The metal in the filling process can be considered as incompressible fluid, which follows the hydrodynamics law and can be described by the conservation of mass and momentum. The mathematical forms of the metal are continuous equation and N-S equation.

Continuity equation:

For the incompressible fluid, ρ: V, V are the three-dimensional components of the fluid

ρ／ ｔ＝０。

N-S equation:

Among them, the N-S equation shows the mechanical law of viscous fluid. The rate of change of the momentum of a body to time is directly proportional to the force acting on the body. Where Δ μ Z and ρ W are the velocities of the fluid at any point in the fluid dynamics, where Δ μ Z and ρ W are the velocities of the fluid at any point in the fluid dynamics.

Energy conservation equation:

Where: C is the specific heat capacity of the fluid, s is the original term, t is the temperature of the fluid, λ is the thermal conductivity of the fluid, and u, V, W are the components of the velocity vector at any point in the fluid on X, y, Z, respectively.

In the SOLA-VOF mathematical model, in order to solve the above equations, we need to discretize them first, then use the VOF method to deal with the free surface, and finally use the sola method to solve the pressure field and velocity field.