## 4 Results and Discussion
### Sloshing Force Decomposition
This suggests the possibility of using simpler equivalent mechanical models such as the bouncing ball model described in [54]. In Figure 15, the filtered sloshing force has been plotted with the direct computation of Expression (20) and also with both versions of Expression (25) for the water and glycerine. As can be observed, the agreement between the SPH computation and the derived versions either considering the \(\gamma\) and \(\beta\) parameters or the force maxima \(F\) and the shift angle \(\phi\) are both close. This means that the apparently complex sloshing force that comes from a violent and turbulent flow described in Section 4.1 can be reconstructed using a simple analytical expression based on a cosine function. This fact confirms the possibility of using reduced order models (ROMs) for this kind of problem as has been demonstrated in [53]. Figure 14: Nondimensional amplitude of the relative motion of the liquid center of gravity with respect to the tank over time for water \(f=0.5\) (**left**) and glycerine \(f=0.39\) (**right**). Comparation between SPH calculation and the expression given by Equation (26), \(-\Delta F_{slosh}/(m_{l}\omega_{0}^{2})\). In order to understand the difference in terms of Reynolds numbers and the impact on the sloshing force, the pressure and viscous contributions to the sloshing force have been separated according to Expression (19) for the water and glycerine. It can be observed that the global sloshing force being similar for both fluids according to Figure 15, the viscous contribution is much smaller in the case of water when both fluids are compared; see Figure 16. This is an expected result and confirms the negligible role of the viscous forces when the Reynolds number is high enough \(Re\sim 10^{5}\) as in the case of water. The pressure forces are on the same order of magnitude for both fluids, being smaller in the case of glycerine. It is also interesting to mention that the viscous forces for the glycerine start having a significant role after \(t>0.52\) s of simulation. This can be accounted for by the fact that the violent relative movement of the glycerine inside the tank starts after three cycles of the tank and not since the beginning, the first impact being at \(t=0.41\) s; see Section 4.1.2. The period of time between \(t=0.41\) s and \(t=0.52\) s is the time required for the velocity gradients to grow on the wall boundary layers that flow over the surfaces. The energy dissipated by the different physical components of the sloshing force can also be studied, where \(F_{slosh}^{p}\) and \(F_{slosh}^{v}\) denote the pressure and viscous parts, respec  
Figure 16: Nondimensional sloshing force components separated into their pressure and viscous contributions, for both the water and the glycerine cases. Figure 15: Filtered nondimensional sloshing force computed directly from the SPH formulation and also using the alternative force reconstruction given by the Equation (25). tively, and also considering the dissipated energy given by the structural damping force \(F_{D}=B_{0d}sign(v_{t})+B_{1d}v_{t}\). These energies are computed using the Expression (27) and then normalized by \(\varepsilon_{0}\). \[\varepsilon_{diss}=\int F\quad dy \tag{27}\]  
In Equation (27), \(F\) could be any of the three forces mentioned in the previous paragraph. It was observed that the sum of the structural and the pressure forces are above 90%. The energy dissipated by the viscous part of the sloshing force is below 4% in all cases; nevertheless, the glycerine viscous dissipation is notably larger compared to water. The total dissipated energy quantified by these three forces sums to 97.4% for the water and to 96.7% for the glycerine; see Figure 17. The rest of the dissipation comes from the additional numerical terms that are introduced into the model for stability, such as the \(\delta\)-SPH.