# Deformation of Air Bubbles Near a Plunging Jet Using a Machine Learning Approach
## 1 Introduction  
The study of the dynamics of air bubbles in a liquid flow field is of practical interest in many areas of environmental, chemical, naval and ocean engineering. The interest in phenomena such as cavitation, air entrainment and foaming is particularly relevant. The theoretical characterization of motion and deformation of isolated bubbles is effective only in simple special cases. The most basic theoretical model of the dynamics of a single bubble is represented by the Rayleigh-Plesset equation [1, 2], which allows for a description of the variation in the bubble radius as a function of four forcers: the external pressure, the internal pressure of the bubble, the surface tension of the liquid and the viscosity of the liquid. It is valid in the case of a perfectly spherical bubble in an infinite liquid domain that is at rest far from the bubble. Moreover, temperature gradients are not considered, while the pressure is a known input governing the bubble deformation. In most situations of practical interest, these assumptions cannot be justified. Later, various authors proposed other theoretical models under different hypotheses [3, 4, 5, 6]. More complex situations are generally modelled by experimental [7; 8; 9; 10] or CFD-based approaches [11; 12; 13; 14]. However, the dynamics of individual bubbles in a water volume are governed by the physical characteristics of water and air, which can be expressed in terms of water density \(\rho_{w}\), bubble density \(\rho_{b}\), water dynamic viscosity \(\mu\) and water surface tension \(\sigma\). The rising of an air bubble depends also on the buoyancy, which is a function of the pressure gradient \(\partial P/\partial z\), where \(z\) indicates the water depth, and that of the gravitational acceleration \(g\). The buoyancy also depends on the air bubble volume, related to the bubble equivalent diameter (\(D_{eq}\)) (i.e., the diameter of a sphere of volume equal to the bubble volume). Furthermore, the relative motion between the air bubble and the surrounding water, which moves with velocity \(V_{w}\), involves a resistance related to the bubble velocity \(V_{b}\)[15]. In general, the evolution of a single bubble that moves in the water can be described according to:  
\[f(\rho_{w},\rho_{b},\mu,\sigma,D_{eq},g,\frac{\partial P}{\partial z},V_{w},V_ {b})=0 \tag{1}\]  
Based on the well-known dimensional analysis [16; 17; 18], four essential dimensionless numbers affecting the evolution of individual bubbles can be obtained:  
\[Re_{b}=\frac{V_{b}\cdot D_{eq}}{v} \tag{2}\]  
\[Fr_{b}=\frac{V_{b}}{\sqrt{g\cdot D_{eq}}} \tag{3}\]  
\[We_{b}=\frac{\rho_{w}\cdot V_{b}^{2}\cdot D_{eq}}{\sigma} \tag{4}\]  
\[Eo=\frac{(\rho_{w}-\rho_{b})\cdot g\cdot D_{eq}^{2}}{\sigma} \tag{5}\]  
where \(Re_{b}\) is the bubble Reynolds number, \(v\) is the water cinematic viscosity, \(Fr_{b}\) is the bubble Froude number, \(We_{b}\) is the bubble Weber number and \(Eo\) is the Eotvos number. A somewhat complex phenomenon that induces the movement of bubbles in a water volume is observed in the presence of a plunging jet. When a water jet plunges into a water pool, air bubbles are generally entrained and carried away in the water volume (Figure 1) if the jet impact velocity exceeds a critical value [19]. The impact of the jet induces a fluctuating pressure field that is very difficult to characterize [20]. Moreover, in the turbulent flow induced by the plunging jet, the evolution of the bubbles is also governed by other mechanisms, e.g., turbulence fluctuation, drag, turbulent shear, etc. [21]. The knowledge of the pressure and shear stress is essential to accurately predict the evolution of the size and shape of the bubbles near the plunging jet, since the shape of a single bubble can fluctuate in response to oscillations in the pressure and shear stress in the liquid surrounding the bubble [22; 23].