Navier-Stokes Equation

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

Continuity Equation

Conservation of mass. $\nabla \cdot \vec{v}=0$

where $\vec{v}=\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$ is the velocity of the fluid.

This equation essentially takes the derivative of each component of $\vec{v}$,

$\nabla \cdot \vec{v}=\frac{du}{dx}+\frac{dv}{dy}+\frac{dw}{dz}$

Momentum Equation

Conversation of momentum. $\rho \frac{d\vec{v}}{dt}=-\nabla p+\mu {\nabla }^2\vec{v}+\rho \vec{F}$

$-\nabla p+\mu {\nabla }^2\vec{v}$ are the internal forces of the equation

$\nabla p$ is the change in pressure that causes a fluid to move. $\mu {\nabla }^2\vec{v}$ is the viscosity.

$\rho \vec{F}$ are the external forces of the equation

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

$\vec{F}$ is the sum of external forces (gravitional force, electromagnetism)

1. Solution must exist
2. Solution must be unique
3. Solution must be smooth

We don’t even know if a solution doesn’t always exist. Based on very standard of physics.