Page 74 - Bird R.B. Transport phenomena

§2.6 Creeping Flow Around a Sphere 59

proaches the fixed sphere vertically upward in the z direction with a uniform velocity v .
x
For this problem, "creeping flow" means that the Reynolds number Re = Dv^p/ /л is less
than about 0.1. This flow regime is characterized by the absence of eddy formation
downstream from the sphere.
The velocity and pressure distributions for this creeping flow are found in Chapter 4
to be
-K (2.6-1)

.,=4-1 •i / 4 W J (2.6-2)

V = 0 (2.6-3)

3
P Po Pgz- c o s (2.6-4)
2 R \y/

In the last equation the quantity p is the pressure in the plane z = 0 far away from the
0
sphere. The term -pgz is the hydrostatic pressure resulting from the weight of the fluid,
and the term containing v x is the contribution of the fluid motion. Equations 2.6-1,2, and
3 show that the fluid velocity is zero at the surface of the sphere. Furthermore, in the
limit as r -» °°, the fluid velocity is in the z direction with uniform magnitude p ; this fol-
M
lows from the fact that v z = v r cos в - v e sin 0, which can be derived by using Eq. A.6-33,
and v = v = 0, which follows from Eqs. A.6-31 and 32.
x y
The components of the stress tensor т in spherical coordinates may be obtained from
the velocity distribution above by using Table B.I. They are

(2.6-5)

(2.6-6)

and all other components are zero. Note that the normal stresses for this flow are
nonzero, except at r = R.
Let us now determine the force exerted by the flowing fluid on the sphere. Because
of the symmetry around the z-axis, the resultant force will be in the z direction. There-
fore the force can be obtained by integrating the z-components of the normal and tan-
gential forces over the sphere surface.

Integration of the Normal Force
At each point on the surface of the sphere the fluid exerts a force per unit area — (p +
r )\ r=R on the solid, acting normal to the surface. Since the fluid is in the region of
rr
greater r and the sphere in the region of lesser r, we have to affix a minus sign in
accordance with the sign convention established in §1.2. The z-component of the force