Basics of Fluid Mechanics, 2014a

8.3. CONSERVATION OF GENERAL QUANTITY 239
The last term on the RHS can be converted using the divergence theorem (see the
appendix 8 ) from a surface integral into a volume integral (alternatively, the volume
integral can be changed to the surface integral) as
∫
∫
ρφU · dA = ∇·(ρφU) dV (8.24)
A
Substituting equation (8.24) into equation (8.23) yields
∫
D
φρdV = d ∫ ∫
φρdV + ∇·(ρφU) dV (8.25)
Dt sys dt cv
cv
Since the volume of the control volume remains independent of the time, the derivative
can enter into the integral and thus combining the two integrals on the RHS results in
∫
∫ ( )
D
d (φρ)
φρdV =
+ ∇·(ρφU) dV (8.26)
Dt
dt
sys
cv
The definition of equation (8.21) LHS can be changed to simply the derivative
of Φ. The integral is carried over arbitrary system. For an infinitesimal control volume
the change is
( ) dV
D Φ
Dt ∼ d (φρ)
{ }} {
= + ∇·(ρφU) dx dy dz (8.27)
dt
8.3.2 Examples of Several Quantities
8.3.2.1 The General Mass Time Derivative
Using φ =1is the same as dealing with the mass conservation. In that case D Φ
Dt
which is equal to zero as
⎧ - ⎛ ⎛ ⎞
φ
⎞
{}}{
d ⎝ 1 ρ⎠
dV
⎛ ⎞
φ
⎜
{}}{
{ }} {
⎟ dx dy dz =0 (8.28)
⎝
+ ∇· ⎝ρ 1 U⎠⎠
⎪⎭ dt
Using equation (8.21) leads to
-
Dρ
Dt
=0−→
∂ρ
∂t
Equation (8.29) can be rearranged as
V
= Dρ
Dt
+ ∇·(ρU) =0 (8.29)
∂ρ
∂t + U ∇·ρ + ρ ∇·U =0 (8.30)
8 These integrals are related to RTT. Basically the divergence theorem relates the flow out (or) in
and the sum of the all the changes inside the control volume.