fluid_mechanics

9.3 Drag 499
TABLE 9.4
Low Reynolds Number Drag Coefficients (Ref. 7) ( Re RUDM, A PD 2 4)
C D d
( RU 2 A2)
Object ( for Re f 1) Object
a. Circular disk normal 20.4Re
c. Sphere
to flow
C D
24.0Re
U
D
U
D
b. Circular disk parallel 13.6Re
d. Hemisphere
to flow
22.2Re
U
U
D
D
For very small
Reynolds number
flows, the drag coefficient
varies inversely
with the
Reynolds number.
where Re rU/m. The use of the dynamic pressure, rU 2 2, in the definition of the drag coefficient
is somewhat misleading in the case of creeping flows 1Re 6 12 because it introduces the
fluid density, which is not an important parameter for such flows 1inertia is not important2. Use of
this standard drag coefficient definition gives the 1Re dependence for small Re drag coefficients.
Typical values of C D for low Reynolds number flows past a variety of objects are given in
Table 9.4. It is of interest that the drag on a disk normal to the flow is only 1.5 times greater than
that on a disk parallel to the flow. For large Reynolds number flows this ratio is considerably larger
1see Example 9.12. Streamlining 1i.e., making the body slender2 can produce a considerable drag
reduction for large Reynolds number flows; for very small Reynolds number flows it can actually
increase the drag because of an increase in the area on which shear forces act. For most objects,
the low Reynolds number flow results are valid up to a Reynolds number of about 1.
E XAMPLE 9.10
Low Reynolds Number Flow Drag
GIVEN A small grain of sand, diameter D 0.10 mm and
specific gravity SG 2.3, settles to the bottom of a lake after
having been stirred up by a passing boat.
FIND
Determine how fast it falls through the still water.
SOLUTION
A free-body diagram of the particle 1relative to the moving particle2
is shown in Fig. E9.10a. The particle moves downward with a constant
velocity U that is governed by a balance between the weight
of the particle, w, the buoyancy force of the surrounding water, F B ,
and the drag of the water on the particle, d.
F B
From the free-body diagram, we obtain
w d F B
where
w g sand V SG g p H2 O
6 D3
and
F B g H2 O V g p H2 O
6 D3
(1)
(2)
U
F I G U R E E9.10a
We assume 1because of the smallness of the object2 that the
flow will be creeping flow 1Re 6 12 with C D 24 Re 1see Table
9.42 so that
d 1 2 r H 2 OU 2 p 4 D2 C D 1 2 r H 2 OU 2 p 24
4 D2 a
b
r H2 OUDm H2 O