Fluids
where \(\dot{\gamma}\) is the shear rate defined by [98]  
\[\dot{\gamma} = \sqrt{2}\|\mathbf{D}\|, \tag{13}\]  
and where \(\|\cdot\|\) denotes the Hilbert-Schmidt norm of a matrix \(\mathbf{A}\) defined as  
\[\|\mathbf{A}\|=\sqrt{\sum_{i,j=1,\ldots,n}a_{ij}^{2}}\]  
The following viscosity models \(\mu_{s}\) for generalized Newtonian fluids could be considered [93; 99]:  
**Newtonian viscosity** model  
\[\mu_{s}^{(0)}=\mu_{\infty}^{(0)}=\mu_{s},\]
**Power-Law** model  
\[\mu_{s}^{(1)}=\mu_{\infty}^{(1)}\dot{\gamma}^{n-1},\]
**Cross** model  
\[\mu_{s}^{(2)}=\mu_{\infty}^{(2)}+\Big{(}\mu_{0}^{(2)}-\mu_{\infty}^{(2)} \Big{)}\Big{[}1+(\lambda\dot{\gamma})^{b}\Big{]}^{-1},\]
**Modified Cross** model  
\[\mu_{s}^{(3)}=\mu_{\infty}^{(3)}+\Big{(}\mu_{0}^{(3)}-\mu_{\infty}^{(3)} \Big{)}\Big{[}1+(\lambda\dot{\gamma})^{b}\Big{]}^{-a},\]
**Carreau** model  
\[\mu_{s}^{(4)}=\mu_{\infty}^{(4)}+\Big{(}\mu_{0}^{(4)}-\mu_{\infty}^{(4)} \Big{)}\Big{[}1+(\lambda\dot{\gamma})^{2}\Big{]}^{\frac{n-1}{2}},\]
**Carreau-Yasuda** model  
\[\mu_{s}^{(5)}=\mu_{\infty}^{(5)}+\Big{(}\mu_{0}^{(5)}-\mu_{\infty}^{(5)} \Big{)}\big{[}1+(\lambda\dot{\gamma})^{m}\Big{]}^{\frac{n-1}{n}},\]  
where \(\mu_{0}^{(\theta)}\) and \(\mu_{\infty}^{(\theta)}\) are the asymptotic viscosity values at zero and infinite shear rates (where \(\theta=0,1,2,4,5\)). The parameters of the non-Newtonian viscosity models are \(a\), \(b\), \(m\), and \(n\), and the relaxation time is denoted by the symbol \(\lambda\). The Power-Law model [93] may explain shear-thinning behavior for \(n<1\), shear-thickening behavior for \(n>1\), and Newtonian behavior for \(m=1\). Several constitutive rheological models are available in the literature with polymeric stresses \(\mathbf{T}_{p}\)[93; 97], which are typically functions A number of constitutive rheological models with polymeric stresses \(\mathbf{T}_{p}\) are available in the literature [93; 97]. These stresses are usually functions \(\mathbf{f}_{S}(\cdot)\) of the conformation tensor \(\mathbf{C}\),  
\[\mathbf{T}_{p}=\frac{\mu_{p}\mathbf{f}_{S}(\mathbf{C})}{\lambda} \tag{14}\]  
where \(\mu_{p}\) the polymeric viscosity. According to [97], the conformation tensor \(\mathbf{C}\in\Re^{n\times n}\) can be viewed as an internal state variable that describes the molecular deformation of the polymer chains. Assuming symmetry and positive definiteness, the conformation tensor \(\mathbf{C}\) behaves in accordance with the following equation:  
\[\dot{\gamma} = \sqrt{2}\|\mathbf{D}\|, \tag{13}\]0where the relaxation function, which varies for each specific constitutive model, is \(f_{R}(\cdot)\). The operator _upper-connected derivative_, described in [97], is indicated by \(\bigtriangledown\). With the symbol \(\bullet\), we intend to represent the matrix-matrix operation. \[\dot{\gamma} = \sqrt{2}\|\mathbf{D}\|, \tag{13}\]1  
where  
\[\
abla\bullet(\mathbf{v}\mathbf{C}) = \left[ \begin{array}{ccc}\sum_{k=1,\ldots,n}v_{k}\frac{\partial \mathbf{c}_{1,k}}{\partial\mathbf{x}_{k}}&\cdots&\sum_{k=1,\ldots,n}v_{k}\frac{ \partial\mathbf{c}_{1,n}}{\partial\mathbf{x}_{k}}\\ \vdots&\ddots&\vdots\\ \sum_{k=1,\ldots,n}v_{k}\frac{\partial\mathbf{c}_{k,1}}{\partial\mathbf{x}_{k}}& \cdots&\sum_{k=1,\ldots,n}v_{k}\frac{\partial\mathbf{c}_{k,n}}{\partial\mathbf{x}_{k}} \end{array} \right] \tag{17}\]  
Table 4 collects the expressions of the strain \(f_{S}(\mathbf{C})\) and relaxation functions \(f_{R}(\mathbf{C})\) for some constitutive models [97]. Classically, in the case of the Oldroyd-B model, it is usual to skip the use of \(C\) by combining Equations (1) and (15). Then, the constitutive equations in terms of the viscoelastic stress tensor, \(T_{p}\) is written as  
\[\lambda T_{p}+\overset{\mathbb{C}}{T_{p}}=2\mu_{p}\mathbf{D}. \tag{18}\]  
The summation of \(\mu_{s}\) and \(\mu_{p}\) of the fluid is defined as total viscosity and denoted as \(\mu_{0}=\mu_{s}+\mu_{p}\).