A continuity equation in physics is an equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations. Continuity equations are a stronger, local form of conservation laws. For example, it is true that "the total energy in the universe is conserved". But this statement does not immediately rule out the possibility that energy could disappear from Earth while simultaneously appearing in another galaxy. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities which are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of living humans; it has a "source term" to account for people giving birth, and a "sink term" to account for people dying. 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. Continuity equations underlie more specific transport equations such as the convection--diffusion equation, Boltzmann transport equation, and Navier-Stokes equations. General equation Preliminary description As stated above, the idea behind the continuity equation is the flow of some property, such as mass, energy, electric charge, momentum, and even probability, through surfaces from one region of space to another. The surfaces, in general, may either be open or closed, real or imaginary, and have an arbitrary shape, but are fixed for the calculation (i.e. not time-varying, which is appropriate since this complicates the maths for no advantage). Let this property be represented by just one scalar variable, q, and let the volume density of this property (the amount of q per unit volume V) be ρ, and the union of all surfaces be denoted by S. Mathematically, ρ is a ratio of two infinitesimal quantities: which has the dimension −3 (where L is length). There are different ways to conceive the continuity equation: either the flow of particles carrying the quantity q, described by a velocity field v, which is also equivalent to a flux j of q (a vector function describing the flow per unit area per unit time of q), or in the cases where a velocity field is not useful or applicable, the flux j of the quantity q only (no association with velocity). In each of these cases, the transfer of q occurs as it passes through two surfaces, the first S1 and the second S2. The flux j should represent some flow or transport, which has dimensions −1−2. In cases where particles/carriers of quantity q are moving with velocity v, such as particles of mass in a fluid or charge carriers in a conductor, j can be related to v by: This relation is only true in situations where there are particles moving and carrying q - it can't always be applied. To illustrate this: if j is electric current density (electric current per unit area) and ρ is the charge density (charge per unit volume), then the velocity of the charge carriers is v. However - if j is heat flux density (heat energy per unit time per unit area), then even if we let ρ be the heat energy density (heat energy per unit volume) it does not imply the "velocity of heat" is v (this makes no sense, and is not practically applicable). In the latter case only j (with ρ) may be used in the continuity equation. Elementary vector form Consider the case when the Image source and copyright details: http://wikipedia.org/wiki/File:BernoullisLawDerivationDiagram.svg Original article available at http://en.wikipedia.org/wiki/Continuity_equation

• published: 23 May 2013
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