Pustam Raut
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$$\color{blue}{\textbf{A lifelong philomath, STEM enthusiast, and hodophile!}}$$ ⚛️🚀🌌✈️🧳

The Navier–Stokes equations, which describe fluid motion, are among my favourite equations (PDEs) in physics.

$$\color{green}{\boxed{\frac {\partial}{\partial t} (\rho\,\mathbf{u}) + \nabla \cdot (\rho\,\mathbf{u} \otimes \mathbf{u}) = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{f}}}$$ $$\color{green}{\texttt{Navier–Stokes equations (conservation form)}}$$

$$\color{red}{\boxed{\rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \underbrace{\overbrace{- \nabla p}^\text{internal} + \overbrace{\nabla \cdot\left\{ \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] \right\}}^\text{Cauchy stress tensor term} + \overbrace{\nabla[\zeta (\nabla\cdot\mathbf{u})]}^\text{bulk viscocity term} + \overbrace{\rho\mathbf{f}}^\text{external}}_{\mathbf R}}}$$ $$\color{red}{\texttt{convective form}}$$

$$\color{blue}{\boxed{\rho \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \overbrace{\rho \left(\underbrace{ \frac{\partial \mathbf{u}}{\partial t}}_\text{variation} + \underbrace{(\mathbf{u} \cdot \nabla) \mathbf{u}}_\text{convection} \right)}^\text{inertia} = \mathbf R -\overbrace{ \rho \left[\underbrace{2\mathbf\Omega\times\mathbf u}_\text{Coriolis} + \underbrace{\mathbf\Omega\times(\mathbf\Omega\times\mathbf r)}_\text{centrifugal} + \underbrace{\frac{\mathrm{d} \mathbf \Omega}{\mathrm{d} t}\times\mathbf r}_\text{Euler/angular}+ \underbrace{\frac{\mathrm{d} \mathbf U}{\mathrm{d} t}}_\text{linear}\right]}^{\text{pseudo/inertial/fictitious forces}}}}$$ $$\color{blue}{\texttt{non-inertial frame}}$$

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