Navier-Stokes equations

It is a PDE that indeed reflects Newton second law $m \cdot a = F$ . It shows how a fluid varies with time:

ρ (\frac{\partial u}{\partial t} + u \cdot \nabla u) = - \nabla P + μ \nabla^{2} u + ρ F

It is also required $\nabla \cdot u = 0$ (divergence), which meas that mass is conserved.

Often, it is used the operator

\frac{D ◻}{D t} := \frac{\partial ◻}{\partial t} + u \cdot \nabla ◻

Expanded, Navier-Stokes shows up as

\begin{aligned} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} & = - \frac{\partial p}{\partial x} + μ (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) \\ \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} & = - \frac{\partial p}{\partial y} + μ (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}}) \\ \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} & = - \frac{\partial p}{\partial z} + μ (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} w}{\partial z^{2}}) \\ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} & = 0 \end{aligned}

if there is no external force.

The case $μ = 0$ (no viscosity) is called Euler equation.