Navier–Stokes equations

Precise mathematical model of Fluid flow in nature. Used to simulate both Newtonian and non-Newtonian fluids.

The momentum equation can be thought of as Newton’s second law but for fluids. It consider fluids as discrete “blobs” or particles that have mass and volume. Ideally, we take the limit as the number of particles goes to infinity, and the mass/volume of each particle goes to zero. It is given by

$$\frac{\partial \vec{u}}{\partial t}+\vec{u}\cdot \nabla \vec{u}+\frac{1}{\rho }\nabla p=\vec{F}+\nu \nabla \cdot \nabla \vec{u}$$

where

• $\vec{u}$ is the fluid velocity
  • It’s consistently treated as a vector here but nothing stops it from being a scalar. Usually represented as a vector field dependent on time $t$
• $\nabla \vec{u}$ is the Gradient of $\vec{u}$
• $\nu$ (nu, not the letter “v”) is coefficient of kinematic viscosity.
  • It measures how “thick” the fluid is, how resistant it is to motion, related to the Laplacian.
• $p$ is pressure, the “normal” contact force.
  • If pressure is equal in every direction, the net force is zero. We consider the pressure gradient force given by $\frac{1}{\rho }\nabla p$, which is in the equation.
• $\rho$ is fluid density
• $\partial t$ represents one time step
• $\vec{F}$ are the forces acting on the fluid.
  • Forces from pressure and viscosity are already present in the equation, and $\vec{F}$ is often simply just gravity: $\vec{F}=m\vec{g}$.

Incompressibility condition

When we can assume the fluid is incompressible, it is said to meet the incompressibility condition: $\nabla \cdot \vec{u}=0$. It implies $\nabla =0$, simplifying the momentum equation a little:

$$\begin{array}{rl}\frac{\partial \vec{u}}{\partial t}+\vec{u}+\frac{1}{\rho }\nabla p& =\vec{F}+\nu \nabla \cdot \nabla \vec{u}\\ \nabla \cdot \vec{u}& =0\end{array}$$

Inviscid Navier–Stokes (Euler Equations)

Building on incompressibility, when we don’t consider viscosity, the equations are further simplified. Often in Graphics these are used instead of the full version.

$$\begin{array}{r}\frac{\partial \vec{u}}{\partial t}+\frac{1}{\rho }\nabla p=\vec{F}\\ \nabla \cdot \vec{u}=0\end{array}$$

They are also called the Euler Equations.