Continuity equation for the flow of conserved entities

We will repeatedly encounter in this book processes that involve the flow of conserved quantities. An easily visualized example is the diffusion of nonreactive particles, but it should be emphasized at the outset that the motion involved can be of any type and the moving object(s) do not have to be particles. The essential ingredient in the following discussion is a conserved entity Q whose distribution in space is described by some time-dependent density function pg(r, t) so that its amount within some finite volume V is given by 

The conservation of Q implies that any change in g(Z) can result only from flow of Q through the boundary of volume V. Let S be the surface that encloses the volume L, and dS—a vector surface element whose direction is normal to the surface in the outward direction. Denote by Jg(r, Z) the flux of Q, that is, the amount of Q moving in the direction of Jg per unit time and per unit area of the surface perpendicular to Jg. The Q conservation law can then be written in the following mathematical form 

Equation (1.44), the local form of Eq. (1.42), is the continuity equation for the conserved Q. Note that in terms of the velocity field v(r, t) = f (r, z) associated with the Q motion we have 

It is important to realize that the derivation above does not involve any physics. It is a mathematical expression of conservation of entities that that can change their position but are not created or destroyed in time. Also, it is not limited to entities distributed in space and could be applied to objects moving in other dimensions. For example, let the function p (r, v, Z) be the particles density in position and velocity space (i.e. p (r, v, t)(fir(fiv is the number of particles whose position and velocity are respectively within the volume element d r about r and the velocity element d v about v). The total number, N = f d r f d vp r, v, Z), is fixed. The change of p in time can then be described by Eq. (1.44) in the form 

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