*2.2. Computational Modeling*

In order to study the acoustic propagation in liquid metal, a simulation model was developed using the COMSOL v5.2a Multiphysics module—Acoustic Piezoeloectric Interaction, Frequency Domain, according to Figure 2. Considering that the acoustic wave propagation is linear, and the shear stresses are negligible for fluids, the acoustic pressure can be calculated by applying the following wave equation [6,7]:

$$\nabla \left(\frac{1}{\rho} \nabla P\right) - \frac{1}{\rho c^2} \frac{\partial^2 P}{\partial t^2} = 0 \tag{3}$$

where ρ is the density of the liquid metal, *c* is the sound velocity in the liquid metal and *t* is time. For the case of a harmonic wave in time, the pressure varies according to:

$$P(r,t) = p(r)^{i\alpha t} \tag{4}$$

where, ω = 2π*f* is the angular frequency and *p* is the acoustic pressure. Assuming that the same harmonic is time dependent, in the same terms, Equation (3) can be reduced, through Equation (4), to the Helmholtz equation:

$$
\nabla \left( \frac{1}{\rho} \nabla p \right) - \frac{\alpha^2}{\rho c^2} p = 0 \tag{5}
$$

**Figure 2.** Geometry modeled using COMSOL Multiphysics—Acoustic Piezoeloectric (PZT) Interaction, Frequency Domain. (1) acousticmedium, (2) Ti6Al4V acoustic radiator, (3) Piezoeloectric (PZT) polarization.

The Acoustic Piezoeloectric Interaction module in COMSOL Multiphysics was used to perform an analysis in the frequency domain. This combines the effects of (i) sound pressure and (ii) piezoelectric, linking the variations of acoustic pressure with the solids that are actuated by the piezoelectric effect of PZT. The physical interface also includes electrostatic elements to solve the electric field in the piezoelectric material. The Helmholtz equation is solved in the fluid domain and the structural equations in the solid domain, together with the constructive relations necessary for the piezoelectric modeling. The physical interface that solves the Helmholtz equation is suitable for the present study, in the domain of linear frequencies with harmonic variation of the pressure field.

In order to evaluate the profile of the acoustic pressure during the deformation of the solids actuated by the PZT piezoelectric effect, water was used since this is a suitable liquid medium to simulate the refinement / modification mechanism that occurs in melts of aluminum alloys at 660–700 ◦C [29]. With the appropriate boundary conditions, the Helmholtz equation can be solved through a range of numerical methods [6,8]. The accuracy of the numerical solution of the Helmholtz equation depends significantly on the wave number *k*(*k* = ω/*c*). The main boundary conditions are described as, (i) *p* = 0 (condition of total waves reflection, attributed to the liquid-air interfaces); (ii) *p* = *p*<sup>0</sup> (interface of the acoustic radiator with the liquid metal); (iii) δ*p*/δ*n* = 0 (condition of "rigid walls" attributed to the lateral walls of the acoustic radiator); (iv) (1/ρ)(δ*p*/δ*n*) + *i*ω*p*/*Z* = 0 (acoustic impedance limit condition *Z* attributed to the container walls).
