Continuity equation - Citizendia

A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product

All the examples of continuity equations below express the same idea. Continuity equations are the (stronger) local form of conservation laws. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves

Any continuity equation has a "differential form" (in terms of the divergence operator) and an "integral form" (in terms of a flux integral). In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks 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". In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

1 General
2 Electromagnetic theory
- 2.1 Derivation from Maxwell's equations
- 2.2 Interpretation
3 Fluid dynamics
4 Quantum mechanics
5 Four-currents
6 See also
7 References

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 $\scriptstyle\varphi$ . The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks 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. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such This equation also generalizes the advection equation. Advection, in mechanical and chemical engineering is a transport mechanism of a substance or a conserved property with a moving Fluid.

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. Charge conservation is the principle that Electric charge can neither be created nor destroyed In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Charge conservation is the principle that Electric charge can neither be created nor destroyed In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric 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. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the Current density is a measure of the Density of flow of a conserved charge. The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

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. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Steady state is a more general situation than Dynamic equilibrium. ^[1] In fluid dynamics, the continuity equation is analogous to Kirchhoff's Current Law in electric circuits. For other laws named after Gustav Kirchhoff, see Kirchhoff's laws.

The differential form of the continuity equation 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 density ( $ρ$ ) 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. In Fluid mechanics or more generally Continuum mechanics, an incompressible flow is Solid or Fluid flow in which the Divergence of 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. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such

Quantum mechanics

In quantum mechanics, the conservation of probability also yields a continuity equation. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In Quantum mechanics, the probability current (sometimes called probability flux) is a concept describing the flow of Probability density.

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. In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In special and General relativity, the four-current is the Lorentz covariant Four-vector that replaces the Electromagnetic Current The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. Current density is a measure of the Density of flow of a conserved charge.

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:

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

References

^ Clancy, L. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves The groundwater energy balance is the energy balance of a Groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body Fluid mechanics is the study of how Fluids move and the Forces on them In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability J. (1975), Aerodynamics, Section 3. 3, Pitman Publishing Limited, London

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