Continuity Equation

A continuity equation in physics is an equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

Continuity equations are a stronger, local form of conservation laws. For example, it is true that "the total energy in the universe is conserved". But this statement does not immediately rule out the possibility that a lot of energy could disappear from Earth while simultaneously appearing in another galaxy. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.

Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities which are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of living humans; it has a "source term" to account for people giving birth, and a "sink term" to account for people dying.

Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.

Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier-Stokes equations.