## 1 Introduction
Barrel finishing (BF) is the most ancient mass finishing operation, which allows conditioning the part's surface without fixturing: this degree of freedom meets AM's major aim of being almost independent of part geometry [20]. Notwithstanding its slow process, BF is vastly used because it is a versatile conditioning method with low-cost equipment [21]. The process is typically performed via an octagonal rotating barrel; the charge, composed of the parts, media, water, and a compound, is moved upward by rotation until a critical angle is reached; hence, a layer slides down. Within this region, namely, the active layer, a relative velocity between parts and media takes place, leading to a delicate abrasive conditioning of the part surfaces. Unfortunately, a specific motion is required for this behavior; thus, particular conditions must be satisfied by using adequate processing parameters [21]. The difficulty lies in the prediction of the flow properties in granular material motions, characterized by an intrinsic complexity of the element laws and a very large number of elements interacting with each other [22]. The elements are discontinuous and highly heterogeneous, and a dilatant behavior under shearing modifies the interactions, leading to an extreme variety of natural phenomena [23]. There is an extensive experimental database on the properties of non-cohesive granular material flow [24]. As early as 1954, [PERSON] stated that the tangential stresses in an intensive shear flow are mainly caused by two factors: particle collisions and impulse interchange between different layers [25]. In his pioneering work, a relevant characterization of dynamic regimes was carried out. He found that the ratio between interparticle collision forces and interstitial viscous forces leads to two different regimes, namely, macroviscous and grain-inertial regimes. This ratio, called the Bagnold number (\(Ba\)), is defined as follows:  
\[Ba=\frac{m\gamma}{(2\lambda_{e}\eta)}, \tag{1}\]  
where \(m\) is the particle mass, \(\gamma\) is the average velocity gradient along the active layer depth, \(\lambda_{e}\) is the characteristic length, and \(\eta\) is the dynamic viscosity of the interstitial fluid. In the case of small \(Ba\) (less than 40), the flow is macroviscous, such as flows of slurries or mud, where the interstitial fluid viscosity overcomes the grain inertia. For \(Ba\) greater than 450, the fluid viscosity can be neglected in comparison with the interparticle collisions; then, a grain-inertial flow takes place. A transitional regime falls between these two values [26]. Six different motions are defined in the literature, i.e., sliding, slumping, rolling, cascading, cataracting, and centrifuging, which are based on variations in the filling and the rotational speed of the barrel. No model exists for the prediction of process parameters to obtain these motions, because they are strictly dependent on the used charge in terms of material and shape, filling percentage, quantity of water and compound, etc. For this reason, the study of BF motions and process parameters is typically developed using bed behavior (BB) diagrams [27]. These are maps of the six motions and their transitions as a function of the filling percentage and the rotational speed of the barrel. These maps must be experimentally obtained because they change with the employed charge. The individuation of the motions and the relative process parameters is fundamental for the design of the BF process because only two motions, i.e., rolling and cascading, permit having an efficient material removal. These motions are characterized by a delicate continuous flow in the form of an avalanche that realizes relative motions between the media and parts being processed, causing a soft mechanical removal of material. The action was modeled in Reference [28], where the authors underlined that, during the BF process, the micro-geometrical profile is gradually cut away from peak to valley without plastic deformation.