Let y be a point in three dimensional space and let

(y1,y2,y3)

be Cartesian coordinates of this point. Let
V be a region in three dimensional space and suppose a fluid having density ρ

(y,t)

and velocity,
v

(y, t)

is flowing through this region. Then the mass of fluid leaving V per unit time is given by
the area integral ∫_{∂V }ρ

(y,t)

v

(y,t)

⋅ ndA while the total mass of the fluid enclosed in V at a
given time is ∫_{V }ρ

(y, t)

dV . Also suppose mass originates at the rate f

(y,t)

per cubic unit per
unit time within this fluid. Then the conclusion which can be drawn through the use of the
divergence theorem is the following fundamental equation known as the mass balance equation.

∂ρ
---+ ∇ ⋅(ρv) = f (y,t) (17.8)
∂t

(17.8)

To see this is so, take an arbitrary V for which the divergence theorem holds. Then the time rate of
change of the mass in V is

∂ ∫ ∫ ∂ρ(y,t)
-- ρ(y,t)dV = -------dV
∂t V V ∂t

where the derivative was taken under the integral sign with respect to t. (This is a physical derivation and
therefore, it is not necessary to fuss with the hard mathematics related to the change of limit operations.
You should expect this to be true under fairly general conditions because the integral is a sort of sum and
the derivative of a sum is the sum of the derivatives.) Therefore, the rate of change of mass

-∂
∂t

∫_{V }ρ

(y,t)

dV , equals

--rate at-which mass enters
∫ ∂ ρ(y,t) ◜ ∫ ◞◟ ◝ ∫
---∂t--dV = − ρ(y,t)v (y,t)⋅ndA + f (y,t)dV
V ∫∂V V
= − (∇ ⋅(ρ(y,t)v(y,t))+ f (y,t))dV.
V

Since this holds for every sample volume V it must be the case that the equation of continuity holds.
Again, there are interesting mathematical questions here which can be explored but since it
is a physical derivation, it is not necessary to dwell too much on them. If all the functions
involved are continuous, it is certainly true but it is true under far more general conditions than
that.

Also note this equation applies to many situations and f might depend on more than just y and t. In
particular, f might depend also on temperature and the density ρ. This would be the case for example if
you were considering the mass of some chemical and f represented a chemical reaction. Mass balance is a
general sort of equation valid in many contexts.