Mechanics of Fluids

shared among the occupants of bb, and thus layer bb as a whole is speeded
up. Similarly, molecules from the slower layer bb cross to aa and tend to
retard the layer aa. Every such migration of molecules, then, causes forces
of acceleration or deceleration in such directions as to tend to eliminate the
differences of velocity between the layers.
In gases this interchange of molecules forms the principal cause of viscosity,
and the kinetic theory of gases (which deals with the random motions
of the molecules) allows the predictions – borne out by experimental observations
– that (a) the viscosity of a gas is independent of its pressure (except
at very high or very low pressure) and (b) because the molecular motion
increases with a rise of temperature, the viscosity also increases with a rise
of temperature (unless the gas is so highly compressed that the kinetic theory
is invalid).
The process of momentum exchange also occurs in liquids. There is, however,
a second mechanism at play. The molecules of a liquid are sufficiently
close together for there to be appreciable forces between them. Relative
movement of layers in a liquid modifies these inter-molecular forces, thereby
causing a net shear force which resists the relative movement. Consequently,
the viscosity of a liquid is the resultant of two mechanisms, each of which
depends on temperature, and so the variation of viscosity with temperature
is much more complex than for a gas. The viscosity of nearly all liquids
decreases with rise of temperature, but the rate of decrease also falls. Except
at very high pressures, however, the viscosity of a liquid is independent of
pressure.
The variation with temperature of the viscosity of a few common fluids is
given in Appendix 2.
1.6.3 The dimensional formula and units of dynamic viscosity
Dynamic viscosity is defined as the ratio of a shear stress to a velocity gradient.
Since stress is defined as the ratio of a force to the area over which it
acts, its dimensional formula is [FL −2 ]. Velocity gradient is defined as the
ratio of increase of velocity to the distance across which the increase occurs,
thus giving the dimensional formula [L/T]/[L] ≡ [T −1 ]. Consequently the
dimensional formula of dynamic viscosity is [FL −2 ]/[T −1 ] ≡ [FTL −2 ]. Since
[F] ≡ [MLT −2 ], the expression is equivalent to [ML −1 T −1 ].
The SI unit of dynamic viscosity is Pa · s, or kg · m −1 · s −1 , but no special
name for it has yet found international agreement. (The name poiseuille,
abbreviated Pl, has been used in France but must be carefully distinguished
Fig. 1.6
Viscosity 25