Conservation equation

In various Physique disciplines, when a quantity is supposed to be preserved (typically the Masse, the Charge or the baryon Nombre) in spite of its displacement, one can establish an equation connecting the variation of this quantity in time to its variation in space, called conservation equation of the size.

Expression

One can establish it in two forms, integral and local.

Form integral

That is to say a size ϕ presumedly preserved, moving according to a vector v . Then for any volume V , of closed surface Σ , one a:

$\backslash \; oint\_\; \{\backslash \; Sigma\}\; \backslash \; phi\; \backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; \{dS\}\; +\; \backslash \; int\_V\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; phi\}\; \{\backslash \; partial\; T\}\; D\; \backslash \; tau\; =\; 0$

In other words, the Flux of ϕ through surface implies a variation of ϕ in volume.

Proof

Volume V contains, at one moment T, the quantity:

$\backslash \; Phi\; =\; \backslash \; iiint\_V\; \backslash \; phi\; D\; \backslash \; tau$

This quantity varies, during dt , because of the contributions or the losses outsides:

$d\; \backslash \; Phi\; =\; D\; \backslash \; Phi\_\; \{E\}\; -\; D\; \backslash \; Phi\_\; \{S\}$

It entered and left, by faiant an algebraic assessment:

$d\; \backslash \; Phi\_e\; -\; D\; \backslash \; Phi\_s\; =\; \backslash \; oint\_\; \{\backslash \; Sigma\}\; \backslash \; phi\; \backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; \{dS\}\; dt$

Lastly, by differentiating Φ and while identifying, one obtains the formula given well.

Local form

That is to say a size ϕ presumedly preserved, moving according to a vector v . Then for any element of volume $d^3\; \backslash \; tau$, one a:

$\backslash \; mathrm\; \{div\}\; \backslash \; phi\; \backslash \; vec\; \{v\}\; +\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; phi\}\; \{\backslash \; partial\; T\}\; =\; 0$

Proof

Starting from the integral form, by using the theorem of Green-Ostrogradski, for an infinitesimal volume d³τ, one finds the formula given well.

Source term

Sometimes, one can establish a “conservation equation” even if there exist sources which vary the size. The conservation equation is not cancelled then more, and its value depends on the production of the sources, called source term. By noting p algebraic production in an infinitesimal volume:

$\backslash \; oint\_\; \{\backslash \; Sigma\}\; \backslash \; phi\; \backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; \{dS\}\; +\; \backslash \; int\_V\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; phi\}\; \{\backslash \; partial\; T\}\; D\; \backslash \; tau\; =\; \backslash \; int\_V\; p\; D\; \backslash \; tau$

$\backslash \; mathrm\; \{div\}\; \backslash \; phi\; \backslash \; vec\; \{v\}\; +\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; phi\}\; \{\backslash \; partial\; T\}\; =\; p$

See too

Related articles

• Mechanical traditional;
• Electromagnetism;

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