## 2 Model  
### Smoothed Particle Hydrodynamics  
Smoothed Particle Hydrodynamics is a mesh-free particle method originally developed by [PERSON] [21] and [PERSON] [22] and used in a wide range of applications: astrophysics [23], shock waves [24; 25; 26; 27], explosion [28; 29; 30], thermo-fluid flows [31], thermo-capillary flows [32], multiphase flow [33; 34], Biological flows [35], non-Newtonian fluid flows [36; 37].  
Thanks to its particle nature SPH is part of the Discrete Multi-Physics framework where, coupled with other particle methods [38; 39; 40], is used to address multi-complex physics phenomena [41; 42; 43; 44; 45] to overcome the single weakness of each method.  
The idea behind the SPH formulation lies in the integral representation of any continuum function \(f(\mathbf{r})\) depending on the three-dimensional position vector \(\mathbf{r}\)  
\[f(\mathbf{r})\approx\iiint f(\mathbf{r}^{\prime})W(\mathbf{r}-\mathbf{r}^{ \prime},h)d\mathbf{r}^{\prime}, \tag{1}\]  
where \(W\) is the smoothing function or kernel and \(h\) is the smoothing length. Dividing the domain in a finite number of computational particles with their own mass \(m=\rho dr^{3}\), is possible approximate any continuum function \(f(\mathbf{r})\) as  
\[f(\mathbf{r})\approx\sum\frac{m_{i}}{\rho_{i}}f(\mathbf{r}_{i})W(\mathbf{r}- \mathbf{r}_{i},h), \tag{2}\]  
where \(m_{i},\rho_{i}\) and \(r_{i}\) are mass, density and position of the \(i\)-th particle. Equation (2) is known as particle approximation in SPH literature [13].  
We used the SPH particle approximation to discretize continuity, momentum and energy conservation equation:  
\[\begin{cases}\frac{d\rho}{dt}=-\rho\frac{\partial\mathbf{v}^{\beta}}{ \partial\mathbf{v}^{\beta}},\\ \frac{d\mathbf{v}^{\alpha}}{dt}=-\frac{1}{\rho}\frac{\partial\mathbf{r}}{ \partial\mathbf{v}^{\beta}},\\ \frac{d\mathbf{v}^{\alpha}}{dt}=-\frac{p}{\rho}\frac{\partial\mathbf{v}^{ \beta}}{\partial\mathbf{v}^{\beta}},\end{cases}\begin{cases}\frac{d\rho_{i}}{ dt}=\sum_{j}m_{i}\mathbf{v}_{ij}^{\beta}\frac{\partial W_{ij}}{\partial \mathbf{v}_{ij}^{\beta}},\\ m_{i}\frac{d\mathbf{v}_{i}^{\alpha}}{dt}=\sum_{l}m_{i}m_{j}\left(\frac{P_{i}}{ \rho_{i}^{2}}+\frac{P_{i}}{\rho_{j}^{2}}+\Pi_{ij}\right)\frac{\partial W_{ij} }{\partial\mathbf{v}_{i}^{\alpha}},\\ m_{i}\frac{d\rho_{i}}{dt}=\frac{1}{2}\sum_{j}m_{i}m_{j}\left(\frac{P_{i}}{ \rho_{i}^{2}}+\frac{P_{i}}{\rho_{j}^{2}}+\Pi_{ij}\right)\mathbf{v}_{ij}^{ \beta}\frac{\partial W_{ij}}{\partial\mathbf{v}_{i}^{\beta}},\end{cases} \tag{3}\]  
where \(\mathbf{v}\) is the velocity vector with \(\mathbf{v}_{ij}=\mathbf{v}_{1}-\mathbf{v}_{j},e\) internal energy and \(\Pi_{ij}\) is the artificial viscosity introduced by [PERSON] [24]. The [PERSON] artificial viscosity depends on a constant parameter called dimensionless dissipation factor, \(\alpha\), and from the speed of sound of particle \(i\) and \(j\), \(c_{i}\) and \(c_{j}\), with the following relationship.  
\[\Pi_{ij}=-\alpha h\frac{c_{i}+c_{j}}{\rho_{i}+\rho_{j}}\frac{\mathbf{v}_{ij} \cdot\mathbf{r}_{ij}}{r_{ij}^{2}+\epsilon h^{2}}. \tag{4}\]  
To solve the set of Equations shown before, an Equation Of State (EOS) that links pressure \(P\) and density \(\rho\) is required.