## Applying Reynold's Theorem to the Continuity Equation

Reynold's Transport Theorem states that

$\frac{d}{dt} \int \int \int_{V_t} f dV =\int \int int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v} \cdot d \mathbf{S}$

where
$f$
is a function of
$\mathbf{r}, \:t$
defined on a volume
$V$
consisting of a sphere expanding at a constant speed
$v$
so that
$r=vt$

The Continuity Equation states
$\frac{\partial \rho}{\partial t} + \mathbf{\nabla} (\rho \mathbf{v}) =0$

Set
$f = \rho$
in Reynold's Theorem gives
$\frac{d}{dt} \int \int \int_{V_t} \rho dV =\int \int int_{V_t} \frac{\partial \rho}{\partial t} dV + \int \int_{S_t} \rho \mathbf{v} \cdot d \mathbf{S}$

Now apply the Divergence Theorem
$\int \int_S \mathbf{F} \cdot d \mathbf{S} = \int \int \int_V \mathbf{\nabla} \cdot \mathbf{F} dV$

with
$\mathbf{F} = \rho \mathbf{v}$
to get
$\frac{d}{dt} \int \int \int_{V_t} \rho dV =\int \int \int_{V_t} \frac{\partial \rho}{\partial t} dV +\int \int \int_{V_t} \mathbf{\nabla} \cdot (\rho \mathbf{v}) dV$

Hence
$\frac{d}{dt} \int \int \int_{V_t} \rho dV =\int \int \int_{V_t}( \frac{\partial \rho}{\partial t} + \mathbf{\nabla} \cdot (\rho \mathbf{v})) dV=0$

Then the left hand side is zero and
$\int \int \int_{V_t} \rho dV =constant$