Basics of Fluid Mechanics, 2014a

228 CHAPTER 8. DIFFERENTIAL ANALYSIS
of the Add Mass and the Add Force, which are easier to discuss when the viscosity
is ignored, and will be presented in the Ideal Flow chapter. It has to be pointed
out that the Add Mass and Add Force appear regardless to the viscosity. Historically,
complexity of the equations, on one hand, leads to approximations and consequently to
the ideal flow approximation (equations) and on the other hand experimental solutions
of Navier–Stokes equations. The connection between these two ideas or fields was done
via introduction of the boundary layer theory by Prandtl which will be discussed as well.
Even for simple situations, there are cases when complying with the boundary
conditions leads to a discontinuity (shock or choked flow). These equations cannot
satisfy the boundary conditions in other cases and in way the fluid pushes the boundary
condition(s) further downstream (choked flow). These issues are discussed in Open
Channel Flow and Compressible Flow chapters. Sometimes, the boundary conditions
create instability which alters the boundary conditions itself which is known as Interfacial
instability. The choked flow is associated with a single phase flow (even the double
choked flow) while the Interfacial instability associated with the Multi–Phase flow. This
phenomenon is presented in Multi–phase chapter and briefly discussed in this chapter.
8.2 Mass Conservation
Fluid flows into and from a three
dimensional infinitesimal control
volume depicted in Figure 8.1. At
a specific time this control volume
can be viewed as a system.
The mass conservation for this infinitesimal
small system is zero
thus
∫
D
ρdV =0 (8.1)
Dt V
x
A
ρ Ux dy dz
C
ρ Uy dx dz
(
ρ + dρ )(
dz
E
G
ρ Uz dx dy
Uz + dUz
dz
D
B
(
)
dx dy
F
)(
ρ + dρ
dy
H
)
dy
Uy + dUy
(
ρ + dρ )(
dx
dx dz
Ux + dUx
dx
)
dy dz
Fig. -8.1. The mass balance on the infinitesimal control
volume.
However for a control volume using
Reynolds Transport Theorem (RTT), the following can be written
∫
D
ρdV = d ∫ ∫
ρdV + U rn ρdA=0 (8.2)
Dt V dt V
A
For a constant control volume, the derivative can enter into the integral (see also for
the divergence theorem in the appendix A.1.2) on the right hand side and hence
dρ
dt dV
{ ∫ }} {
V
dρ
dt
∫A
dV + U rn ρdA=0 (8.3)