Mathematical form of Mass Conservation for continuum body:

Integral form

Differential form

(1)	$\begin{array}{l}\displaystyle \frac{d}{dt} \iiint_\Omega \rho \, dV = \frac{dm_\Omega}{dt}\end{array}$

(2)	$\begin{array}{l}\displaystyle \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \frac{d\rho (t, {\bf r})}{dt}\end{array}$

where

$\begin{array}{l}t\end{array}$	time	$\begin{array}{l}\rho(t, {\bf r})\end{array}$	continuum body spatial density distribution
$\begin{array}{l}{\bf r }\end{array}$	position vector	$\begin{array}{l}{\bf u}(t, {\bf r)\end{array}$	continuum body spatial velocity distribution
$\begin{array}{l}\Omega\end{array}$	space volume (could be finite or infinite)	$\begin{array}{l}\displaystyle \frac{dm_\Omega}{dt}\end{array}$	mass generation rate with the space volume $\begin{array}{l}\Omega\end{array}$
$\begin{array}{l}\nabla\end{array}$	gradient operator	$\begin{array}{l}\displaystyle \frac{d\rho(t, {\bf r})}{dt}\end{array}$	volume-specific mass generation rate at a given point in space $\begin{array}{l}{\bf r }\end{array}$

For the specific case of stationary process when density is not explicitly dependent on time:

(3)	$\begin{array}{l}\displaystyle \frac{\partial \rho}{\partial t} = 0 \rightarrow \nabla (\rho \, {\bf u}) = 0\end{array}$

For the specific case of finite number of mass generation locations the differential equation (2) takes form:

(4)	$\begin{array}{l}\displaystyle \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \sum_k \dot m_k(t) \cdot \delta({\bf r}-{\bf r}_k)\end{array}$

where

$\begin{array}{l}{\bf r }_k\end{array}$	position vector of the $\begin{array}{l}k\end{array}$ -th source/sink
$\begin{array}{l}\dot m_k(t)\end{array}$	mass generation rate at $\begin{array}{l}k\end{array}$ -th source/sink: $\begin{array}{l}\displaystyle \dot m_k(t) = \frac{dm_k}{dt}\end{array}$
$\begin{array}{l}\delta ( \bf r )\end{array}$	Dirac delta function

Alternatively it can be written as:

(5)	$\begin{array}{l}\displaystyle \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \sum_k \rho(t, {\bf r}) \cdot \dot q_k(t) \cdot \delta({\bf r}-{\bf r}_k)\end{array}$

wehre

$\begin{array}{l}q_k(t)\end{array}$

volumetric value of body mass generation rate at $\begin{array}{l}k\end{array}$ -th source/sink: $\begin{array}{l}\displaystyle q_k(t) = \frac{dV_k}{dt}\end{array}$

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Continuity equation @model

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