fluid_mechanics

1.6 Viscosity 15
U
δa
P
u
B
B'
b
y
A
δβ
Fixed plate
F I G U R E 1.5 Behavior of a fluid
placed between two parallel plates.
u = 0 on surface
Solid body
u = u(y)
V1.5 Capillary tube
viscometer
Dynamic viscosity
is the fluid property
that relates shearing
stress and fluid
motion.
y
as that shown by the photograph in the margin, this is not true. The experimental observation that
the fluid “sticks” to the solid boundaries is a very important one in fluid mechanics and is usually
referred to as the no-slip condition. All fluids, both liquids and gases, satisfy this condition.
In a small time increment, dt, an imaginary vertical line AB in the fluid would rotate through
an angle, db, so that
Since da U dt it follows that
We note that in this case, db is a function not only of the force P 1which governs U2 but also of
time. Thus, it is not reasonable to attempt to relate the shearing stress, t, to db as is done for solids.
Rather, we consider the rate at which db is changing and define the rate of shearing strain, g # , as
which in this instance is equal to
A continuation of this experiment would reveal that as the shearing stress, t, is increased by increasing
P 1recall that t PA2, the rate of shearing strain is increased in direct proportion—that is,
t r g #
or
tan db db da
b
db U dt
b
g # db
lim
dtS0 dt
g # U b du
dy
t r
This result indicates that for common fluids such as water, oil, gasoline, and air the shearing stress
and rate of shearing strain 1velocity gradient2 can be related with a relationship of the form
t m du
dy
where the constant of proportionality is designated by the Greek symbol m 1mu2 and is called the absolute
viscosity, dynamic viscosity, or simply the viscosity of the fluid. In accordance with Eq. 1.9,
plots of t versus dudy should be linear with the slope equal to the viscosity as illustrated in Fig. 1.6.
The actual value of the viscosity depends on the particular fluid, and for a particular fluid the viscosity
is also highly dependent on temperature as illustrated in Fig. 1.6 with the two curves for water.
Fluids for which the shearing stress is linearly related to the rate of shearing strain 1also referred
to as rate of angular deformation2 are designated as Newtonian fluids after I. Newton (1642–1727).
Fortunately most common fluids, both liquids and gases, are Newtonian. A more general formulation
of Eq. 1.9 which applies to more complex flows of Newtonian fluids is given in Section 6.8.1.
du
dy
(1.9)