Consider a finite volume of fluid V which is of arbitrary size and shape but fixed. Furthermore, it is supposed that this volume is wholly contained within the fluid. In view of the principle of conservation of mass, the only way that the mass of the volume V can change is by mass transport across its boundaries. That is, if T were zero then the mass M of the volume must be constant. In general, M can vary, but its variation must be in accord with the equation
where is the outward unit normal to the surface of V.
However, from the definition of fluid density
or since V is fixed
Inserting this in Eq. (2) and transforming the surface integral of Eq. (2) by the divergence theorem (Eq. 154 of Art. 1.071) yields
Since V is arbitrary, the only way that Eq. (5) can be satisfied in general is that the integrand vanish, consequently
This is one form of the equation of continuity.
A second form is obtained as follows. Note that
where is the specific volume. Thus we see that the fluid can only be divergenceless if the material rate of change of the density vanishes.
A convenient integral form of the equation of continuity is obtained by making use of the three-dimensional analog to Leibnitz' rule. If f is any arbitrary fucntion of x, y, z, t then
where V(t) denotes any arbitrary vollume where boundaries move with the arbitrary velocity .
If we consider a volume of fluid whose boundaries move with the fluid velocity ( ) and replace f by then Eq. (9) takes the form
or, using the divergence theorem,
The integrand on the right can be expressed as
but in view of Eq. (I-1) the first term in brackets vanishes, and the second can be expressed as a material derivative. Thus, Eq. (11) becomes
where the boundaries of V move with the fluid velocity. Under this condition there can be no net mass transport across the boundaries and therefore the mass must be conserved. Thus Eq. (12) can be written
for a moving volume of given mass M.