# Navier-Stokes Equation

This tells you everything about how a fluid behaves. https://www.youtube.com/watch?v=ERBVFcutl3M

### Continuity Equation

Conservation of mass. $ββv=0$

where $v=βuvwββ$ is the velocity of the fluid.

This equation essentially takes the derivative of each component of $v$,

$ββv=dxduβ+dydvβ+dzdwβ$

### Momentum Equation

Conversation of momentum. $Οdtdvβ=ββp+ΞΌβ_{2}v+ΟF$

$ββp+ΞΌβ_{2}v$ are the internal forces of the equation

$βp$ is the change in pressure that causes a fluid to move. $ΞΌβ_{2}v$ is the viscosity.

$ΟF$ are the external forces of the equation

total derivative = pressure gradient + body force term + diffusion term

$F$ is the sum of external forces (gravitional force, electromagnetism)

- Solution must exist
- Solution must be unique
- Solution must be smooth

We donβt even know if a solution doesnβt always exist. Based on very standard of physics.