1. . Conservation of Mass -- Equation of Continuity
(x,y,z)
(x+x, y+y, z+z) x y z x z y
[Accumulation] = [Input] - [Output] Or
[Rate of [Rate of [Rate of
Accumulation] = Influx] Efflux]
Mass is neither created nor destroyed in ordinary
chemical processes. Net flow across the sixfaces
of our control volume will add to or take away
from the mass inside.
Rate of Accumulation of Mass:
 x y z t
Any addition or loss to the mass in the control volume
during time interval t will result in a change in density 
Volumetric flow across the face at x+x: (vx y z)x+x
Flux of mass across the face at x+x: -(vx y z)x+x
Flux of mass across the face at x: (vx y z)x
“Right face”
“Left face”
Front--Flux of mass across the face at y: (vy x z)y
Back--Flux of mass across the face at y+y: -(vy x z)y+y
Top-- Flux of mass across the face at z+z: -(vz x y)z+z
Bottom--Flux of mass across the face at z: (vz x y)z

2. . DIVERGENCE Net flux of mass across all six faces:
(vx yz)x - (vx yz)x+x + (vy xz)y - (vy xz)y+y + (vz xy)z - (vz xy)z+z
Substituting into the word equation and rearranging:
 xyz + [(vx) x+x -(vx)x]yz + [(vy) y+y -(vy)y] xz + t
[(vz)z+z-(vz)z] xy = 0
Divide by xyz and take the limit as x, y, z  0
Lim { + [(vx) x+x -(vx)x] + [(vy) y+y -(vy)y] + [(vz)z+z-(vz)z] } = 0
x0 t x y z
y0
z0
So,
 + (vx) + (vy) + (vz) = 0
t x y z
Using the definition of the divergence operator in rectangular coordinates:
 = -  (v) t .
DIVERGENCE