It is a PDE that indeed reflects Newton second law $m\cdot a=F$. It shows how a fluid varies with time:
$$ \rho \left(\frac{\partial u}{\partial t} +u\cdot \nabla u\right)=-\nabla P+\mu \nabla^2 u+\rho F $$It is also required $\nabla \cdot u=0$ (divergence), which meas that mass is conserved.
Often, it is used the operator
$$ \frac{D \square}{Dt}:=\frac{\partial \square}{\partial t}+u\cdot \nabla \square $$named material derivative in the context of Continuum Mechanics.
Expanded, Navier-Stokes shows up as
$$ \begin{aligned} \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z} &=-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}\right) \\ \frac{\partial v}{\partial t}+u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z} &=-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}+\frac{\partial^{2} v}{\partial z^{2}}\right) \\ \frac{\partial w}{\partial t}+u \frac{\partial w}{\partial x}+v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z} &=-\frac{\partial p}{\partial z}+\mu\left(\frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial y^{2}}+\frac{\partial^{2} w}{\partial z^{2}}\right) \\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} &=0 \end{aligned} $$if there is no external force.
The case $\mu=0$ (no viscosity) is called Euler equation.
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Author of the notes: Antonio J. Pan-Collantes
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