*2.1. Model Setup*

Let us consider the cooling and solidification of liquid metal in a thick-walled cylindrical chill mold subjected to low-frequency mechanical vibration (Figure 1). Mold walls are a source of horizontal vibration. Initially, the mold is uniformly heated to a temperature *Ts*, and a molten metal in the mold has a temperature *T*<sup>0</sup> above the liquidus temperature *Tm*1. Then the "liquid metal–hill mold" system cools down. The solidification starts at mold walls where the melt temperature at some point in time *tcol* becomes equal to *Tm*1. From this moment, the metal solidification starts and the solidification front moves from the mold walls to the cylinder axis. Solidification ends at a time *tfroz*. Then, the "solid metal–chill mold" system continues to cool down.

**Figure 1.** Scheme for the numerical setup of metal solidification with vibration: 1—thick-walled cylindrical chill mold, 2—liquid metal, 3—vibration generator, 4—molten zone, 5—transition zone, 6—zone of solid metal, 7—mold wall.

We made the following assumptions:


The mathematical model of the phase transition under these assumptions is limited to solving a one-dimensional Stefan problem in cylindrical coordinates [19,20].

$$\begin{cases} \frac{\partial T\_1}{\partial t} + v \frac{\partial T\_1}{\partial r} = \frac{a\_1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T\_1}{\partial r} \right), & r \le r\_{s2} \\ \frac{\partial T\_{12}}{\partial t} = \frac{a\_{12}(T)}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T\_{12}}{\partial r} \right), & r\_{s1} > r > r\_{s2} \\ \frac{\partial T\_2}{\partial t} = \frac{a\_2}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T\_2}{\partial r} \right), & r\_0 > r \ge r\_{s1} \\ \frac{\partial T\_3}{\partial t} = \frac{a\_3}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T\_3}{\partial r} \right), & r > r\_0 \end{cases} \tag{1}$$

with initial and boundary conditions:

$$\begin{array}{rcl} t = 0: r\_{s1} = r\_0, T = T\_0 \text{ under } r < r\_0; \ T = T\_s \text{ under } r = r\_0\\ \qquad r = r\_{s2}: \frac{\partial T\_1}{\partial r} = \frac{\partial T\_{12}}{\partial r}, T\_1 = T\_{12} = T\_{m2} \\\ \qquad r = 0: \frac{\partial T}{\partial r} = 0; r = r\_0: \frac{\partial T\_2}{\partial r} = \frac{\partial T\_3}{\partial r}, T\_2 = T\_3 \\\ \qquad r \to \infty: T\_3 = T\_s \end{array} \tag{2}$$

where *r* is the cylindrical coordinate, *t* is the time, *a* = *<sup>c</sup>*<sup>ρ</sup> *<sup>λ</sup>* is the thermal diffusivity, index 1 refers to the melt, index 2 to the solid phase, index 12 to the transition zone, and index 3 refers to the chill mold.

The effective coefficient of thermal diffusivity in the transition zone is defined as *a*12(*T*) = *a*<sup>1</sup> + *fsa*2, where *fs* corresponds to the volume fraction of the solid phase in the transition (mushy) zone: *fs* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>T</sup>*−*Tm*<sup>1</sup> *Tm*2−*Tm*<sup>1</sup> .

The velocity of solidification front *<sup>∂</sup>rs*<sup>1</sup> *<sup>∂</sup><sup>t</sup>* is determined from the conditions at the border of solidification *rs*<sup>1</sup> (that is, corresponding to the liquidus temperature *Tm*1) [19–21]:

$$r = r\_{s1} \colon \lambda\_1 \frac{\partial T\_1}{\partial r} - \lambda\_2 \frac{\partial T\_2}{\partial r} = \rho\_1 L \frac{dr\_{s1}}{dt}, \ T\_1 = T\_{12} = T\_{m1\prime} \tag{3}$$

where *L* is the latent heat of solidification, λ<sup>1</sup> and <sup>1</sup> are the thermal conductivity and liquid phase density, respectively, and λ<sup>2</sup> is the thermal conductivity of the solid phase.

Define an offset of fluid microscopic volumes in the environment by solving the set of wave equations:

$$\begin{cases} \frac{\partial^2 S\_1}{\partial t^2} = \frac{c\_1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial S\_1}{\partial r} \right), & r \le r\_{s2} \\\frac{\partial^2 S\_{12}}{\partial t^2} = \frac{c\_{12}(T)}{r} \frac{\partial}{\partial r} \left( r \frac{\partial S\_{12}}{\partial r} \right), & r\_{s1} > r > r\_{s2} \\\frac{\partial^2 S\_2}{\partial t^2} = \frac{c\_2}{r} \frac{\partial}{\partial r} \left( r \frac{\partial S\_2}{\partial r} \right), & r \ge r\_{s1} \end{cases} \tag{4}$$

with initial and boundary conditions:

$$\begin{array}{l} t = 0: r\_s = r\_0, S = 0; \\ r = r\_{s2}: \frac{\frac{\partial S\_1}{\partial r}}{\frac{\partial r}{\partial r}} = \frac{\frac{\partial S\_{12}}{\partial r}}{\frac{\partial S\_2}{\partial r}}, S\_1 = S\_{12}; \\ r = r\_{s1}: \frac{\frac{\partial S\_{12}}{\partial r}}{\frac{\partial S\_1}{\partial r}} = \frac{\frac{\partial S\_2}{\partial r}}{\frac{\partial r}{\partial r}}, S\_{12} = S\_{22}; \\ r = 0: \frac{\frac{\partial S\_1}{\partial r}}{\frac{\partial r}{\partial r}} = 0; r = r\_0: S = A \sin \omega t; \end{array} \tag{5}$$

where *S* is the particle displacement relative to an equilibrium position, *c* is the sound speed, *A* and ω are the vibration amplitude and frequency, respectively. An effective speed of sound in the transition zone is defined as *c*12(*T*) = *c*<sup>1</sup> + *fsc*2. The velocity of microscopic volumes of the liquid is calculated as: *v*<sup>1</sup> = *dS*<sup>1</sup> *dt* . The deformation and stress are defined as: *<sup>ε</sup>*<sup>1</sup> <sup>=</sup> *dS*<sup>1</sup> *dr* and *σ*<sup>1</sup> = *E*1*ε*<sup>1</sup> = *E*<sup>1</sup> *dS*<sup>1</sup> *dr* , respectively, where *E*<sup>1</sup> is the volumetric modulus of melt elasticity which is determined from the known ratio in the longitudinal wave: *E*<sup>1</sup> = *c*<sup>2</sup> <sup>1</sup>ρ1. The values *v*12, ε12, σ<sup>12</sup> are also calculated in the transition zone.

Consider an integral characteristic of melt stresses during the time of solidification:

$$Z\_{\sigma} = \frac{1}{r\_0} \int\_{t\_{col}}^{t\_{fvw}} |r\_1| \,. \tag{6}$$

*Z<sup>σ</sup>* has a dimensionality of mechanical impedance *Zs* = *c*11. This value can be considered as a stress integral during solidification related to the unit element of solidification front trajectory from the mold walls to its axis *r*0. It also can be considered as the total value of stresses of the vibration field directed at overcoming the mechanical impedance during solidification.

It can be assumed that the integral value *Zσ* determines the effectiveness of vibration and provides a comparative characteristic for evaluating process conditions.
