Page 76 - Bird R.B. Transport phenomena

§2.6 Creeping Flow Around a Sphere 61

The first term is the buoyant force, which would be present in a fluid at rest; it is the mass
of the displaced fluid multiplied by the gravitational acceleration. The second term, the
kinetic force, results from the motion of the fluid. The relation
F k = бтгрКи* (2.6-15)
1
is known as Stokes' law. It is used in describing the motion of colloidal particles under an
electric field, in the theory of sedimentation, and in the study of the motion of aerosol
particles. Stokes' law is useful only up to a Reynolds number Re = Dv p/fi of about 0.1.
x
At Re = 1, Stokes' law predicts a force that is about 10% too low. The flow behavior for
larger Reynolds numbers is discussed in Chapter 6.
This problem, which could not be solved by the shell balance method, emphasizes
the need for a more general method for coping with flow problems in which the stream-
lines are not rectilinear. That is the subject of the following chapter.

EXAMPLE 2.6-1 Derive a relation that enables one to get the viscosity of a fluid by measuring the terminal ve-
locity v of a small sphere of radius R in the fluid.
Determination of t
Viscosity front the SOLUTION
Terminal Velocity
If a small sphere is allowed to fall from rest in a viscous fluid, it will accelerate until it reaches
of a Falling Sphere
a constant velocity—the terminal velocity. When this steady-state condition has been reached
the sum of all the forces acting on the sphere must be zero. The force of gravity on the solid
acts in the direction of fall, and the buoyant and kinetic forces act in the opposite direction:
3
57rR g = \irR?pg + 6ir/jiRv t (2.6-16)
Ps
Here p and p are the densities of the solid sphere and the fluid. Solving this equation for the
s
terminal velocity gives
2
fi = lR (p s - p)g/v t (2.6-17)
This result may be used only if the Reynolds number is less than about 0.1.
This experiment provides an apparently simple method for determining viscosity. How-
ever, it is difficult to keep a homogeneous sphere from rotating during its descent, and if it
does rotate, then Eq. 2.6-17 cannot be used. Sometimes weighted spheres are used in order to
preclude rotation; then the left side of Eq. 2.6-16 has to be replaced by m, the mass of the
sphere, times the gravitational acceleration.

QUESTIONS FOR DISCUSSION
1. Summarize the procedure used in the solution of viscous flow problems by the shell balance
method. What kinds of problems can and cannot be solved by this method? How is the defin-
ition of the first derivative used in the method?
2. Which of the flow systems in this chapter can be used as a viscometer? List the difficulties
that might be encountered in each.
3. How are the Reynolds numbers defined for films, tubes, and spheres? What are the dimen-
sions of Re?
4. How can one modify the film thickness formula in §2.2 to describe a thin film falling down
the interior wall of a cylinder? What restrictions might have to be placed on this modified for-
mula?
5. How can the results in §2.3 be used to estimate the time required for a liquid to drain out of a
vertical tube that is open at both ends?
6. Contrast the radial dependence of the shear stress for the laminar flow of a Newtonian liquid
in a tube and in an annulus. In the latter, why does the function change sign?