PX285 - I2a - conservation of momentum

considering a lagrangian fluid element of volume, $V$ , that moves at the fluid velocity, $\vec{u}$
- lagrangian: its mass is constant, ie: $ρ d V =$ constant
the rate of change of momentum:

\frac{d}{d t} (ρ d V \vec{u}) = ρ d V \frac{d \vec{u}}{d t}

where, $\frac{d \vec{u}}{d t} = \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla) \vec{u}$
PX285 - I2b - advective derivative

momentum conservation is newton's second law:

\vec{F} = m \vec{a} = ρ d V \frac{D \vec{u}}{D t}

$d V$ can be dropped to calculate the force per unit volume/force density, that can be called force in shorthand
the pressure force is $- \vec{\nabla} P$
the force due to gravity is $ρ \vec{g}$
combining the above, and the viscous force:

⟹ ρ \frac{d \vec{u}}{d t} = \sum \frac{\vec{F}}{d V} = - \vec{\nabla} P + {\vec{F}}_{v} + ρ \vec{g} + \dots

the viscous force:

{\vec{F}}_{v} = μ \nabla^{2} \vec{u} + λ \vec{\nabla} (\vec{\nabla} \cdot \vec{u})

where, $μ$ and $λ$ arise from the shear and volume viscosities, respectively

often, $λ \approx μ / 3$
hence, the navier-stokes equation for incompressible fluids $(\vec{\nabla} \cdot \vec{u} = 0) :$

ρ \frac{\partial \vec{u}}{\partial t} + ρ (\vec{u} \cdot \vec{\nabla}) \vec{u} = - \vec{\nabla} P + μ \nabla^{2} \vec{u} - ρ g \hat{k}

note:
- the full continuity equation for $ρ$ involves $\vec{u}$
- the NS for $\vec{u}$ involves $P$ , so additional equation for $P$ is needed, which comes from the conservation of energy