# A Primer on CFD-DEM for Polymer-Filled Suspensions
## 1 Introduction  
Filled suspensions are commonly used in various industries, such as the chemical industry, mining, food processing, and cosmetics. The properties of the particles suspended in the base fluid (e.g., size, shape, and surface properties) are important for the development of new and better materials/products. For example, numerous polysaccharides and proteins can combine to generate colloidal particles that can create a viscous solution [1], an elastic gel, or a liquid crystalline phase (which are frequently utilized as structuring or thickening ingredients in meals because of this). There is also particle erosion behavior in viscoelastic surfactant abrasive slurry jetting [2], where viscoelastic surfactant fluids are used due to the advantages of low friction, environmental friendliness, and microstructures recoverability after mechanical degradation. Other applications could be cited here, but the focus should be on the viscous and elastic nature of the fluids. This system's rheological behavior is extremely complicated and strongly non-linear, because the system's microstructure changes as a result of the applied deformation. The most simple model that takes into account such changes is the power law model. Therefore, in this work, we aim at understanding the behavior of such a complex system, where viscosity changes with the rate of deformation. This understanding is crucial in order to study more complex flows where elasticity plays a fundamental role. For low and weak deformations/flows, the model can be used to study viscoelastic materials; for high and strong deformations/flows, a more complex viscoelastic model should be used instead. Before we proceed to such complexflows, it is imperative to study the controlled deformation of particle flows modeled by a power law fluid. Therefore, to understand the behaviors of the materials under different deformations and to identify the influences of the particles on the final product, rheological characterization is often performed. This characterization is done with the help of rheometers (see Figure 1), which allow the fluid properties to be measured under shear stress or shear rate conditions (considering controlled flows). Figure 1 shows a schematic of a rheometer consisting of an upper plate (\(z=H\)), a lower plate (\(z=0\)), and a gap between the plates, which is filled with a sample of the material to be tested. Both plates have a radius of \(r\). The lower plate is fixed and the upper plate can rotate with an angular velocity \(\Omega\), resulting in a deformation of the material sample (\(\dot{\gamma}=r\Omega/H\), with \(\dot{\gamma}\) being the shear rate). This deformation in a controlled environment allows the properties of the material to be studied based on the material's response to the imposed deformation. This characterization is commonly supported by numerical methods, which allow simulations of filled suspensions to be carried out. In this way, it is possible to determine the behaviors of the suspensions at any time and any point in the domain. Regarding the vast scientific literature about filled suspensions, early research studies focused on the relationship between relative viscosity and particle volume fraction, such as those presented by [PERSON] [3,4] and [PERSON] [5]. A review of the literature on experimental studies of the rheology of suspensions shows that only the apparent viscosity of the suspension (\(\eta=\tau/\dot{\gamma}\), where \(\tau\) is the shear stress) is presented. However, to capture the non-Newtonian rheology, such as shear-thinning and yield stress, this one-dimensional measure is not sufficient. Some theoretical and experimental studies have attributed shear-thinning to the dependence of viscosity on the Peclet number, particle Reynolds number, or Stokes number of the suspension [6,7]. Other works, such as [8] investigated the dependence of the shear-thinning degree on the particle volume fraction \(\phi\).