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Continuity equation

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 scriptstylevarphi is some quantity, ƒ is a function describing the flux of scriptstylevarphi, and s describes the generation (or removal) rate of scriptstylevarphi. 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(xt) 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
ρ 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

See also

References

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