Solidification in material processing has recently been extended to include the effects of fluid flow in the melt. The Navier-Stokes equations govern the motion of fluids. Liquid metals for example, undergoes through both cooling and solidification phase change. In order to accurately track the changes of the liquid metal, these equations, which are used to examine changes of properties during dynamic and thermal interactions. In addition, these equations are broadly applied mathematical models and are adjustable to fit the content of the problem and expressed based on the conservation of mass, momentum, and energy 1. They can be seen as Newton’s second law of motion for fluids. Newton’s second law states that the rate of change of momentum of a body is directly proportional to the force applied. As a result, this change in momentum takes place in the direction of the applied force. There is an assumption that must be made when dealing with the Navier-Stokes equations. The assumption is that the stress in the fluid is equal to the sum of a diffusing viscous term and a pressure term. The diffusing viscous term is proportional to the gradient of velocity.

The momentum in a control volume is kept constant. This implies the conservation of momentum. In the case of a compressible Newtonian fluid, this yields:

? is noted as the fluid density, u is the fluid velocity, p is the fluid pressure, ? is the fluid dynamic viscosity, I is the identity matrix, and F is force. The different numbered terms correspond to the different types of forces. Term 1 are the inertial forces, term 2 are the pressure forces, term 3 are the viscous forces, and term 4 are the external forces applied to the fluid. These equations are always solved together with the continuity equation:

The Navier-Stokes equations represent the conservation of momentum, whereas the continuity equation represents the conservation of mass. In addition, the energy equation, or the first law of thermodynamics represents the conservation of energy.

The Navier-Stokes equations are the heart of fluid flow modeling. Solving them for a particular set of boundary conditions predicts the fluid velocity and its pressure in a given geometry. These boundary condition include the time domain, compressibility, the low and high Reynold’s number, and turbulence. The analysis of fluid flow can be conducted in either steady or unsteady condition depending on the physical incident. Steady condition refers to time independent whereas unsteady condition refers to time dependent. If the fluid flow is steady, it indicates that the motion of the fluid and parameters do not rely on the change in time. The term ?()/?t is equal to 0. As a result, the continuity and momentum equations have to be re-derived. The new continuity equation is shown below:

The new Navier-Stokes equation in x direction is shown below:

While the steady flow assumption negates the effect of some non-linear terms and provides a convenient solution, variation of density is an obstacle which keeps the equation in a complex formation 2.

The compressibility of particles is a significant issue due to the malleable structure of fluids. All types of fluid flow are compressible in a various range regarding molecular structure. However, most of them can be assumed to be incompressible in the cases where the density changes are negligible. As a result, the term ??/?t=0 can be eliminated regardless of whether the flow is steady or unsteady. The continuity equation changes once again:

The new Navier-Stokes equation in x direction is shown below:

Although incompressible flow assumption provides new and reasonable equations, the application of steady flow assumption enables us to ignore non-linear terms. This is where ?()/?t=0. Furthermore, the density of fluid in high speed cannot be accepted as incompressible since the changes in density are important. The Mach number is a dimensionless number which is used to investigate fluid flow in order to determine if it is incompressible or compressible. The equation for the Mach number is shown below:

The Ma represents the Mach number, V is the velocity of flow, and a is the speed of sound at sea level. It was found online that the value for the speed of sound at sea level was 340.29ms. When the Mach number is lower than 0.3, it is acceptable to assume incompressibility. On the other hand, the change in density cannot be neglected because the density must be considered as a significant parameter. For example, if the velocity of a car is higher than 100ms, the suitable approach to conduct credible numerical analysis is the compressible flow. Aside from the velocity, the effect of thermal properties on the density changes has to be taken into consideration for geophysical flows 3.

The Reynolds number is the ratio of inertial and viscous effects. It is also very effective on Navier-Stokes equations to shorten the mathematical model. While Re approaches ?, the viscous effects are accepted to be negligible, and as a result, the viscous terms in Navier-Stokes equations can be eliminated. The new simplified form of Navier-Stokes equation in the x-direction is shown below:

Although the viscous effects are important for fluids, the inviscid flow model partially provides a reliable model to predict real process for some certain cases. An example of this is high-speed external flow over bodies is a broadly used approximation where inviscid approach works. While Re?1, the inertial effects are assumed to be negligible and as a result, the related terms in Navier-Stokes equations can be eliminated. The new simplified form of Navier-Stokes equation in the x-direction is called either creeping flow or Stokes flow and is shown below:

Having tangible viscous effects, creeping flow is a suitable approach to investigate certain situations such as the flow of lava, swimming of microorganisms, flow of polymers, lubrication, etc.

The behavior of the fluid under dynamic conditions is a challenging obstacle that’s characterized under laminar and turbulent. The laminar flow is in an orderly fashion in which the motion of fluid can be predicted easily and precisely. Turbulent flow, on the other hand, has various hindrances, and as a result, it is hard to predict the fluid flow. Therefore, it shows a chaotic behavior. The Reynolds number predicts the behavior of fluid flow to be whether laminar or turbulent through several properties such as velocity, length, viscosity, and also the type of flow. Turbulent flow can be applied to the Navier-Stokes equations in order to conduct solutions to chaotic behavior of the fluid flow. Aside from the laminar, transport quantities of the turbulent flow is driven by instantaneous values. Instead of using the instantaneous values which cause non-linearity, a numerical solution with mean values can be used to provide an appropriate mathematical model which is named the Reynolds-averaged Navier-Stokes (RANS). These fluctuations can be negligible for most engineering cases and as a result, the new general form of The Reynolds-averaged Navier-Stokes (RANS) equation in the x-direction is shown below