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G.E. Leblanc, et. al.. "Viscosity Measurement."
Copyright 2000 CRC Press LLC. <http://www.engnetbase.com>.
Viscosity Measurement
G.E. Leblanc
McMaster University
R.A. Secco
30.1
Shear Viscosity
Newtonian and Non-Newtonian Fluids • Dimensions and
Units of Viscosity • Viscometer Types • Capillary
Viscometers • Falling Body Methods • Oscillating Method •
Ultrasonic Methods
The University of Western Ontario
M. Kostic
Northern Illinois University
30.1Shear Viscosity
. Physical systems and applications as diverse as
fluid flow in pipes, the flow of blood, lubrication of engine parts, the dynamics of raindrops, volcanic
eruptions, planetary and stellar magnetic field generation, to name just a few, all involve fluid flow and
are controlled to some degree by fluid viscosity.
An important mechanical property of fluids is
viscosity
is defined as the internal friction of a fluid. The
microscopic nature of internal friction in a fluid is analogous to the macroscopic concept of mechanical
friction in the system of an object moving on a stationary planar surface. Energy must be supplied (1) to
overcome the inertial state of the interlocked object and plane caused by surface roughness, and (2) to
initiate and sustain motion of the object over the plane. In a fluid, energy must be supplied (1) to create
viscous flow units by breaking bonds between atoms and molecules, and (2) to cause the flow units to
move relative to one another. The resistance of a fluid to the creation and motion of flow units is due
to the viscosity of the fluid, which only manifests itself when motion in the fluid is set up. Since viscosity
involves the transport of mass with a certain velocity, the viscous response is called a
Viscosity
momentum transport
process
. The velocity of flow units within the fluid will vary, depending on location. Consider a liquid
between two closely spaced parallel plates as shown in
Figure 30.1
.
A force,
, applied to the top plate
causes the fluid adjacent to the upper plate to be dragged in the direction of
F
. The applied force is
communicated to neighboring layers of fluid below, each coupled to the driving layer above, but with
diminishing magnitude. This results in the progressive decrease in velocity of each fluid layer, as shown
by the decreasing velocity vector in Figure 30.1, away from the upper plate. In this system, the applied
force is called a
F
shear
(when applied over an area it is called a
shear stress
), and the resulting deformation
·
. The
mathematical expression describing the viscous response of the system to the shear stress is simply:
rate of the fluid, as illustrated by the
velocity gradient
dU
/
dz
, is called the
shear strain rate
,
g
x
zx
h
dU
dz
˙
t
=
x
=
hg
(30.1)
zx
zx
where
t
, the shear stress, is the force per unit area exerted on the upper plate in the
x
-direction (and
zx
hence is equal to the force per unit area exerted by the fluid on the upper plate in the
x
-direction under
the assumption of a no-slip boundary layer at the fluid–upper plate interface);
dU
/
dz
is the gradient of
x
the
x
-velocity in the
z
-direction in the fluid; and
h
is the
coefficient of viscosity
. In this case, because one
is concerned with a shear force that produces the fluid motion,
h
is more specifically called the
shear
© 1999 by CRC Press LLC
System for defining Newtonian viscosity. When the upper plate is subjected to a force, the fluid
between the plates is dragged in the direction of the force with a velocity of each layer that diminishes away from
the upper plate. The reducing velocity eventually reaches zero at the lower plate boundary.
FIGURE 30.1
. In fluid mechanics, diffusion of momentum is a more useful description of viscosity
where the motion of a fluid is considered without reference to force. This type of viscosity is called the
dynamic viscosity
kinematic viscosity
,
n,
and is derived by dividing dynamic viscosity by
r
, the mass density:
h
r
n
=
(30.2)
(i.e., layered or sheet-like) or
streamline flow as depicted in Figure 30.1, and it refers to the molecular viscosity or
The definition of viscosity by Equation 30.1 is valid only for
laminar
.
The molecular viscosity is a property of the material that depends microscopically on bond strengths,
and is characterized macroscopically as the fluid’s resistance to flow. When the flow is turbulent, the
diffusion of momentum is comprised of viscous contributions from the motion, sometimes called the
intrinsic viscosity
eddy viscosity
, in addition to the intrinsic viscosity. Viscosities of turbulent systems can be as high as 10
6
times greater than viscosities of laminar systems, depending on the Reynolds number.
, depending on the
type of strain involved. Shear viscosity is a measure of resistance to isochoric flow in a shear field, whereas
volume viscosity is a measure of resistance to volumetric flow in a three-dimensional stress field. For
most liquids, including hydrogen bonded, weakly associated or unassociated, and polymeric liquids as
well as liquid metals,
Molecular viscosity
is separated into
shear viscosity
and bulk or
volume viscosity
,
h
v
h
/
h
»
1, suggesting that shear and structural viscous mechanisms are closely related
v
[1].
The shear viscosity of most liquids decreases with temperature and increases with pressure, which is
opposite to the corresponding responses for gases. An increase in temperature usually causes expansion
and a corresponding reduction in liquid bond strength, which in turn reduces the internal friction.
Pressure causes a decrease in volume and a corresponding increase in bond strength, which in turn
enhances the internal friction. For most situations, including engineering applications, temperature
effects dominate the antagonistic effects of pressure. However, in the context of planetary interiors where
the effects of pressure cannot be ignored, pressure controls the viscosity to the extent that, depending
on composition, it can cause fundamental changes in the molecular structure of the fluid that can result
in an anomalous viscosity decrease with increasing pressure [2].
Newtonian and Non-Newtonian Fluids
Equation 30.1 is known as Newton’s law of viscosity and it formulates Sir Isaac Newton’s definition of
the viscous behavior of a class of fluids now called Newtonian fluids.
© 1999 by CRC Press LLC
FIGURE 30.2
Flow curves illustrating Newtonian and non-Newtonian fluid behavior.
If the viscosity throughout the fluid is independent of strain rate, then the fluid is said to be a
Newtonian
fluid
. The constant of proportionality is called the coefficient of viscosity, and a plot of stress vs. strain
rate for Newtonian fluids yields a straight line with a slope of
, as shown by the solid line flow curve
in
Figure 30.2
.
Examples of Newtonian fluids are pure, single-phase, unassociated gases, liquids, and
solutions of low molecular weight such as water. There is, however, a large group of fluids for which the
viscosity is dependent on the strain rate. Such fluids are said to be non-Newtonian fluids and their study
is called
h
. In differentiating between Newtonian and non-Newtonian behavior, it is helpful to
consider the time scale (as well as the normal stress differences and phase shift in dynamic testing)
involved in the process of a liquid responding to a shear perturbation. The velocity gradient,
rheology
dU
/
dz
, in
x
·
the fluid is equal to the shear strain rate,
g
, and therefore the time scale related to the applied shear
·
perturbation about the equilibrium state is
t
, where
t
=
g
–1
. A second time scale,
t
,
called the
relaxation
s
s
r
time
characterizes the rate at which the relaxation of the strain in the fluid can be accomplished and is
related to the time it takes for a typical flow unit to move a distance equivalent to its mean diameter.
For Newtonian water,
,
are rare in practice, the
time required for adjustment of the shear perturbation in water is much less than the shear perturbation
period (i.e.,
t
~ 10
–12
s and, because shear rates greater than 10
6
s
–1
r
). However, for non-Newtonian macromolecular liquids like polymeric liquids, for
colloidal and fiber suspensions, and for pastes and emulsions, the long response times of large viscous
flow units can easily make
t
«
t
r
s
. An example of a non-Newtonian fluid is liquid elemental sulfur, in
which long chains (polymers) of up to 100,000 sulfur atoms form flow units that are easily entangled,
which bind the liquid in a “rigid-like” network. Another example of a well-known non-Newtonian fluid
is tomato ketchup.
With reference to Figure 30.2, the more general form of Equation 30.1 also accounts for the nonlinear
response. In terms of an initial shear stress required for flow to start,
t
>
t
r
s
t
(0), an initial linear term in the
xy
•
•
Newtonian limit of a small range of strain rate,
g¶t
(0)/
¶
g
, and a nonlinear term
O
(
g
2
), the shear stress
xy
•
dependence on strain rate,
t
(
g
) can be described as:
xy
© 1999 by CRC Press LLC
()
˙
gt
¶
0
()
()
=
()
+
xy
˙
˙
2
tgt
0
+
O
g
(30.3)
xy
xy
˙
¶
g
•
For a Newtonian fluid, the initial stress at zero shear rate is zero and the nonlinear function
O
(
g
2
) is zero,
•
so Equation 30.3 reduces to Equation 30.1, since
¶
t
(0)/
¶
g
then equals
h
. For a non-Newtonian fluid,
xy
•
) is nonzero. This characterizes fluids in which shear stress
increases disproportionately with strain rate, as shown in the dashed-dotted flow curve in Figure 30.2,
or decreases disproportionately with strain rate, as shown in the dashed flow curve in Figure 30.2. The
former type of fluid behavior is known as
t
(0) may be zero but the nonlinear term
O
(
g
2
xy
or dilatancy, and an example is a concentrated
solution of sugar in water. The latter, much more common type of fluid behavior, is known as
shear thickening
shear
thinning
or pseudo-plasticity; cream, blood, most polymers, and liquid cement are all examples. Both
behaviors result from particle or molecular reorientations in the fluid that increase or decreases, respec-
tively, the internal friction to shear. Non-Newtonian behavior can also arise in fluids whose viscosity
changes with time of applied shear stress. The viscosity of corn starch and water increases with time
duration of stress, and this is called
. Conversely, liquids whose viscosity decreases with
time, like nondrip paints, which behave like solids until the stress applied by the paint brush for a
sufficiently long time causes them to flow freely, are called
rheopectic behavior
.
Fluid deformation that is not recoverable after removal of the stress is typical of the purely viscous
response. The other extreme response to an external stress is purely elastic and is characterized by an
equilibrium deformation that is fully recovered on removal of the stress. There are an infinite number
of intermediate or combined viscous/elastic responses to external stress, which are grouped under the
behavior known as
thixotropic fluids
. Fluids that behave elastically in some stress range require a limiting or
yield stress before they will flow as a viscous fluid. A simple, empirical, constitutive equation often used
for this type of rheological behavior is of the form:
viscoelasticity
˙
n
ttgh
yx
=+
(30.4)
y
p
where
t
is the
yield stress
,
h
is an
apparent viscosity
called the plastic viscosity, and the exponent
n
allows
y
p
for a range of non-Newtonian responses:
n
= 1 is pseudo-Newtonian behavior and is called a
Bingham
fluid
> 1 is shear thickening behavior. Interested readers should
consult [3–9] for further information on applied rheology.
;
n
< 1 is shear thinning behavior; and
n
Dimensions and Units of Viscosity
From Equation 30.1, the dimensions of dynamic viscosity are M L
–1
T
–1
and the basic SI unit is the Pascal
second (Pa·s), where 1 Pa·s = 1 N s m
–2
. The c.g.s. unit of dyn s cm
–2
is the poise
(P). The dimensions
of kinematic viscosity, from Equation 30.2, are L
2
T
–1
and the SI unit is m
2
s
–1
. For most practical situations,
this is usually too large and so the c.g.s. unit of cm
s
–1
, or the
stoke (St), is preferred.
Table 30.1
li
sts
some common fluids and their shear dynamic viscosities at atmospheric pressure and 20
2
°
C.
TABLE 30.1
Shear Dynamic Viscosity of Some Common Fluids
at 20
°
C and 1 atm
Fluid
Shear dynamic viscosity (Pa·s)
Air
1.8
´
10
–4
Water
1.0
´
10
–3
Mercury
1.6
´
10
–3
Automotive engine oil (SAE 10W30)
1.3
´
10
–1
Dish soap
4.0
´
10
–1
Corn syrup
6.0
© 1999 by CRC Press LLC
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