Mechanics of Fluids

this book, it is important to give attention to the fundamental assumptions
on which the solution is based.
First, it is assumed that the motion is such that the inertia forces on the
particles of fluid (i.e. the forces required to accelerate or decelerate them)
may be neglected in comparison with the forces due to viscosity. As we have
seen, the ratio of inertia forces to viscous forces is represented by Reynolds
number, and so this condition is met if the Reynolds number of the flow is
very small.
The measurement of viscosity 213
Other assumptions are that no other boundary surface is sufficiently near Stokes’s law
to affect the flow round the sphere, that the sphere is rigid, that the motion
is steady and that there is no slip between the fluid and the sphere. Using
these hypotheses, Stokes found that the drag force opposing the motion
equals 3πµud, where µ represents the dynamic viscosity, u the velocity of
the sphere relative to the undisturbed fluid and d the diameter of the sphere.
This result is now known as Stokes’s Law. It has been found that, to obtain
good agreement with experimental results, the Reynolds number (expressed
as ρud/µ) must be less than about 0.1. The result of this restriction is that, for
ordinary fluids such as water or air, the sphere must be almost microscopic in
size. If Stokes’s Law is to be valid for larger spheres, then either the viscosity
must be very large or the velocity exceedingly small.
If a small solid particle is falling through a fluid under its own weight, the
particle accelerates until the net downward force on it is zero. No further
acceleration is then possible and the particle is said to have reached its terminal
velocity. This may be calculated for a small sphere on the assumption
that the Reynolds number is small enough for Stokes’s Law to be valid. If the
density of the fluid is ρ and the mean density of the sphere is ρs, then, when
the terminal velocity has been reached, the force balance on the sphere may
be stated as
Downward force = Weight of sphere − Buoyancy − Drag force on sphere
= π
6 d3ρsg − π
6 d3ρg − 3πµud = 0
whence
u = d2 (ρs − ρ)g
18µ
(6.28)
For the determination of viscosity, a small solid sphere of known weight is
allowed to fall vertically down the centre of a cylinder containing the liquid
under test. The velocity with which the sphere falls is measured; it does not,
however, quite coincide with the terminal velocity as given by eqn 6.28. In
practice the liquid cannot be of infinite extent as assumed in the derivation
of Stokes’s Law, so corrections are necessary to allow for the effect of the
walls of the cylinder. (For this effect to be negligible, the diameter of the
cylinder must be more than about 100 times the diameter of the sphere.)
Moreover, the measurement of velocity must not be begun until the sphere
has reached its terminal velocity, and should not be continued when the
sphere nears the base of the cylinder since this influences the rate of fall of
objects in its vicinity.