## 3 Modelling and Control
### Approaching Flow Stage (AFS)  
In this initial stage, tilting angle \(\theta\) goes from zero to a threshold value of \(\phi\), so the stabilized outflow height reaches a threshold value \(h_{q}\) at a certain outflow bulk velocity \(v_{q}\); see Figure 3.  
While for a high \(v_{q}\), the outflow follows a free falling motion, the flow may attach to the walls of the vessel at low values of \(v_{q}\) and \(\theta\), causing liquid to drip through the vessel walls; see Figure 1 case b. Regarding other factors that may be involved in this kind of flow, however, most of them can be prevented by guaranteeing a minimum value of \(v_{q}\) at the vessel tip. To overcome this initial issue, inspired by intuitive human pouring action, a slosh induction method (SIM) is proposed in this work. SIM consists of creating an anti-symmetrical oscillation of the fluid free surface such that the tilting angle \(\theta\) of the rotated vessel 1 produces an equal angle \(\theta_{1}\) of the liquid-free surface at a certain liquid level condition. This condition is set when the liquid level \(h_{e}\) is equal or lower than the window level \(h_{q}\) at an arbitrary angle of \(\theta=20^{\circ}\). See Figure 4 for reference. SIM will not produce a significant change in the liquid volume shape from the moment of rotation of \(\theta\) till the angle reaches the conditional \(\phi\) value, leaving a free surface inclination of \(\phi\) degrees with respect to horizontal, such as it is indicated in transition 1-2 of Figure 4. That retains a linear model for the slosh. After \(\theta_{1}\) and \(\theta\) become \(\phi\), free surface will freely return to the flat horizontal shape (see Figure 4 transitions 3-6) allowing the slosh to travel back to the opposite vessel wall and run out of the vessel with a velocity close to the fundamental velocity of the induced slosh. When the liquid level condition is not met, there is no need to apply AFS control and CFS can be resumed. In such a case, the outflow will follow free fall motion from the \(0^{\circ}\) and until the final angle, the x-axis position of the fluid fall is predictable, since the \(x\)-component velocity of flow (\(v_{q}\)) is known by SPH computations. This is one advantage of using the SPH method as an emulator engine. Figure 4: Transitions of AFS with SIM. Transition 1 defines initial conditions in hydrostatic equilibrium where the angle of the pouring vessel and the angle of the free surface are zero. Transitions 2-6 define the slosh induction method steps that run only if the height condition (\(h_{e}\leq h_{q}\) at \(20^{\circ}\)) is met. Figure 3: Stages of pouring. To the (**left**), the approaching flow stage (AFS) going from the initial time to \(t_{AFS}\). To the (**right**), the succeeding stage of constant flow (CFS) going from \(t_{AFS}\) to \(t_{CFS}\). [PERSON] [27] studied the moment of inertia of liquid in a vessel, and showed that, as the area of liquid gets more symmetrical, the moment of inertia becomes smaller, and, in the same way, the portion of liquid that moves along the vessel becomes smaller. Therefore, as the moment of inertia states, without using SIM, the location of the center of rotation would be right at the center of mass, making the moment of inertia close to zero and retaining the free surface orthogonal to the gravity vector in such case. In contrast, AFS with SIM uses the \(z\)-axis as the axis of rotation located at the middle bottom of the vessel 1. This increases the moment of inertia to allow the free surface to elevate along the vessel bottom plane. See Figure 4 transition 2. In this work, AFS without SIM Case is not going to be developed; however, there are two steps required to run such case: (1) find the center of mass to locate the center of rotation, and (2) rotate the vessel over this center, and reach the proposed outflow height \(h_{q}\). Recall that in this case, one could have a scenario of a brief volume to pour with respect to the total vessel volume, and thus, CFS could not begin with the initial condition of \(h_{q}\).