fluid_mechanics

172 Chapter 4 ■ Fluid Kinematics
Inflow
portion of
control surface
CS in
δ n
^
n
θ
δA
V
^
n
θ
δV = δ n δA
δ = V δt
V
^
n
θ
δ
V
(a)
F I G U R E 4.15
(b)
Inflow across a typical portion of the control surface.
(c)
By integrating over the entire outflow portion of the control surface, CS out , we obtain
B # out
cs out
dB # out
cs out
rbV cos u dA
The flowrate of a
parameter across
the control surface
is written in terms
of a surface integral.
The quantity V cos u is the component of the velocity normal to the area element dA. From the
definition of the dot product, this can be written as V cos u V nˆ . Hence, an alternate form of
the outflow rate is
B # out
csout
rbV nˆ dA
(4.16)
In a similar fashion, by considering the inflow portion of the control surface, CS in , as shown
in Fig. 4.15, we find that the inflow rate of B into the control volume is
B # in
csin
rbV cos u dA
csin
rbV nˆ dA
(4.17)
We use the standard notation that the unit normal vector to the control surface, nˆ , points out from the
control volume. Thus, as is shown in Fig. 4.16, 90° 6 u 6 90° for outflow regions 1the normal
component of V is positive; V nˆ 7 02. For inflow regions 90° 6 u 6 270° 1the normal component
of V is negative; V nˆ 6 02. The value of cos u is, therefore, positive on the CV out portions of the
control surface and negative on the CV in portions. Over the remainder of the control surface, there is
no inflow or outflow, leading to V nˆ V cos u 0 on those portions. On such portions either V 0
1the fluid “sticks” to the surface2 or cos u 0 1the fluid “slides”along the surface without crossing it2
1see Fig. 4.162. Therefore, the net flux 1flowrate2 of parameter B across the entire control surface is
B # B# rbV nˆ dA a
out in cs out
rbV nˆ dAb
cs in
cs
rbV nˆ dA
(4.18)
where the integration is over the entire control surface.
CS in
CS
V • ^
n < 0
^
n
V = 0
^
n
V • ^
n = 0
(a) (b) (c)
^
n
CS out
^
V • n > 0
F I G U R E 4.16
Possible velocity configurations
on portions of the control surface:
(a) inflow, (b) no flow
across the surface, (c) outflow.