## 3 Fundamentals of CFD in the Context of Mechanical Processes
### CFD Models for Non-Newtonian and Viscoelastic Food Materials  
Many food materials exhibit complex rheological behaviors that deviate significantly from the assumptions of Newtonian fluid dynamics. Products such as doughs, batters, purees, emulsions, and protein gels often demonstrate shear-thinning, viscoplastic, thixotropic, or viscoelastic properties. Accurate modeling of these behaviors is essential for realistic CFD simulations of food processing operations, including mixing, shaping, extrusion, and transport. Proper incorporation of non-Newtonian rheology into CFD models ensures reliable predictions of pressure drop, shear rate distribution, residence time, energy input, and temperature profiles. Failing to account for these behaviors can lead to incorrect process design and suboptimal product quality [32]. Food systems are typically classified into several categories of non-Newtonian behavior. Shear-thinning or pseudoplastic behavior, where apparent viscosity decreases with increasing shear rate, is common in sauces, dairy emulsions, fruit pulps, and starch-based formulations. Conversely, shear-thickening or dilatant behavior, characterized by an increase in viscosity with shear rate, is less common but can occur in concentrated starch slurries. Yield stress fluids--such as ketchup, mayonnaise, and chocolate--require a minimum stress threshold to initiate flow. Thixotropic materials, including certain gels and pastes, exhibit a time-dependent decrease in viscosity under constant shear. Finally, viscoelastic materials display both elastic and viscous responses, as seen in protein gels, gluten networks, and high-protein extrudates [33]. To accurately simulate these behaviors, CFD platforms must incorporate appropriate rheological models. One of the most widely used formulations is the power-law model:  
\[\mu_{app}=K\cdot\dot{\gamma}^{n-1}.\]  
where \(\mu_{app}\) is the apparent viscosity, \(K\) is the consistency index, \(\dot{\gamma}\) is the shear rate, and \(n\) is the flow behavior index. Although computationally efficient and suitable for shear-thinning fluids, this model lacks a yield stress component and becomes inaccurate at very low shear rates [34]. For materials that exhibit both shear-thinning and yield stress behavior, the Herschel-Bulkley model is preferred:  
\[\tau=\tau_{y}+K\cdot\dot{\gamma}^{n}\]  
Here, \(\tau\) is the shear stress and \(\tau_{y}\) is the yield stress. This model is appropriate for viscoplastic food systems such as tomato paste, meat emulsions, and chocolate. A simpler alternative is the Bingham plastic model:  
\[\tau=\tau_{y}+\mu_{p}\cdot\dot{\gamma}\]  
which assumes linear behavior beyond the yield stress and is often used to approximate the flow of materials that behave as solids until a threshold is exceeded where \(\tau\)--shearstress [Pa], \(\gamma_{y}\)--yield stress [Pa], i.e., the minimum stress required to initiate flow, \(\mu_{p}\)--plastic viscosity [Pa\(\cdot\)s], representing the slope of the stress-shear rate curve beyond yield \(\dot{\gamma}\)--shear rate [1/s], the rate of deformation of the fluid. Additional models such as the Casson model, commonly used for chocolate, and the Carreau-Yasuda and Cross models, which capture shear-thinning behavior across a broad range of shear rates, are also implemented in commercial CFD software like ANSYS Fluent, COMSOL Multiphysics, and OpenFOAM. The selection of a rheological model should be informed by experimental data obtained via rotational rheometry under conditions relevant to the process being modeled [33; 35; 36]. In systems where elasticity plays a significant role, such as during dough kneading, extrusion of gelatinous materials, or structuring of plant-based meat analogues, viscoelastic models are necessary to capture the material response accurately. The Maxwell model is used to describe linear viscoelastic fluids with stress relaxation characteristics, while the Kelvin-Voigt model accounts for time-dependent strain and instantaneous elastic recovery.