Navier Stokes Equations

The Navier-Stokes equations are a set of Partial Differential Equations that describe almost all fluid dynamics. They describe how the velocity, pressure, temperature, and density of a moving fluid are related.

Incompressible Navier Stokes Equations

For incompressible fluids; if one part of the fluid is displaced, the opposite side is also displaced equally and instantly. Incompressibility means constant density $\rho$.

$$\begin{array}{c}\frac{\partial }{\partial t}\vec{u}=-(\vec{u}\cdot \nabla )\vec{u}-\frac{1}{\rho }\nabla p+\nu {\nabla }^2\vec{u}+{\vec{F}}_{\text{external}}\\ \nabla \cdot \vec{u}=0\end{array}$$

• where $\vec{u}$ is the velocity vector
• where $t$ is time
• where $\rho$ is the density of the fluid
• where $p$ is the pressure of the fluid
• where $\nu$ is the kinematic viscosity
• where ${\vec{F}}_{\text{external}}$ is the external force

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• The first equation is the moment equation.
  • The Advection term
    • $-(\vec{u}\cdot \nabla )\vec{u}$
    • Says that the fluid moves on its own, and moves faster in regions of higher velocity. Shocker.
  • The Pressure term
    • $-\frac{1}{\rho }\nabla p$
    • Says that the fluid flow is along a pressure gradient.
    • This term causes divergence, which is illegal is zero-divergence land, and can be omitted.
    • We can correct for this omission through Projection, which uses Helmholtz-Hodge Decomposition to isolate the zero-divergence component for a given point in the field. This involves solving the Poisson Equation and then subtracting the Gradient.
  • The Diffusion term
    • $\nu {\nabla }^2\vec{u}$
    • Says that the fluid flow tends to diffuse along the velocity gradient, and that the more the fluid diffuses, the smooth the velocity field gets.
  • The External Force term
    • ${\vec{F}}_{\text{external}}$
    • Says “Hello, I am external force”
• The second equation is the continuity equation, which enforces incompressibility.
  • At any point, the velocity flow into that point and out of that point must net to zero, i.e. the Divergence is zero.
• These equations generally cannot be solved analytically, and instead require Numerical Methods