Continuity and Gauges

Continuity

The integral form of the continuity equation states that:

• The amount of q in a region increases when additional q flows inward through the surface of the region, and decreases when it flows outward;
• The amount of q in a region increases when new q is created inside the region, and decreases when q is destroyed;
• Apart from these two processes, there is no other way for the amount of q in a region to change.

Mathematically, the integral form of the continuity equation expressing the rate of increase of q within a volume V is:

${\frac {dq}{dt}}+$ | intsubscpt = $\scriptstyle S$ | integrand = $\mathbf {j} \cdot d\mathbf {S} =\Sigma$

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

where

• S is any imaginary closed surface, that encloses a volume V,
• Template:Oiint denotes a surface integral over that closed surface,
• q is the total amount of the quantity in the volume V,
• j is the flux of q,
• t is time,
• Σ is the net rate that q is being generated inside the volume V per unit time. When q is being generated, it is called a source of q, and it makes Σ more positive. When q is being destroyed, it is called a sink of q, and it makes Σ more negative. This term is sometimes written as $dq/dt|_{\text{gen}}$ or the total change of q from its generation or destruction inside the control volume.

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, Σ > 0), and decreases when someone in the building dies (a sink, Σ < 0).

Licensing

Content obtained and/or adapted from:

• Continuity equation, Wikipedia under a CC BY-Sa license