Conservation
of Mass
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The
principle of conservation of mass refers to the fact that when the amount of
mass within a region of space decreases then the amount of mass outside the
volume increases. The total mass in the universe thus remains constant. To place
this in a mathematical form define mass current vector, J, as a
vector whose direction is the direction of the flow of matter and whose
magnitude is mass per unit time per unit area. See Fig. 1 below
Mass
density, r, is defined as mass per unit volume. I.e.
The
magnitude, J, of the mass current vector, J, is then given by
From the
above diagram we know that
Substituting into Eq. (2) we find that
This is
just momentum density, i.e.
The
total mass within a volume, V, of space in term of mass density is
Let S be
the surface bounding the volume V. The time rate of decrease of the mass
within the volume must be accounted for by a mass flux through the surface. I.e.
Change
the surface integral into a volume integral using Stoke’s Theorem
Since
this must hold in general the integrands must be equal. Therefore
Substituting
in the momentum density from Eq. (5) and rearranging terms we find
This is
known as Euler’s Equation or the equation of continuity and
is the mathematical expression for the local conservation of mass.