Integral form

Differential form

\frac{d}{dt} \iiint_\Omega \rho \, dV = \frac{dm_\Omega}{dt}

\frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) =   \frac{d\rho (t, {\bf r})}{dt}

where

	time		continuum body spatial density distribution
	position vector		continuum body spatial velocity distribution
	space volume (could be finite or infinite)		mass generation rate with the space volume
	gradient operator		volume-specific mass generation rate at a given point in space

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

\frac{\partial \rho}{\partial t} = 0 \rightarrow  \nabla (\rho \, {\bf u}) = 0

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

\frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) =   \sum_k \dot m_k(t) \cdot \delta({\bf r}-{\bf r}_k)

where

	position vector of the -th source/sink
	mass generation rate at -th source/sink:
	Dirac delta function

Alternatively it can be written as:

\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)

wehre

volumetric value of body mass generation rate at -th source/sink:

See also