## 5 Mathematical Models
### Models at the Macro-Scale
#### 5.1.1 The Navier-Stokes Equations for Viscoelastic Fluids  
A fluid is said to be Newtonian if the local rate of change in its deformation over time is linearly proportional to the viscous stresses arising from its flow. On the other hand, non-Newtonian fluids defy this principle, and, for the most part, their viscosity depends on the present or past shear rate. Non-Newtonian fluids include blood, proteins, polymers, suspensions, emulsions, chemical reactants, and most other fluids found in both nature and industry. These fluids typically exhibit a wide range of remarkable properties, such as viscoelasticity, shear-thinning, and shear-thickening properties [93; 94]. Shear-thinning (also called pseudoplastic) fluids are characterized by an apparent viscosity which decreases by increasing the shear rate. Conversely, shear-thickening fluids are those whose apparent viscosity rises as the shear rate decreases [95]. Finally, it should be noted that viscoelastic fluids exhibit time-dependent strain, a characteristic in which the fluids' strain gradually approaches the equilibrium value when stress is applied. Navier-Stokes equations are used to describe the conservation of mass and momentum of incompressible fluid, varying in temporal domain \(\Theta\), in a spatial domain \(\Omega\subseteq\Re^{n}\), \(n=1,2,3\) with boundary \(\delta\Omega\) as the following problem: \begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**Model** & _“Pros”_ & _“Cons”_ \\ \hline The Navier-Stokes equations for Viscoelastic Fluids (Section 5.1.1) & The most comprehensive way to describe complex fluid dynamics. & The definition of model parameters could be very hard. \\ \hline The Level Set Method (LSM) for the two-phase flow problem (Section 5.1.2) & LSM is based on a front-capturing approach that uses a scalar function on a stationary grid to capture the interface evolution. The ability to compute phenomena like bubble breakup and coalescence is made possible by the implicit interface capture. & The main difficulty in using this approach is keeping a clear boundary between the different fluids. \\ \hline The Immersed Method (IBM) for the two-phase flow problem (Section 5.1.2) & Boundary front-capturing and front-tracking approaches. It can be useful to describe fluid-structure interaction when the structure is intended to be an immersed body. & IBM could inherit drawbacks of front-tracking approaches in describing interfaces with complex geometries. Moreover, the correct definition of the distribution function \(d\) could be crucial. \\ \hline The continuum surface (CSF) model for the two-phase flow problem (Section 5.1.2) & The probably most famous approach used to compute the surface tension force needed by the application of front capturing methods. The model considers surface tension force as a continuous, 3D effect across an interface, rather than a boundary value condition on the interface. & The definition of the transition region and of the “color” function \(\mathcal{L}\). \\ \hline Transport and response of biological and chemical species (Section 5.1.3) & Models used to study how the tissue reacts to the chemical and mechanical environment in the culture medium. They can be considered a strict framework of conservation laws for mixes of interacting continua that describe the transfer of mass and momentum between different component phases. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of _“Pros and Cons”_ of the models at the macro-scale.