Continuity Equation
Continuity equation - Wikipedia, the free encyclopedia
Any continuity equation has a "differential form" (in terms of the divergence ... In electromagnetic theory, the continuity equation can either be regarded as an ...
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Continuity Equation -- from Eric Weisstein's World of Physics
A continuity equation expresses a conservation law by equating a net flux over a ... Continuity equations often can be expressed in either integral or ...
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Continuity Equation
Then if J is to exactly satisfy a discrete continuity equation we must define to be ... The derivation of the continuity equation in the continuum case relies on no ...
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Continuity equation
The continuity equation describes a basic concept, namely that a change in ... The diffusion equation ... The diffusion equation will be used to calculate the ...
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Continuity equation@Everything2.com
The fluid continuity equation is based on the Law of Conservation of Mass and ... The continuity equation is used in physics classes and by aeronautical and ...
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Continuity Equation
The general continuity equation accounts for the conservation of any quantity Q ... (2,3) are known as The Continuity Equation for quantity Q. They apply for ...
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PlanetMath: continuity equation
Integral form of continuity equation ... Continuity equation for a control volume. ... This is version 5 of continuity equation, born on 2006-05-31, modified ...
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continuity equation: Definition from Answers.com
continuity equation ( ?känt?n?ü?d? i?kw?zh?n ) ( physics ) An equation obeyed by any conserved, indestructible quantity such as mass, electric charge,
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Continuity Equation - Fluid Flow Hydraulic and Pneumatic, Engineers Edge
The continuity equation is simply a mathematical expression of the principle of ... Example: Continuity Equation - Centrifugal Pump The inlet diameter of the ...
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The Continuity Equation
The presence of in the continuity equation incorporates the physics of the flow ... With the continuity equation determined, all that remains for a complete ...
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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.
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 an equation of conservation of mass. Its differential form is
{\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0
where \rho is density, t is time, and u is fluid velocity. If \rho is a constant, as in the case of incompressible flow, this mass continuity equation simplifies into a volume continuity equation:
\nabla \cdot \mathbf{u} = 0
Further, the Navier-Stokes equations form a vector continuity equation describing the conservation of 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
See also
- Conservation law
- Euler equations
- Incompressible fluid
- Schrödinger equation
- Probability density function
Continuity equation - Wikipedia, the free encyclopedia
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 ...
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Continuity Equation -- from Eric Weisstein's World of Physics
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