Answer
Laminar flow in a Couette apparatus has a practically constant velocity gradient,
which makes calculation much easier. To check whether the flow is laminar, the
Reynolds number has to be calculated; it is roughly given by Table 5.1 for flat plates.
Because one of the ‘‘plates’’ is moving, the maximum velocityvshould be taken; it is
2 ptimes radius times angular velocity, i.e., 2p? 0 : 1? 180 = 60 ¼ 1 :88 m?s^1. Since
d¼ 0 :005 m,r¼920 kg?m^3 , andZ¼ 0 :075 Pa?s, we obtain Re¼220, clearly
below the critical value. The velocity gradient C¼v=d¼375 s^1. The energy
dissipation rate equalsZC^2 ¼1400J?m^3 ?s^1. The specific heatcpof triglyceride oil
equals 2:1kJ?kg^1 ?K^1 or 1: 93? 106 J?m^3 ?K^1. The quotientZC^2 =cpnow yields
a temperature increase rate of ð 1 = 1380 ÞK?s^1. Hence it would take
1380 = 60 ¼23 min to raise the temperature by 1 K.
5.1.2 Viscosity
Molecules in a fluid undergo continuous Brownian or heat motion and thus
have kinetic energy (Section 4.3.1). When the fluid flows, they have some
additional kinetic energy and—owing to the velocity gradient—this energy
varies from place to place. Envisaging simple shear flow, adjacent layers
have a different velocity. During such flow, some molecules will move by
Brownian motion from one layer to another one, which means to one with
another velocity; such a molecule thus is accelerated or decelerated. This
implies that a (small) part of the kinetic energy related to the flow is lost and
converted into heat. This is the classical explanation for the viscosity of
gases. The theory predicts that the viscosity of a gas increases with
temperature (see Table 5.2) and is virtually independent of pressure. Neither
of these two predictions is true for liquids. The explanation of viscosity in a
liquid involves other factors, and has much to do with the limited free
volume between the molecules: it is difficult for them to move past the other
ones and this difficulty is enhanced in the presence of a velocity gradient.
This then would mean that the viscosity is far greater in liquids than in gases
and decreases with increasing temperature (higherT?lower density?more
free space between molecules).
Table 5.2 gives some examples. It is seen that for homologous
compounds, the viscosity increases with molecular size, in accordance with
simple theory. It is also seen that there is a considerable variation among
various types of molecules. This is related to the attractive interaction forces
between molecules, and the existence of hydrogen bonds in water and
alcohols is often held responsible for the relatively high viscosity of these
compounds. However, the molecular explanation of viscosity is intricate.