Basics of Fluid Mechanics, 2014a

8.2. MASS CONSERVATION 231
Combining equation (8.14) with the change in the control volume (8.9) divided
by infinitesimal control volume, drrdθdz yields
Continuity in Cylindrical Coordinates
∂ρ
∂t + 1 ∂ (rρU r )
+ 1 ∂ρU θ
+ ∂ρU z
=0
(8.15)
r ∂r r ∂θ ∂z
Carrying similar operations for the spherical coordinates, the continuity equation
becomes
∂ρ
∂t + 1 ∂ ( Continuity in Spherical Coordinates
)
r 2 ρU r
r 2 + 1 ∂ (ρU θ sin θ)
+ 1 ∂ρU φ
=0 (8.16)
∂r r sin θ ∂θ r sin θ ∂z
The continuity equations (8.8), (8.15) and (8.16) can be expressed in different coordinates.
It can be noticed that the second part of these equations is the divergence
(see the Appendix A.1.2 page 570). Hence, the continuity equation can be written in a
general vector form as
Continuity Equation
∂ρ
+ ∇ · (ρU) =0
(8.17)
∂t
Advance material can be skipped
The mass equation can be written in index notation for Cartesian coordinates.
The index notation really does not add much to the scientific understanding. However,
this writing reduce the amount of writing and potentially can help the thinking about
the problem or situation in more conceptional way. The mass equation (see in the
appendix for more information on the index notation) written as
∂ρ
∂t + ∂ (ρU) i
=0 (8.18)
∂x i
Where i is is of the i, j, and k 4 . Compare to equation (8.8). Again remember that the
meaning of repeated index is summation.
End Advance material
The use of these equations is normally combined with other equations (momentum
and or energy equations). There are very few cases where this equation is used on its
own merit. For academic purposes, several examples are constructed here.
8.2.1 Mass Conservation Examples
Example 8.1:
A layer of liquid has an initial height of H 0 with an uniform temperature of T 0 . At
4 notice the irony the second i is the direction and first i is for any one of direction x(i), y(j), and
z(k).