continuity equation

The continuity equation is a mathematical expression of mass conservation, stating that the rate of change of mass in a given region is equal to the net mass flux entering or leaving that region.

Differential form

$$\frac{\partial }{\partial t}\rho +\nabla \cdot (\rho \mathbf{u})=s$$

where
$\rho (\mathbf{x},t)$	density at a given point of the continuum, measured in kilograms per meter cubed $\frac{\text{kg}}{{\text{m}}^3}$
$\mathbf{x}$	position vector in three-dimensional space, typically expressed as $\mathbf{x}=(x,y,z)$ and measured in meters $\text{m}$
$t$	time, measured in seconds $\text{s}$
$\mathbf{u}(\mathbf{x})$	vector field describing the movement of some quantity $q$, measured in meters per second $\frac{\text{m}}{\text{s}}$,
$\nabla \cdot$	divergence operator which when applied to $\rho \mathbf{u}$ is: $\nabla \cdot (\rho \mathbf{u})=\frac{\partial }{\partial x}(\rho u_x)+\frac{\partial }{\partial y}(\rho u_y)+\frac{\partial }{\partial z}(\rho u_z),$ measured in kilograms per cubic meter per second $\frac{\text{kg}}{{\text{m}}^3\cdot \text{s}}$.
$s$	generation of the quantity $q$ described by $\mathbf{u}(\mathbf{x},t)$ per unit volume per unit time; with terms that generate $q$ (i.e. $\sigma >0$) or remove $q$ (i.e. $\sigma <0$) being referred to as sources and sinks, respectively,