This formula only applies to laminar flow, not
to turbulent motion. It is interesting that warm-blooded animals regulate the heat loss from their
bodies by changing the diameter of their blood vessels (varying r) and hence controlling the rate
of blood flow.
A simple proof of this formula using dimensional analysis is given
below:
Assume that the volume V passing through the tube per second is given by the
equation:
V = kηxryDpz/L
where the quantity Dp/L is called the pressure gradient down the tube. Using dimensional
analysis this gives x = -1, y = 4 and z = 1.
Therefore the volume per second
is:
Volume per second = kr4Δp/ηL
The value of k can be shown to be π/8 and therefore:
The apparatus shown in Figure 2 may be used to determine the viscosity of a liquid
such as water. The water flows from a constant head apparatus through a horizontal capillary
tube and is then collected by a beaker; the time for a given volume to be collected is measured.
The pressure difference between the two ends of the capillary tube is hrg where r is the density of the liquid. The
internal diameter of the capillary tube is measured with a travelling microscope.
It
is important that the temperature of the water is measured and that the water only drips from the
free end of the tube, since in the rigorous proof of the formula the water is assumed to have no
residual kinetic energy. Laminar flow must also be assumed.