Basics of Fluid Mechanics, 2014a

5.6. THE DETAILS PICTURE – VELOCITY AREA RELATIONSHIP 167
For constant density (conservation of volume) equation 6 and (h >z) reduces to
∫
U rn ρdA=0 (5.35)
In the container case for uniform velocity equation 5.35 becomes
A
U z A = U e A e =⇒ U z = − A e
A U e (5.36)
It can be noticed that the boundary is not moving and the mass inside does not change
this control volume. The velocity U z is the averaged velocity downward.
The x component of velocity is obtained
by using a different control volume.
into the page
Y control Volume
The control volume is shown in Figure 5.9.
The boundary are the container far from
A
−
the flow exit with blue line projection into y
page (area) shown in the Figure 5.9. The
mass conservation for constant density of
this control volume is
∫
−
A
∫
U bn ρdA+
A
U rn ρdA=0
(5.37)
X control Volume
A
− x into the page
x y
Ue A e
Fig. -5.9. Control volume and system before
and after the motion.
Usage of control volume not included in the previous analysis provides the velocity at
the upper boundary which is the same as the velocity at y direction. Substituting into
(5.37) results in
∫
∫
A e
A − x
A U e ρdA+ U x ρdA=0 (5.38)
A yz
Where A x − is the area shown the Figure under this label. The area A yz referred to
area into the page in Figure 5.9 under the blow line. Because averaged velocities and
constant density are used transformed equation (5.38) into
A yz
A { }} {
e
A A x − U e + U x Y (x) h =0 (5.39)
Where Y (x) is the length of the (blue) line of the boundary. It can be notice that the
velocity, U x is generally increasing with x because A − x increase with x.
The calculations for the y directions are similar to the one done for x direction.
The only difference is that the velocity has two different directions. One zone is right
to the exit with flow to the left and one zone to left with averaged velocity to right.
If the volumes on the left and the right are symmetrical the averaged velocity will be
zero.
6 The point where (z = h) the boundary term is substituted the flow in term.