378
12 Internal Flows in Marine Organisms
r.n 1.0
.~
C.
's.
s
.g 0.5
8
B
'"
:a ea 0.0
t=:
.9
'" t=:
S
:a -0.5
I
t=:
o
Z
0.0
0.2
- - - - - - - - - -
parabolic flow
piP!'!. ~s_
0.4
0.6
0.8
1.0
Non-dimensional velocity
Fig. 12.1: Velocity distribution in a circular pipe
charge, Q(r), of fluid transported through the pipe can be determined by multiplying the area of an annulus 21rdr by the local velocity u( r) and integrating
across the cross-section from the axis to a given distance, r, i. e.:
Q(r) =
27rru(r)dr =
2 -
- -
.
l o
r
1rt1pD4 [(2r)2 (2r)4]
o
128111
D
D
(12.2)
The non-dimensional cumulative discharge 1281l1Q (r) /1r t1pD 4 is shown in
Fig. 12.2 as a function of non-dimensional distance from the pipe axis, 2r / D.
At r = D /2, the cumulative discharge is equal to the total discharge given in
Eq. (2.103):
1rt1pD 4
Q = 128111
Let us rewrite Eq. (12.3) as follows:
A
_
1281ll
up -
4 Q.
1rD
(12.3)
(12.4)
The term 1281l1/(1rD4) is the resistance of a steady laminar flow in a circular
cylindrical pipe. Thus, the pressure drop, t1p, required to transport volume Q
of the fluid is linearly dependent on the resistance of the flow.