## 2 Fluid Models  
The goal of pouring liquid is to estimate the final spatial location where the flow will drop in, at a certain pouring height and with a certain vessel motion.  
Considering a hydrostatic equilibrium with an orthogonal free surface, a fluid discharge in an open space can be simplified as a free fall of solid motion dynamics in stationary conditions [14].  
However, this is only true if outflow velocity remains within a valid range of laminar flow velocities. Case a in Figure 1 illustrates the free fall fluid motion that is to be aimed for with predictable curvature over time.  
In contrast, unpredictable flow motion in fluid free falls may happen in a fluid with a free surface in motion (not orthogonal to the gravity vector). See Figure 1 case b. Even when initial conditions of hydrostatic equilibrium are met, turbulent outflow can occur during pouring when flow velocity, \(v_{q}\), remains particularly low. The most typical causes of turbulence at low \(v_{q}\) values are the border shape of the vessel (Coanda effect) and the force of surface tension.  
In this work, we desired to address the most visible source of turbulence: the turbulence due to low outflow velocities. To overcome this turbulence, the pouring task was divided into two stages to separate the low \(v_{q}\) flows from the high \(v_{q}\) flows. This work also contributes a new method of slosh induction to link these two stages with a control action. SIM uses the flow velocity of an induced slosh to produce replicable laminar flow at different tilting angles during initial pouring. Section 4 describes this method and its kinematic phases.  
Using dimensional analysis, that which may occur when Weber number \(We=2\rho gh_{q}^{2}/3\sigma\) is small (\(\rho\) is water density; \(g\), gravity acceleration; \(h_{q}\), the liquid film height over the vessel rim; and \(\sigma\), the surface tension of the fluid). For water, the condition for avoiding rim-attached flow roughly corresponds to \(h_{q}>3\) mm.  
A bimanipulated liquid transfer kinematic model is proposed in our simulations. One vessel for pouring and one for catching. Vessel 1, the vessel of pouring, will only have a rotational motion, and vessel 2, the vessel of catching, will only have a translational motion; see Figure 1. A pouring height measured from the vessel 1 tip to the top border of vessel 2 is assumed. As the aim of this work was having a real time simulation to control the robotic pouring, a 2D model was preferred to 3D because its computational cost (time) is much lower. In addition, as the simulation plane was chosen to be perpendicular to the tilting axis of the pouring vessel, we anticipated no 3D flow effects.  
Modelling the fluid statics is one of the most important initial conditions to consider prior to the actual modelling of pouring motion itself. For this reason, first, the two models that set the simulation fundamentals were needed in detail. The Lagrangian fluid dynamics model is described in Section 2.1, as is the detail of its implementation to work as a free surface flow model. The second model's rigid body dynamics and its coupling with fluid are described in Section 3.  
Figure 1: Case a. Expected laminar flow with free fall trajectory slope. Case b. Unexpected no-slope turbulent fall trajectory.