The filling process of liquid metal is a non-isothermal flow process with heat loss and solidification.The metal fluid flow should follow the three conservation laws of mass, momentum and energy. The continuity equation, momentum equation and energy equation can be used to describe the flow process.

#### 1.Continuity equation:

The continuity equation is the basic description of fluid flow by the law of mass conservation:

Where delta is the density of the fluid, t is the time, V is the velocity of the fluid, where the components in the x, y and Z directions are u, V and w respectively.

For incompressible fluids, the continuity equation can be simplified as:

#### 2.N-S equation:

According to Newton’s second law, the rate of change of the object’s momentum with time is proportional to the force acting on the object. Finally, the N-S equation of the incompressible fluid is obtained as follows: