Tensors for Physics

8.4 Further Applications of Volume Integrals 141

d

dt

MV+

∫

∂V

nνjνd^2 s= 0 , (8.86)

wherenis the outer normal vector on the surface of the volumeV. This equation
says: the mass contained in the volume changes with time according to the flux of
mass which goes in and out of the volume, through the surface. There is no creation
or annihilation of mass.
As an example, consider a fluid confined by a pipe, with spatially varying cross
section and open ends. The normal vectors of the open parts of the volumeVare
denotedbyn 1 andn 2 ,thepertainingareasareA 1 andA 2 .Forthisgeometryn 1 =−n 2
applies. Since the side walls are assumed to be impenetrable for the fluid, the time
change of the mass is

dMV

dt

=I 1 +I 2 ,

whereI 1 =−

∫

∂A 1 n^1 ·jd

(^2) sandI 2 =−∫
∂A 2 n^2 ·jd
(^2) sare the fluxes into and out of
V, respectively. One hasI 1 >0 andI 2 <0 whenn 1 andn 2 are anti-parallel and
parallel toj.
With the help of the substantial time derivative (7.50), viz.
d
dt

:=

∂

∂t

+vν∇ν,

the continuity equation can also be written as

dρ

dt

+ρ∇νvμ= 0.

The quantityV =ρ−^1 is the volume per mass, also called specific volume. The
continuity equation is equivalent to

d

dt

lnV=∇νvν. (8.87)

This shows: the specific volume and hence the density does not change with time
when the divergence of the velocity vanishes. A flow with∇·v=0 is referred to as
incompressible flow.
In electrodynamics, the symbolsρandjare used for the charge density and the
electric flux density. The integral ofρover a volume is the chargeQcontained in
this volume. The continuity equation has the same form as (8.85), provided that no
charges are created or annihilated. Then the continuity equation describes the local
charge conservation.