Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are each conserved under its own appropriate condition, 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 equation

Differential form

The differential form for a general continuity equation is

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

where

• $\scriptstyle \varphi$ is some quantity per unit volume,
• $\mathbf {f}$ is a vector function describing the flux (flow per unit area and unit time) of $\scriptstyle \varphi$,
• $\nabla \cdot$ is divergence,
• t is time,
• and $s$ is the generation (negative in the case of removal) per unit volume and unit time of $\scriptstyle \varphi$. Terms that generate ($s>0$) or remove ($s<0$) $\scriptstyle \varphi$ are referred to as a "sources" and "sinks" respectively.

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

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

because $s=0$.

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.

Note the flux f should represent some flow or transport, which has dimensions [quantity][T]^-1[L]^-2, where [quantity]/[L]³ is the dimension of $\varphi$.

Other equations in physics, such as Gauss's law of the electric field and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually called by the term "continuity equation", because f in those cases does not represent the flow of a real physical quantity.

Integral form

In the integral form of the continuity equation, S is any imaginary closed surface that fully encloses a volume V, like any of the surfaces on the left. S can not be a surface with boundaries that do not enclose a volume, like those on the right. (Surfaces are blue, boundaries are red.)

By the divergence theorem, the continuity equation can be rewritten in an equivalent way, called the "integral form":

${\frac {{\rm {d}}q}{{\rm {d}}t}}+\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset (\mathbf {f} \cdot \mathbf {n} )\,{\rm {d}}S=\Sigma$

where

• S is a closed surface that encloses a volume V. S is arbitrary but fixed (unchanging in time) for the calculation;
• $q\equiv \iiint _{V}\varphi \,{\rm {d}}V$ is the total amount of $\phi$ in the volume V (for example, if $\phi$ is the mass density, then q is the total mass in the volume V);
• $\Sigma \equiv \iiint _{V}s\,{\rm {d}}V$ is the total generation (negative in the case of removal) per unit time by the sources and sinks in the volume V;
• $\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset \,{\rm {d}}S$ denotes a surface integral;
• $\mathbf {n}$ is the outward-pointing unit normal to the surface S;

In a simple example, V could be a building, and q could be the number of people in the building. The surface S would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a "source" where $s>0$), and decreases when someone in the building dies (a "sink" where $s<0$).

Electromagnetic theory

Main article: Charge conservation

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),

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

Derivation from Maxwell's equations

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

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

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 is the movement of charge. 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 states 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 the flow velocity vector field. 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.

Energy

By conservation of energy, which can only be transferred and not created/destroyed leads to a continuity equation, an alternative mathematical statement of energy conservation to the thermodynamic laws. Letting u = local energy density (energy per unit volume), q = energy flux (transfer of energy per unit cross-sectional area per unit time) as a vector, then

$\nabla \cdot \mathbf {q} +{\frac {\partial u}{\partial t}}=0$

By Fourier's law for a uniformly conducting medium,

$\mathbf {q} =-k\nabla T$

where k is the thermal conductivity (not Boltzmann constant), this can also be written as:

$\nabla \cdot \left(-k\nabla T\right)+{\frac {\partial u}{\partial t}}=0$

${\frac {\partial u}{\partial t}}=k\nabla ^{2}T.$

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

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

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:

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

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

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

then

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

implies that the current is conserved:

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

See also

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