Page 201 - Bird R.B. Transport phenomena

§6.3 Friction Factors for Flow around Spheres 185

56.3 FRICTION FACTORS FOR FLOW AROUND SPHERES
In this section we use the definition of the friction factor in Eq. 6.1-5 along with the di-
mensional analysis of §3.7 to determine the behavior of /for a stationary sphere in an in-
finite stream of fluid approaching with a uniform, steady velocity v . We have already
x
studied the flow around a sphere in §2.6 and §4.2 for Re < 0.1 (the "creeping flow" re-
gion). At Reynolds numbers above about 1 there is a significant unsteady eddy motion
in the wake of the sphere. Therefore, it will be necessary to do a time average over a time
interval long with respect to this eddy motion.
Recall from §2.6 that the total force acting in the z direction on the sphere can be
written as the sum of a contribution from the normal stresses (F ) and one from the tan-
n
gential stresses (F). One part of the normal-stress contribution is the force that would be
t
present even if the fluid were stationary, F . Thus the "kinetic force," associated with the
s
fluid motion, is
F = (F - F ) + F, = F form + F friction (6.3-1)
k
n
s
The forces associated with the form drag and the friction drag are then obtained from
f ^
2
fform(0 = f >o ^ Г (~Щг=к cos e)R sin в dO d<t> (6.3-2)
Jo
sin в sin
У
Since v is zero everywhere on the sphere surface, the term containing dv /d0 is zero.
r
r
If now we split/into two parts as follows
/ = /form + /friction (63-4)
then, from the definition in Eq. 6.1-5, we get

/form(0 = | \ (-& U cos 0) sin в dO dф (6.3-5)
"Jo Jo
sin в dO dф (6.3-6)
2
f=i
The friction factor is expressed here in terms of dimensionless variables
--
; = ¥ (637)
к
and a Reynolds number defined as

To evaluate f(t) one would have to know Ф and v as functions of г, д, ф, and t.
e
We know that for incompressible flow these distributions can in principle be ob-
tained from the solution of Eqs. 3.7-8 and 9 along with the boundary conditions
B.C. 1: atf = 1, v = 0 and v = 0 (6.3-9)
r
e
B.C. 2: atf = oo r v = 1 (6.3-10)
z
B.C. 3: at f = oo, Ф = о (6.3-11)
and some appropriate initial condition on v. Because no additional dimensionless
groups enter via the boundary and initial conditions, we know that the dimensionless
pressure and velocity profiles will have the following form:
Ф &, в, ф, Ь Re) v = v(r, в, ф, Ь Re) (6.3-12)