PX285 - I4 - core equations

the continuity equation:

\frac{\partial ρ}{\partial t} + \vec{\nabla} \cdot (ρ \vec{u}) = 0

this reduces to $\vec{\nabla} \cdot \vec{u} = 0$ for incompressible fluids
the momentum equation:

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

for incompressible fluids:

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

$(\vec{u} \cdot \vec{\nabla}) \vec{u}$ is called the inertial term
the pressure equation, for adiabatic processes:

\frac{d}{d t} (\frac{P}{ρ^{γ}}) = 0

simplifications

if $μ, λ = 0$ , it is an inviscid fluid
if $\frac{\partial}{\partial t} = 0$ , it is a steady flow
if $\vec{\nabla} \cdot \vec{u} = 0$ , it is an incompressible flow, or $ρ =$ constant