*2.3. Fluid Motion Equation and Discretization*

The governing equations for a weakly compressible viscous flow are based on the relationship between the velocity of sound and the flow density under adiabatic conditions, as well as the Navier-Stokes Equations:

$$\left(\frac{Dp}{D\rho}\right)\_S = c^2\tag{7}$$

$$
\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho + \mathbf{F}\_s \tag{8}
$$

where **v** is the fluid velocity, *p* is the pressure, *c* is the velocity of sound, μ is the viscosity, and **F***<sup>s</sup>* is the interfacial force. Subsequently, Equation (8) can be formulated for each particle as follows:

$$m\_i \frac{D\mathbf{v}\_i}{Dt} = -\sum\_{j=1}^{N} \left( \langle p \rangle\_i \boldsymbol{V}\_i^2 + \langle p \rangle\_j \boldsymbol{V}\_j^2 + \Pi\_{ij} \right) \nabla \mathcal{W}\_{ij} + \sum\_{j=1}^{N} \frac{2\mu\_i \mu\_j}{\mu\_i + \mu\_j} \left( \boldsymbol{V}\_i^2 + \boldsymbol{V}\_j^2 \right) \frac{\mathbf{r}\_{ij}}{\left| \mathbf{r}\_{ij} \right|^2} \mathbf{v}\_{ij} \nabla \mathcal{W}\_{ij} + m\_i \mathbf{g} + \langle \mathbf{F}\_\delta \rangle\_i \tag{9}$$

where Π is the artificial viscosity term, which is usually added to the pressure gradient term to help in diffusing sharp variations in the flow and dissipate the energy of the high-frequency term [35]. To determine the time derivative of pressure from Equation (7), Tait's equation of state can generally be used [28]:

$$
\langle p \rangle\_i = \frac{c^2 \rho\_0}{\mathcal{V}} \left\{ \left( \frac{\rho\_i}{\rho\_0} \right)^{\mathcal{V}} - 1 \right\} \tag{10}
$$

where γ (= 7.0) is the adiabatic exponent and ρ<sup>0</sup> is the true density value of the material. Considering the balance of the time step and the incompressible behavior of the artificial compressible fluid, an optimal value for *c* must exist.
