This expression represents the principle of conservation of linear momentum applied to a general fluid:

The law of conservation of mass is written:

In these equations, ρ represents the density "Density (physics)"), u (i = 1,2,3) the Cartesian components of velocity, F the acceleration field created by forces applied on the body, such as gravity, P the fluid pressure, and μ the dynamic viscosity.

where Δ = e is the divergence "Divergence (mathematical)") of the fluid and δij the Kronecker delta. D/Dt is the total derivative or material time derivative following the fluid:

The nonlinearity of the equations is due precisely to the term related to the total derivative. When μ is uniform over the entire fluid, the fluid equations are simplified as follows:.

Or, in vector form:.

For fluids of zero viscosity, that is, when μ = 0, the resulting equations are called Euler equations "Euler equations (fluids)") which are used in the study of compressible fluids and shock waves:

On the other hand, if a viscous but incompressible fluid is considered, then the density ρ can be considered constant (as in a liquid) and the equations turn out to be:

and the continuity equation takes the following form:

The Navier–Stokes equations are widely used in video games to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games"[1]​ by Jos Stam, which elaborates on one of the methods proposed in Stam's earlier and most famous paper "Stable Fluids"[2]​ from 1999. Stam proposes a stable fluid simulation using a Navier solution method – Stokes of 1968, together with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992.

More recent implementations based on this work run on the graphics processing units (GPU) of gaming systems instead of the central processing unit (CPU) and achieve a much higher degree of performance.[3]​[4]​

Many improvements have been proposed to Stam's original work, which inherently suffers from high numerical dissipation in both speed and mass.

An introduction to interactive fluid simulation can be found in the 2007 Association for Computing Machinery SIGGRAPH course, Fluid Simulation for Computer Animation.[5]​.

This expression represents the principle of conservation of linear momentum applied to a general fluid:

The law of conservation of mass is written:

In these equations, ρ represents the density "Density (physics)"), u (i = 1,2,3) the Cartesian components of velocity, F the acceleration field created by forces applied on the body, such as gravity, P the fluid pressure, and μ the dynamic viscosity.

where Δ = e is the divergence "Divergence (mathematical)") of the fluid and δij the Kronecker delta. D/Dt is the total derivative or material time derivative following the fluid:

The nonlinearity of the equations is due precisely to the term related to the total derivative. When μ is uniform over the entire fluid, the fluid equations are simplified as follows:.

Or, in vector form:.

For fluids of zero viscosity, that is, when μ = 0, the resulting equations are called Euler equations "Euler equations (fluids)") which are used in the study of compressible fluids and shock waves:

On the other hand, if a viscous but incompressible fluid is considered, then the density ρ can be considered constant (as in a liquid) and the equations turn out to be:

and the continuity equation takes the following form:

The Navier–Stokes equations are widely used in video games to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games"[1]​ by Jos Stam, which elaborates on one of the methods proposed in Stam's earlier and most famous paper "Stable Fluids"[2]​ from 1999. Stam proposes a stable fluid simulation using a Navier solution method – Stokes of 1968, together with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992.

More recent implementations based on this work run on the graphics processing units (GPU) of gaming systems instead of the central processing unit (CPU) and achieve a much higher degree of performance.[3]​[4]​

Many improvements have been proposed to Stam's original work, which inherently suffers from high numerical dissipation in both speed and mass.

An introduction to interactive fluid simulation can be found in the 2007 Association for Computing Machinery SIGGRAPH course, Fluid Simulation for Computer Animation.[5]​.