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

Continuity equations are the (stronger) local form of conservation laws. All the examples of continuity equations below express the same idea, which is roughly that: the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary. A conserved quantity cannot increase or decrease, it can only move from place to place.

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. 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

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

where

• ${\displaystyle \scriptstyle \varphi }$ is some quantity,
• ${\displaystyle \mathbf {f} }$ is a vector function describing the flux (flows) of ${\displaystyle \scriptstyle \varphi }$,
• ${\displaystyle \nabla \cdot }$ is divergence,
• and ${\displaystyle s}$ is a function describing the generation and removal of ${\displaystyle \scriptstyle \varphi }$. Terms that generate (${\displaystyle s>0}$) or remove (${\displaystyle s<0}$) ${\displaystyle \scriptstyle \varphi }$ are referred to as a "sources" and "sinks" respectively.

In the case that ${\displaystyle \scriptstyle \varphi }$ is a conserved quantity that cannot be created or destroyed (such as energy), the continuity equation is:

${\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {f} =0\,}$

because ${\displaystyle s=0}$.

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 J (in amperes per square meter) is equal to the negative rate of change of the charge density ρ (in coulombs per cubic meter),

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

Derivation from Maxwell's equations

One of Maxwell's equations, Ampère's law (with Maxwell's correction), states that

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

Taking the divergence of both sides results in

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

but the divergence of a curl is zero, so that

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

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

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

Substitute this into equation (1) to obtain

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

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:

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

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

${\displaystyle \nabla \cdot \mathbf {u} =0,}$

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. If P(x, t) is to be a probability density function, then

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

where j is probability flux.

Summary of Classical Continuity Equations

Continuity equations describe transport of conserved quantities though a local region of space. Note that these equations are not fundamental simply because of conservation; they can be derived.

Continuity Description	Nomenclature	General Equation	Simple Case
Hydrodynamics, Fluid Flow	${\displaystyle j_{\mathrm {m} }\,\!}$ = Mass current current at the cross-section ${\displaystyle \rho \,\!}$ = Volume mass density ${\displaystyle \mathbf {u} \,\!}$ = velocity field of fluid ${\displaystyle \mathbf {A} \,\!}$ = cross-section	${\displaystyle \nabla \cdot (\rho \mathbf {u} )+{\partial \rho \over \partial t}=0\,\!}$	${\displaystyle j_{\mathrm {m} }=\rho _{1}\mathbf {A} _{1}\cdot \mathbf {u} _{1}=\rho _{2}\mathbf {A} _{2}\cdot \mathbf {u} _{2}\,\!}$
Electromagnetism, Charge	${\displaystyle I\,\!}$ = Electric current at the cross-section ${\displaystyle \mathbf {J} \,\!}$ = Electric current density ${\displaystyle \rho \,\!}$ = Volume electric charge density ${\displaystyle \mathbf {u} \,\!}$ = velocity of charge carriers ${\displaystyle \mathbf {A} \,\!}$ = cross-section	${\displaystyle \nabla \cdot \mathbf {J} +{\partial \rho \over \partial t}=0\,\!}$	${\displaystyle I=\rho _{1}\mathbf {A} _{1}\cdot \mathbf {u} _{1}=\rho _{2}\mathbf {A} _{2}\cdot \mathbf {u} _{2}\,\!}$
Quantum Mechnics, Probability	${\displaystyle \mathbf {j} \,\!}$ = probablility current/flux ${\displaystyle P=P(x,t)\,\!}$ = probablility density function	${\displaystyle \nabla \cdot \mathbf {j} +{\frac {\partial P}{\partial t}}=0\,\!}$

Four-currents

Conservation of a current (not necessarily an electromagnetic current) is expressed compactly as the Lorentz invariant divergence of a four-current:

${\displaystyle J^{\mu }=\left(c\rho ,\mathbf {j} \right)}$

where

c is the speed of light

ρ the charge density

j the conventional current density.

μ labels the space-time dimension

so that since

${\displaystyle \partial _{\mu }J^{\mu }={\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} }$

then

${\displaystyle \partial _{\mu }J^{\mu }=0}$

implies that the current is conserved:

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0.}$

See also

• Conservation law
• Euler equations
• Groundwater energy balance
• Incompressible fluid
• Mass flow rate
• Noether's Theorem
• Probability density function
• Schrödinger equation

Notes

1. ^ Pedlosky, Joseph (1987). Geophysical fluid dynamics. Springer. pp. 10–13. ISBN 9780387963877.
2. Clancy, L.J.(1975), Aerodynamics, Section 3.3, Pitman Publishing Limited, London

• Equations of fluid dynamics
• Conservation equations
• Partial differential equations