PX285 - I1 - conservation of mass

considering a small volume, $V$ , of a fluid, bounded by a surface, $S$ , with a total area, $A$
taking a small element of the surface, with a normal vector, $\vec{n}$ , directed outward and through, $d \vec{S}$
the mass flow per second through $d S$ is $ρ \vec{u} \cdot d \vec{S}$ , where $d \vec{S} = \vec{n} d S$
the total mass in/outflow through $S :$

\subset \supset \iint_{S} ρ \vec{u} \cdot d \vec{S}

the mass of the fluid in the volume, $V$

M = ∭_{V} ρ d V

the mass is conserved, so the rate of mass change is equal to the rate of the mass in/outflow

\frac{\partial}{\partial t} ∭_{V} ρ d V = - \subset \supset \iint_{S} ρ \vec{u} \cdot d \vec{S}

using the divergence theorem:

\subset \supset \iint_{S} \vec{A} \cdot d \vec{S} = ∭_{V} (\vec{\nabla} \cdot \vec{A}) d V

\begin{matrix} ∭_{V} \frac{\partial ρ}{\partial t} d V + ∭_{V} \vec{\nabla} \cdot (ρ \vec{u}) d V = 0 \\ \frac{\partial ρ}{\partial t} + \vec{\nabla} \cdot (ρ \vec{u}) = 0 \end{matrix}

this is called the continuity equation
limiting case: if $ρ =$ constant, the fluid is incompressible

0 + ρ \vec{\nabla} \cdot \vec{u} = 0 ⟹ \vec{\nabla} \cdot \vec{u} = 0