Continuity equation

A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.

All the examples of continuity equations below express the same idea. Continuity equations are the (stronger) local form of conservation laws.

Any continuity equation has a "differential form" (in terms of the divergence operator) and an "integral form" (in terms of a flux integral). In this article, only the "differential form" versions will be given; see the article divergence theorem for how to express any of these laws in "integral form".

General
The general form for a continuity equation is


 * $$\frac{\partial \varphi}{\partial t} + \nabla \cdot f = s$$

where $$\scriptstyle\varphi$$ is some quantity, &fnof; is a function describing the flux of $$\scriptstyle\varphi$$, and s describes the generation (or removal) of $$\scriptstyle\varphi$$. This equation may be derived by considering the fluxes into an infinitesimal box. This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation.

Electromagnetic theory
In electromagnetic theory, the continuity equation can either be regarded as an empirical law expressing (local) charge conservation, or can be derived as a consequence of two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density,


 * $$ \nabla \cdot \mathbf{J} = - {\partial \rho \over \partial t}. $$

Derivation from Maxwell's equations
One of Maxwell's equations, Ampère's law, states that


 * $$ \nabla \times \mathbf{H} = \mathbf{J} + {\partial \mathbf{D} \over \partial t}. $$

Taking the divergence of both sides results in


 * $$ \nabla \cdot \nabla \times \mathbf{H} = \nabla \cdot \mathbf{J} + {\partial \nabla \cdot \mathbf{D} \over \partial t}, $$

but the divergence of a curl is zero, so that


 * $$ \nabla \cdot \mathbf{J} + {\partial \nabla \cdot \mathbf{D} \over \partial t} = 0. \qquad \qquad (1) $$

Another one of Maxwell's equations, Gauss's law, states that


 * $$ \nabla \cdot \mathbf{D} = \rho.\, $$

Substitute this into equation (1) to obtain


 * $$ \nabla \cdot \mathbf{J} + {\partial \rho \over \partial t} = 0,\,$$

which is the continuity equation.

Interpretation
Current density is the movement of charge density. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.

Fluid dynamics
In fluid dynamics, the continuity equation is a mathematical statement that, in any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. In fluid dynamics, the continuity equation is analogous to Kirchhoff's Current Law in electric circuits.

The differential form of the continuity equation is:


 * $$ {\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$

where $$ \rho $$ is fluid density, t is time, and u is fluid velocity. If density ($$\rho$$) is a constant, as in the case of incompressible flow, the mass continuity equation simplifies to a volume continuity equation:


 * $$\nabla \cdot \mathbf{u} = 0$$

which means that the divergence of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.

Further, the Navier-Stokes equations form a vector continuity equation describing the conservation of linear momentum.

Quantum mechanics
In quantum mechanics, the conservation of probability also yields a continuity equation. Let P(x, t) be a probability density function and write


 * $$ \nabla \cdot \mathbf{j} = -{ \partial \over \partial t} P(x,t) $$

where J is probability flux.

Four-currents
Conservation of a current (not necessarily an electromagnetic current) is expressed compactly as the Lorentz invariant divergence of a four-current:
 * $$J^a = \left(c \rho, \mathbf{j} \right)$$

where
 * c is the speed of light
 * &rho; the charge density
 * j the conventional current density.
 * a labels the space-time dimension

so that since
 * $$\partial_a J^a = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} $$

then
 * $$\partial_a J^a = 0$$

implies that the current is conserved:
 * $$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0$$