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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.

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1 General
2 Electromagnetic theory
- 2.1 Derivation
- 2.2 Interpretation
3 Fluid dynamics
4 Quantum mechanics
5 Four-currents
6 See also

General

The general form for a continuity equation is

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

where $\scriptstyle\varphi$ is some quantity, ƒ 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 in and on 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 is derived from 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

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, a continuity equation is a mathematical statement for conservation of mass. Its differential form is

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

where $ρ$ is fluid density, t is time, and u is fluid velocity. If $ρ$ 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, it 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 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

ρ the charge density

j the conventional current density.

$\partial_a J^a = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0$