Basics of Fluid Mechanics, 2014a

8.2. MASS CONSERVATION 229
The first term in equation (8.3) for the infinitesimal volume is expressed, neglecting
higher order derivatives, as
∫
V
dρ dρ
dV =
dt dt
dV
{ }} {
dx dy dz +
The second term in the LHS of equation (8.2) is expressed 2 as
∼0
{ ( }} {
d 2 )
ρ
f
dt 2 + ··· (8.4)
∫
A
U rn ρdA=
dA xz
{ }} {
dx dz
dA yz
{ }} {
dy dz [ (ρU x )| x
− (ρU x )| x+dx
]
+
dA xz
] { }} {
[(ρU y )| y
− (ρU y )| y+dy
+ dx dy [ ]
(ρU z )| z
− (ρU z )| z+dz
(8.5)
The difference between point x and x + dx can be obtained by developing Taylor series
as
(ρU x )| x+dx
=(ρU x )| x
+ ∂ (ρU x)
∂x ∣ dx (8.6)
x
The same can be said for the y and z coordinates. It also can be noticed that, for
example, the operation, in the x coordinate, produces additional dx thus a infinitesimal
volume element dV is obtained for all directions. The combination can be divided by
dx dy dz and simplified by using the definition of the partial derivative in the regular
process to be
∫
[ ∂(ρUx )
U rn ρdA= − + ∂(ρU y)
+ ∂(ρU ]
z)
(8.7)
∂x ∂y ∂z
A
Combining the first term with the second term results in the continuity equation
in Cartesian coordinates as
Continuity in Cartesian Coordinates
∂ρ
∂t + ∂ρU x
∂x
+ ∂ρU y
+ ∂ρU z
=0
∂y ∂z
(8.8)
Cylindrical Coordinates
The same equation can be derived in cylindrical coordinates.
change, as depicted in Figure 8.2, in the control volume is
The net mass
2 Note that sometime the notation dA yz also refers to dA x .
d ṁ = ∂ρ
dv
{ }} {
dr dz r dθ (8.9)
∂t