46
Fluid Viscosity
Measurement
1
46.1 Shear Viscosity ................................................................................46-1
46.2 Newtonian and Non-Newtonian Fluids .....................................46-3
Dimensions and Units of Viscosity
46.3 Viscometer Types............................................................................46-5
Rheometer • Rotational Methods/Concentric
Cylinders • Cone-and-Plate Viscometers • Parallel Disks
46.4 Pressure and Gravity Flow Methods .........................................46-11
Capillary Viscometers • Glass Capillary
Viscometers • Orifice/Cup, Short Capillary: Saybolt Viscometer
46.5 Falling Body Methods ..................................................................46-13
Falling Sphere • Falling Cylinder • Falling Methods in Opaque
Liquids • Rising Bubble/Droplet
R.A. Secco
The University of
Western Ontario
J.R. deBruyn
Northern Illinois University
M. Kostic
Northern Illinois University
46.6 Oscillation Methods .....................................................................46-17
46.7 Acoustic Methods .........................................................................46-20
46.8 Microrheology ...............................................................................46-22
Particle-Tracking Microrheology • Dynamic Light Scattering
46.9 High-Pressure Rheology..............................................................46-25
High-Pressure Rheometry
References ..................................................................................................46-29
Further Information.................................................................................46-31
46.1 Shear Viscosity
An important mechanical property of fluids is 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, and planetary and stellar magnetic field generation, to name just a few, all involve fluid flow
and are controlled to some degree by fluid viscosity. Viscosity is the tendency of a fluid to resist flow and
can be thought of as the internal friction of a fluid. Microscopically, viscosity is related to molecular diffusion and depends on the interactions between molecules or, in complex fluids, larger-scale flow units.
Diffusion tends to transfer momentum from regions of high momentum to regions of low momentum,
thus smoothing out variations in flow velocity. In this sense, the internal friction of a fluid is analogous
to the macroscopic mechanical friction, which causes an object sliding across a planar surface to slow
down. In the mechanical system, energy must be supplied to sustain the motion of the object over the
plane, while in a fluid, energy must be supplied to sustain a flow. Since viscosity is related to the diffusive transport of momentum, the viscous response of a fluid is called a momentum transport process.
The flow velocity within a fluid will vary, depending on location. Consider a viscous fluid at constant
pressure between two closely spaced parallel plates as shown in Figure 46.1. A force, F, applied to the
46-1
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46-2
Mechanical Variables
F
dU
x /d
z
Ux
z
y
x
FIGURE 46.1 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.
top plate causes the fluid adjacent to the upper plate to move in the direction of F. The fluid adjacent to
the top plate is constrained by the no-slip boundary condition to move at the same speed as the plate.
Similarly, the fluid next to the stationary bottom plate must be stationary. The motion of the top plate
thus causes the fluid to flow with a velocity profile across the liquid that decreases linearly from the
upper to the lower plate, as shown in Figure 46.1. This arrangement is referred to as simple shear. The
applied force is called a shear, and the force per unit area a shear stress. The resulting deformation rate of
the fluid, or equivalently the velocity gradient dUx/dz, is called the shear strain rate, g zx . The mathematical expression describing the viscous response of the system to the shear stress is simply
t zx =
dU x
hdU x
⎛
⎞
= hg zx ⎜ separate and multiply h from
term⎟
⎝
⎠
dz
dz
(46.1)
where
τzx, the shear stress, is the force per unit area exerted on the upper plate in the x-direction (and
hence is equal to the force per unit area exerted by the fluid on the upper plate in the negative
x-direction)
dUx/dz is the gradient of the x-velocity in the z-direction in the fluid, that is, the shear strain rate
η is the coefficient of viscosity
Note that in general, the shear strain rate is more complex function of the fluid velocity-gradient tensor.
In this case, because one is concerned with a shear force that produces the fluid motion, η is more specifically called the shear dynamic viscosity. In fluid mechanics, where the motion of a fluid is considered
without reference to force, it is common to define the kinematic viscosity, ν, which is simply given by
n=
h
r
(46.2)
where ρ is the mass density of the fluid.
The viscosity defined by Equation 46.1 is relevant only for laminar (i.e., layered or sheetlike) or streamline flow as depicted in Figure 46.1, and it refers to the molecular viscosity or intrinsic viscosity. The
molecular viscosity is a property of the material that depends microscopically on interactions between
individual molecules and is manifested macroscopically as the fluid’s resistance to flow. When the flow
is turbulent, small-scale turbulent vortices can contribute to the overall diffusion of momentum, resulting in an effective viscosity, sometimes called the eddy viscosity, that, depending on the Reynolds number, can be as much as 106 times larger than the intrinsic viscosity.
Molecular viscosity is separated into shear viscosity and bulk or volume viscosity, η v, 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 3D stress field.
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Fluid Viscosity Measurement
For most liquids, including hydrogen bonded weakly associated or unassociated, and polymeric
liquids as well as liquid metals, η/η v ≈ 1, suggesting that shear and structural viscous mechanisms
are closely related [1].
The shear viscosity of most liquids decreases with temperature, which is opposite to what occurs
in gases. Under most conditions, including those relevant in engineering applications, viscosity
can be considered to be independent of pressure, as temperature effects are very much larger. In
the context of planetary interiors, however, the effects of pressure cannot be ignored, and 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].
46.2 Newtonian and Non-Newtonian Fluids
Equation 46.1 is known as Newton’s law of viscosity. 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 versus strain rate for Newtonian fluids yields a straight
line with a slope of η, as shown by the solid line flow curve in Figure 46.2. 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 and their study is called rheology. 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. In general, the shear strain rate g is more complex function of the second invariant
of the fluid velocity-gradient tensor; however, in a simple shear flow, like on Figure 46.1, it reduces to the
velocity gradient, dUx/dz. Therefore, the time scale related to the applied shear perturbation about the
equilibrium state is ts, where t s = g −1. A second time scale, tr, called the relaxation 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,
tr ∼ 10−12 s, and, because shear rates greater than 106 s−l are rare in practice, the time required for the
fluid to respond to the strain perturbation in water is much less than the time scale of the perturbation
itself (i.e., tr ≪ ts). 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 (e.g., polymer molecules or aggregates of interacting particles) can easily make tr > ts. An example
tic
m
D
at
ila
er
ta
ial
nt
s
m
at
er
ia
ls
Ps
e
ud
op
las
Shear stress
Non-Newtonian
n liq
tonia
New
eη
Slop
uids
Strain rate
FIGURE 46.2 Flow curves illustrating Newtonian and non-Newtonian fluid behavior.
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46-4
Mechanical Variables
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 46.2, Equation 46.1 can be written in a more general form:
t xy (g ) = t xy (0) +
g ∂t xy (0)
+ O(g 2 )
∂g
(46.3)
Here τxy(0) is a yield stress required for flow to start, the term proportional to g is the usual linear
Newtonian term in limit of small strain rates, and the nonlinear term O(g 2 ) accounts for the nonlinear
viscous response of some materials.
For a Newtonian fluid, the initial stress at zero shear rate and the nonlinear function O(g 2 ) are both
negligible (zero), so Equation 46.3 reduces to Equation 46.1, since ∂t xy (0)/∂g then equals η. For a nonNewtonian fluid, τxy(0) may be zero but the nonlinear term O(g 2 ) is nonzero. This characterizes fluids in
which shear stress varies nonlinearly with strain rate, as shown by the dashed-dotted and dashed flow
curves in Figure 46.2. The former type of fluid behavior is known as shear thickening or dilatancy, and
an example is a concentrated suspension of cornstarch in water. The latter, much more common type of
fluid behavior is known as shear thinning or pseudoplasticity; cream, blood, most polymers, and liquid
cement are all examples. Both behaviors result from particle or molecular reorientations or rearrangements in the fluid that increase or decrease, respectively, the internal friction to shear. Non-Newtonian
behavior can also arise in fluids whose viscosity changes with time of applied shear stress. Fluids whose
viscosity increases over time when a shear stress is applied are called rheopectic. Conversely, liquids
whose viscosity decreases with time are called thixotropic fluids. Nondrip paint, which will not flow
until a shear stress is applied by the paint brush for a sufficiently long time, is a common example of a
thixotropic fluid.
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 viscoelasticity. 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
t yx = t y + g nhp
(46.4)
where τy is the yield stress, ηp is an apparent viscosity called the plastic viscosity, and the exponent n
allows for a range of non-Newtonian responses: n = 1 describes pseudo-Newtonian behavior and is
called a Bingham fluid; n < 1 describes shear thinning behavior; and n > 1 describes shear thickening
behavior. Interested readers should consult [3–9] for further information on applied rheology.
46.2.1 Dimensions and Units of Viscosity
From Equation 43.1, 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 43.2, 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 2 s−1, or the stoke (St), is
preferred. Table 46.1 lists some common fluids and their shear dynamic viscosities at atmospheric
pressure and 20°C.
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Fluid Viscosity Measurement
TABLE 46.1 Shear Dynamic Viscosity of Some
Common Fluids at 20°C and 1 atm
Fluid
Air
Water
Mercury
Automotive engine oil (SAE 10W30)
Dish soap
Corn syrup
Shear Dynamic
Viscosity (Pa · s)
1.8 × 10−4
1.0 × 10−3
1.6 × 10−3
1.3 × 10−1
4.0 × 10−1
6.0
46.3 Viscometer Types
46.3.1 Rheometer
A viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For
liquids with viscosities, which vary with flow conditions, an instrument called a rheometer is used.
Viscometers only measure under one flow condition. The instruments for viscosity measurements
are designed to determine “a fluid’s resistance to flow,” a fluid property defined earlier as viscosity.
The fluid flow in a given instrument geometry defines the strain rates, and the corresponding stresses
are the measure of resistance to flow. If strain rate or stress is set and controlled, then the other one
will, everything else being the same, depend on the fluid viscosity. If the flow is simple (one dimensional, if possible) such that the strain rate and stress can be determined accurately from the measured
quantities, the absolute dynamic viscosity can be determined; otherwise, the relative viscosity will
be established. For example, the fluid flow can be set by dragging the fluid with a sliding or rotating
surface, or a body falling through the fluid, or by forcing the fluid (by external pressure or gravity) to
flow through a fixed geometry, such as a capillary tube, annulus, a slit (between two parallel plates), or
orifice. The corresponding resistance to flow is measured as the boundary force or torque or pressure
drop. The flow rate or efflux time represents the fluid flow for a set flow resistance, like pressure drop
or gravity force. The viscometers are classified, depending on how the flow is initiated or maintained,
as in Table 46.2.
The basic principle of all viscometers is to provide as simple flow kinematics as possible, preferably 1D
(isometric) flow, in order to determine the shear strain rate accurately, easily, and independent of fluid
type. The resistance to such flow is measured, and thereby the shearing stress is determined. The shear
viscosity is then easily found as the ratio between the shearing stress and the corresponding shear strain
rate. Practically, it is never possible to achieve desired 1D flow or ideal geometry, and a number of errors,
listed in Table 46.3, can occur and need to be accounted for [4–8]. A sample list of manufacturers/
distributors of commercial viscometers/rheometers is given in Table 46.4. There is no implied endorsement of any manufacturer on this list.
46.3.2 Rotational Methods/Concentric Cylinders
The main advantage of the rotational as compared to many other viscometers is its ability to operate
continuously at a given shear rate, so that other steady-state measurements can be conveniently performed. That way, time dependency, if any, can be detected and determined. Also, subsequent measurements can be made with the same instrument and sampled at different shear rates, temperature, etc.
For these and other reasons, rotational viscometers are among the most widely used class of instruments
for rheological measurements.
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Mechanical Variables
TABLE 46.2
Viscometer/Rheometer Classification and Basic Characteristics
Type/Geometry
Basic Characteristics/Comments
Drag flow types: Flow set by motion of instrument boundary/surface using external or gravity force
Rotating concentric cylinders (Couette)
Good for low viscosity, high shear rates; for R2/R1 ≅ 1, see Figure 46.3;
hard to clean thick fluids
Rotating cone and plate
Homogeneous shear, best for non-Newtonian fluids and normal
stresses; need good alignment, problems with loading
and evaporation
Rotating parallel disks
Similar to cone and plate but inhomogeneous shear; shear varies with
gap height, easy sample loading
Sliding parallel plates
Homogeneous shear, simple design, good for high viscosity; difficult
loading and gap control
Falling body (ball, cylinder)
Very simple, good for high temperature and pressure; need density
and special sensors for opaque fluids, not good for viscoelastic fluids
Rising bubble oscillating body
Similar to falling body viscometer; for transparent fluids
Oscillating body
Needs instrument constant, good for low-viscous liquid metals
Pressure flow types: Fluid set in motion in fixed instrument geometry by external or gravity pressure
Long capillary (Poiseuille flow)
Simple, very high shears and range but very inhomogeneous shear,
bad for time dependency, and is time consuming
Orifice/cup (short capillary)
Very simple, reliable but not for absolute viscosity and
non-Newtonian fluids
Slit (parallel plates) pressure flow
Similar to capillary but difficult to clean
Axial annulus pressure flow
Similar to capillary, better shear uniformity but more complex,
eccentricity problem and difficult to clean
Others/miscellaneous
Ultrasonic
Good for high-viscosity fluids, small sample volume, gives shear and
volume viscosity, and elastic property data; problems with surface
finish and alignment, complicated data reduction
Source: Adapted from Macosko, C. W., Rheology: Principles, Measurements, and Applications, VCH, New York, 1994.
TABLE 46.3
Different Causes of Viscometer/Rheometer Errors
Error/Effect
End/edge effect
Kinetic energy losses
Secondary flow
Nonideal geometry
Shear rate nonuniformity
Temperature variation and viscous heating
Turbulence
Surface tension
Elastic effects
Miscellaneous effects
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Cause/Comment
Energy losses at the fluid entrance and exit of main test
geometry
Loss of pressure to kinetic energy
Energy loss due to unwanted secondary flow, vortices,
etc.; increases with Reynolds number
Deviations from ideal shape, alignment, and finish
Important for non-Newtonian fluids
Variation in temperature, in time, and in space, influences
the measured viscosity
Partial and/or local turbulence often develops even at low
Reynolds numbers
Difference in interfacial tensions
Structural and fluid elastic effects
Depends on test specimen, melt fracture, thixotropy,
rheopexy
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46-7
Fluid Viscosity Measurement
TABLE 46.4 Viscometer/Rheometer Manufacturers and Distributors
(for Measurement at 1 atm)
A&D Engineering Co. Ltd
Ametek
Ankersmid Lab
Anton Parr
ATS
Avenisense
Bartec
Brookfield Eng. Labs, Inc.
BYK Instr.
Cannon Inst. Co.
Ceramic Instr.
Cole-Palmer Inst. Co.
Dong Jin Inst. Corp.
Fungilab
Galvanic Applied Sciences, Inc.
Gottfert Werkstorf-Prufmaschinen GmbH
Hydramotion
Kinematica AG
Kittiwake
Koehler Inst. Co., Inc.
Lamy Rheology
Lauda
Lemis
Malvern
Marimex
Micro Motion
Nametre Co.
Norcross
Orion
Paar Physica USA, Inc.
PAC
Parker
Perten
Physica Messtechnik GmbH
REL
Reologica Instr.
Rheometric Scientific, Inc.
Rheotec
Santam Eng. Energy Co. Ltd
SI Analytics
So Fraser
Stanhope-Seta
T.A. Instruments, Inc.
Tanaka
Thermo Scientific
TMI
Toyo Seiki Seisaku-Sho Ltd.
Weschler Instr.
Note: All the previously mentioned manufacturers/distributors can be found on the Internet, along
with their most recent contact information, product description, and, in some cases, pricing. A variety
of products using different viscosity measuring methods are available from this list.
Concentric cylinder-type viscometers/rheometers are usually employed when absolute viscosity needs
to be determined, which in turn requires knowledge of well-defined shear rate and shear stress data.
Such instruments are available in different configurations and can be used for almost any fluid. There
are models for low and high shear rates. More complete discussion on concentric cylinder viscometers/
rheometers is given elsewhere [4–9]. In the Couette-type viscometer, the rotation of the outer cylinder, or cup, minimizes centrifugal forces, which cause Taylor vortices. The latter can be present in the
Searle-type viscometer when the inner cylinder, or bob, rotates.
Usually, the torque on the stationary cylinder and rotational velocity of the other cylinder are measured for determination of the shear stress and shear rate, respectively, which is needed for viscosity
calculation. Once the torque, T, is measured, it is simple to describe the fluid shear stress at any point
with radius r between the two cylinders, as shown in Figure 46.3:
t rq (r ) =
T
2pr 2Le
(46.5)
In Equation 46.5, L e = (L + L c) is the effective length of the cylinder at which the torque is measured. In
addition to the cylinder’s length L, it takes into account the end-effect correction L c [4–8].
For a narrow gap between the cylinders (β = R 2/R1 ≅ 1), regardless of the fluid type, the velocity profile
can be approximated as linear, and the shear rate within the gap will be uniform:
g (r ) ≅
WR
(R2 − R1)
(46.6)
where
Ω = (ω2 − ω1) is the relative rotational speed
R– = (R1 + R 2)/2 is the mean radius of the inner (1) and outer (2) cylinders
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46-8
Mechanical Variables
D2 = 2R2
T
Outer
cylinder
D1 = 2R1
L
Fluid
sample
z
r
Inner
cylinder
ω2
FIGURE 46.3 Concentric cylinders viscometer geometry.
Actually, the shear rate profile across the gap between the cylinders depends on the relative rotational
speed, radii, and the unknown fluid properties, which seems an “open-ended” enigma. The solution of
this complex problem is given elsewhere [4–8] in the form of an infinite series and requires the slope
of a logarithmic plot of T as a function of Ω in the range of interest. Note that for a stationary inner
cylinder (ω1 = 0), which is the usual case in practice, Ω becomes equal to ω2. However, there is a simpler
procedure [10] that has also been established by German standards [11]. For any fluid, including nonNewtonian fluids, there is a radius at which the shear rate is virtually independent of the fluid type for
a given Ω. This radius, being a function of geometry only, is called the representative radius, RR, and is
determined as the location corresponding to the so-called representative shear stress, τ R = (τ1 + τ2)/2, the
average of the stresses at the outer and inner cylinder interfaces with the fluid, that is,
1/ 2
⎧ [2b 2 ] ⎫
RR = R1 ⎨
2 ⎬
⎩[1 + b ]⎭
1/ 2
⎧ 2 ⎫
= R2 ⎨
2 ⎬
⎩[1 + b ]⎭
(46.7)
Since the shear rate at the representative radius is virtually independent on the fluid type (whether
Newtonian or non-Newtonian), the representative shear rate is simply calculated for Newtonian fluid
(n = 1) and r = RR, according to [10]:
⎧[b 2 + 1]⎫
g R = g r =RR = w 2 ⎨ 2
⎬
⎩[b − 1]⎭
(46.8)
The accuracy of the representative parameters depends on the geometry of the cylinders (β) and fluid
type (n).
It is shown in [10] that, for an unrealistically wide range of fluid types (0.35 < n < 3.5) and cylinder
geometries (β = 1–1.2), the maximum errors are less than 1%. Therefore, the error associated with the
representative parameter concept is virtually negligible for practical measurements.
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46-9
Fluid Viscosity Measurement
Finally, the (apparent) fluid viscosity is determined as the ratio between the shear stress and corresponding shear rate using Equations 46.5 through 46.8, as
h = hR =
t R ⎧ [b 2 − 1] ⎫ T ⎧ [b 2 − 1] ⎫ T
=⎨
=⎨
⎬
⎬
g R ⎩[4pb 2R12Le ]⎭ w 2 ⎩[4pR22Le ]⎭ w 2
(46.9)
For a given cylinder geometry (β, R 2, and L e), the viscosity can be determined from Equation 46.9 by
measuring T and ω2.
As already mentioned, in Couette-type viscometers, the Taylor vortices within the gap are virtually
eliminated. However, vortices at the bottom can be present, and their influence becomes important
when the Reynolds number reaches the value of unity [10,11]. Furthermore, flow instability and turbulence will develop when the Reynolds number reaches values of 103–104. The Reynolds number, Re, for
the flow between concentric cylinders is defined [11] as
⎧[rw 2R12 ]⎫ 2
Re = ⎨
⎬[b − 1]
⎩ 2h ⎭
(46.10)
46.3.3 Cone-and-Plate Viscometers
The simple cone-and-plate viscometer geometry provides a uniform rate of shear and direct measurements of the first normal stress difference. It is the most popular instrument for measurement of nonNewtonian fluid properties. The working shear stress and shear strain rate equations can be easily
derived in spherical coordinates, as indicated by the geometry in Figure 46.4, and are, respectively,
t qf =
3T
[2pR 3 ]
(46.11)
and
W
g =
q0
(46.12)
θ
Ω
Cone
θ0
Fluid
sample
R
Plate
T
FIGURE 46.4
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Cone-and-plate viscometer geometry.
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46-10
Mechanical Variables
where R and θ 0 < 0.1 rad (≈6°) are the cone radius and angle, respectively. The viscosity is then easily
calculated as
h=
t qf [3Tq 0 ]
=
g
2pW R 3
(46.13)
Inertia and secondary flow increase, while shear heating decreases the measured torque (Tm). For more
details, see [4,5], and the torque correction is given as
Tm
= 1 + 6 ⋅ 10 −4 Re2
T
(46.14)
where
Re =
{r[Wq R] }
0
2
(46.15)
h
46.3.4 Parallel Disks
This geometry (Figure 46.5), which consists of a disk rotating in a cylindrical cavity, is similar to the
cone-and-plate geometry, and many instruments permit the use of either one. However, the shear rate
is no longer uniform but depends on radial distance from the axis of rotation and on the gap h, that is,
rW
g (r ) =
h
(46.16)
For Newtonian fluids, after integration over the disk area, the torque can be expressed as a function of
viscosity, so that the latter can be determined as
h=
2Th
[p WR 4 ]
(46.17)
R
Ω, θ
Upper
plate
Fluid
sample
h
Lower
plate
T
z
r
FIGURE 46.5
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Parallel disks viscometer geometry.
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46-11
Fluid Viscosity Measurement
46.4 Pressure and Gravity Flow Methods
46.4.1 Capillary Viscometers
The capillary viscometer is based on the fully developed laminar tube flow theory (Hagen–Poiseuille
flow) and is shown in Figure 46.6. The capillary tube length is many times larger than its small diameter,
so that entrance flow is neglected or accounted for in more accurate measurement or for shorter tubes.
The expression for the shear stress at the wall is
⎡ ΔP ⎤ ⎡ D ⎤
⋅
tw = ⎢
⎣ L ⎥⎦ ⎢⎣ 4 ⎥⎦
(46.18)
and
ΔP = (P1 − P2 ) + (z 1 − z 2 ) −
[CrV 2 ]
2
(46.19)
where C ≅ 1.1, P, z, V = 4Q/[πD2], and Q are correction factor, pressure, elevation, the mean flow velocity,
and the fluid volume flow rate, respectively. The subscripts 1 and 2 refer to the inlet and outlet, respectively.
The expression for the shear rate at the wall is
⎧[3n + 1]⎫ ⎧ 8V ⎫
g = ⎨
⎬⋅⎨ ⎬
⎩ 4n ⎭ ⎩ D ⎭
(46.20)
Sample
reservoir
Fluid
sample
Inlet
P1, z1
z
r
Capillary
tube
L
V
Outlet
P2, z2
D
FIGURE 46.6 Capillary viscometer geometry.
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46-12
Mechanical Variables
where n = d[logτw]/d[log(8V/D)] is the slope of the measured log(τw) – log(8V/D) curve. Then, the viscosity is simply calculated as
h=
t w ⎧ 4n ⎫ ⎧ ΔPD 2 ⎫ ⎧ 4n ⎫ ⎧[ΔPD 4p]⎫
=⎨
⎬=⎨
⎬⋅⎨
⎬
⎬⋅⎨
g ⎩[3n + 1]⎭ ⎩[32LV ]⎭ ⎩[3n + 1]⎭ ⎩ [128QL] ⎭
(46.21)
Note that n = 1 for a Newtonian fluid, so the first term [4n/(3n + 1)] becomes unity and disappears from
the earlier equations. The advantages of capillary over rotational viscometers are low cost, high accuracy
(particularly with longer tubes), and the ability to achieve very high shear rates, even with high-viscosity
samples. The main disadvantages are high residence time and variation of shear across the flow, which
can change the structure of complex test fluids, as well as shear heating with high-viscosity samples.
46.4.2 Glass Capillary Viscometers
Glass capillary viscometers are very simple and inexpensive. Their geometry resembles a U-tube with at
least two reservoir bulbs connected to a capillary tube passage with inner diameter D. The fluid is drawn
up into one bulb reservoir of known volume, V0, between etched marks. The efflux time, ∆t, is measured
for that volume to flow through the capillary under gravity.
From Equation 46.21 and taking into account that V0 = (∆t)VD2 π/4 and ∆P = ρg(z1 − z2), the kinematic viscosity can be expressed as a function of the efflux time only, with the last term, K/∆t, added to
account for error correction, where K is a constant [7], that is,
n=
h ⎧ ⎡ 4n ⎤ [pg (z 1 − z 2)D 4] ⎫
⋅
=⎨
⎬ (Δt − K Δt)
128LVo
r ⎩ ⎢⎣ (3n + 1) ⎥⎦
⎭
(46.22)
Note that for a given capillary viscometer and n ≅ 1, the curl-bracketed term is a constant. The last
correction term is negligible for a large capillary tube ratio, L/D, where kinematic viscosity becomes
linearly proportional to measured efflux time. Various kinds of commercial glass capillary viscometers,
like Cannon–Fenske type or similar, can be purchased from scientific and/or supply stores. They are
the modified original Ostwald viscometer design in order to minimize certain undesirable effects, to
increase the viscosity range, or to meet specific requirements of the tested fluids, like opacity. Glass
capillary viscometers are often used for low-viscosity fluids.
46.4.3 Orifice/Cup, Short Capillary: Saybolt Viscometer
The principle of these viscometers is similar to glass capillary viscometers, except that the flow through
a short capillary (L/D ≪ 10) does not satisfy or even approximate the Hagen–Poiseuille, fully developed, pipe flow. The influences of entrance end effect and changing hydrostatic heads are considerable.
The efflux time reading, ∆t, represents relative viscosity for comparison purposes and is expressed as
“viscometer seconds,” like the Saybolt seconds or Engler seconds or degrees. Although the conversion
formula, similar to glass capillary viscometers, is used, the constants k and K in Equation 46.23 are
purely empirical and dependent on fluid types:
n=
h
K
= k Δt −
r
Δt
(46.23)
where k = 0.00226, 0.0216, 0.073 and K = 1.95, 0.60, 0.0631, for Saybolt universal (∆t < 100 s), Saybolt
Furol (∆t > 40 s), and Engler viscometers, respectively [12,13]. Due to their simplicity, reliability, and
low cost, these viscometers are widely used for Newtonian fluids, in the oil and other industries, where
simple correlations between the relative properties and desired results are needed. However, these viscometers are not suitable for absolute viscosity measurement nor for non-Newtonian fluids.
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46.5 Falling Body Methods
46.5.1 Falling Sphere
The falling sphere viscometer is one of the earliest and simplest methods to determine the absolute shear
viscosity of a Newtonian fluid. In this method, a sphere is allowed to fall freely a measured distance
through a viscous liquid medium, and its velocity is determined. The viscous drag of the falling sphere
results in the creation of a restraining force, F, described by Stokes’ law:
F = 6phrU
s t
(46.24)
where
rs is the radius of the sphere
Ut is the terminal velocity of the falling body
If a sphere of density ρ2 is falling through a fluid of density ρ1 in a container of infinite extent, then
by balancing Equation 46.24 with the net force of gravity and buoyancy exerted on a solid sphere, the
resulting equation of absolute viscosity is
h = 2grs2
(r2 − r1)
9U t
(46.25)
Equation 46.25 shows the relation between the viscosity of a fluid and the terminal velocity of a sphere
falling within it. Having a finite container volume necessitates the modification of Equation 46.25 to correct for effects on the velocity of the sphere due to its interaction with container walls (W) and ends (E).
Considering a cylindrical container of radius r and height H, the corrected form of Equation 46.25 can
be written as
h = 2grs2
(r2 − r1)W
(9U tE )
(46.26)
where
3
⎛r ⎞
⎛r ⎞
⎛r ⎞
W = 1 − 2.104 ⎜ s ⎟ + 2.09 ⎜ s ⎟ − 0.95 ⎜ s ⎟
⎝r⎠
⎝r⎠
⎝r⎠
⎛r ⎞
E = 1 + 3.3 ⎜ s ⎟
⎝H⎠
5
(46.27)
(46.28)
The wall correction was empirically derived [15] and is valid for 0.16 ≤ rs/r ≤ 0.32. Beyond this range,
the effects of container walls significantly impair the terminal velocity of the sphere, thus giving rise to
a false high-viscosity value.
Figure 46.7 is a schematic diagram of the falling sphere method and demonstrates the attraction of
this method—its simplicity of design. The simplest and most cost-effective approach in applying this
method to transparent liquids would be to use a sufficiently large graduated cylinder filled with the liquid. With a distance marked on the cylinder near the axial and radial center (the region least influenced
by the container walls and ends), a sphere (such as a ball bearing or a material that is nonreactive with
the liquid) with a known density and sized to within the bounds of the container correction free falls the
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Mechanical Variables
ρ1
rs
ρ2
H
d
Ut
2r
FIGURE 46.7 Schematic diagram of the falling sphere viscometer. Visual observations of the time taken for the
sphere to traverse the distance d are used to determine a velocity of the sphere. The calculated velocity is then used
in Equation 46.24 to determine a shear viscosity.
length of the cylinder. As the sphere passes through the marked region of length d at its terminal velocity, a measure of the time taken to traverse this distance allows the velocity of the sphere to be calculated.
Using Equation 46.26, the shear viscosity of the liquid can be determined.
This method is useful for liquids with viscosities between 10−3 and 105 Pa · s.
46.5.2 Falling Cylinder
The falling cylinder method is similar in concept to the falling sphere method except that a flat-ended,
solid circular cylinder freely falls vertically in the direction of its longitudinal axis through a liquid sample within a cylindrical container. A schematic diagram of the configuration is shown in Figure 46.8.
Taking an infinitely long cylinder of density ρ2 and radius rc falling through a Newtonian fluid of density
ρ1 with infinite extent, the resulting shear viscosity of the fluid is given as
h = grc2
(r2 − r1)
2U t
(46.29)
Just as with the falling sphere, a finite container volume necessitates modifying Equation 46.29 to
account for the effects of container walls and ends. A correction for container wall effects can be analytically deduced by balancing the buoyancy and gravitational forces on the cylinder, of length L, with the
shear force on the sides and the compressional force on the cylinder’s leading end and the tensile force
on the cylinder’s trailing end. The resulting correction term, or geometrical factor, G(k) (where k = rc/r),
depends on the cylinder radius and the container radius, r, and is given by
G (k ) =
[k 2 (1 − ln k ) − (1 + ln k )]
(1 + k 2 )
(46.30)
Unlike the fluid flow around a falling sphere, the fluid motion around a falling flat-ended cylinder
is very complex. The effects of container ends are minimized by creating a small gap between the
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ρ2
L
ρ1
rc
d
Ut
2r
FIGURE 46.8 Schematic diagram of the falling cylinder viscometer. Using the same principle as the falling
sphere, the velocity of the cylinder is obtained, which is needed to determine the shear viscosity of the fluid.
cylinder and the container wall. If a long cylinder (here, a cylinder is considered long if ψ ≥ 10,
where ψ = L/r) with a radius nearly as large as the radius of the container is used, then the effects of
the walls would dominate, thereby reducing the end effects to a second-order effect. A major drawback with this approach is, however, if the cylinder and container are not concentric, the resulting
inhomogeneous wall shear force would cause the downward motion of the cylinder to become
eccentric. The potential for misalignment motivated the recently obtained analytical solution to
the fluid flow about the cylinder ends [16]. An analytical expression for the end correction factor
(ECF) was then deduced [17] and is given as
1
⎛ 8k ⎞ ⎛ G(k ) ⎞
= 1+ ⎜
⎝ pC w ⎟⎠ ⎜⎝ y ⎟⎠
ECF
(46.31)
where Cw = 1.003852 − 1.961019k + 0.9570952k2. Cw was derived semiempirically [17] as a disk wall correction factor. This is based on the idea that the drag force on the ends of the cylinder can be described
by the drag force on a disk. Equation 46.31 is valid for ψ ≤ 30 and agrees with the empirically derived
correction [16] to within 0.6%.
With wall and end effects taken into consideration, the working formula to determine the shear viscosity of a Newtonian fluid from a falling cylinder viscometer is
h=
[grc2 (r2 − r1)G(k )]
(2U t /ECF)
(46.32)
In the past, this method was primarily used as a method to determine relative viscosities between transparent fluids. It has only been since the introduction of the ECF [16,17] that this method could be rigorously used as an absolute viscosity method. With a properly designed container and cylinder, this
method is now able to provide accurate absolute viscosities from 10 −3 to 107 Pa · s.
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46.5.3 Falling Methods in Opaque Liquids
The falling body methods described earlier have been extensively applied to transparent liquids where
optical (often visual) observation of the falling body is possible. For opaque liquids, however, falling
body methods require the use of some sensing technique to determine, often in situ, the position of the
falling body with respect to time. Techniques have varied, but they all have in common the ability to
detect the body as it moves past the sensor. A recent study at high pressure [18] demonstrated that the
contrast in electric conductivity between a sphere and opaque liquid could be exploited to dynamically
sense the moving sphere if suitably placed electrodes penetrated the container walls as shown in the
schematic diagram in Figure 46.9. References to other similar in situ techniques are given in [18].
46.5.4 Rising Bubble/Droplet
For many industrial processes, the rising bubble viscometer has been used as a method of comparing the
relative viscosities of transparent liquids (such as varnish, lacquer, and beer) for decades. Although its
use was widespread, the actual behavior of the bubble in a viscous liquid was not well understood until
long after the method was introduced [19]. The rising bubble method has been thought of as a derivative
of the falling sphere method; however, there are fundamental physical differences between the two. The
major physical differences are as follows: (1) The density of the bubble is less that of the surrounding
liquid, the compressibility of the bubble leads to a change in bubble volume depending on its rise position in the fluid column, and (2) the bubble itself has some unique viscosity. Each of these differences
Electrodes
d
Resistance
Passage of sphere through
the electrode planes
Time
FIGURE 46.9 Diagram of one type of apparatus used to determine the viscosity of opaque liquids in situ. The
electrical signal from the passage of the falling sphere indicates the time to traverse a known distance (d) between
the two sensors.
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can, and do, lead to significant and extremely complex rheological problems that have yet to be fully
explained. If a bubble of gas or droplet of liquid with a radius, r b, and density, ρ′, is freely rising in some
enclosing viscous liquid of density ρ, then the shear viscosity of the enclosing liquid is determined by
⎛ 1 ⎞ [2grb (r − r ′)]
h=⎜ ⎟
⎝e⎠
9U t
2
(46.33)
where
e=
(2h + 3h′)
3(h + h′)
(46.34)
where η′ is the viscosity of the bubble. It must be noted that when η′ ≫ η, ε (as for solid spheres in a
fluid), ε = 1, which reduces Equation 46.33 to 46.25. For η′ ≪ η (as for gas bubbles in a fluid), ε becomes
2/3, and the viscosity calculated by Equation 46.33 is 1.5 times greater than the viscosity calculated by
Equation 46.25.
During the rise, great care must be taken to avoid contamination of the bubble and its surface with
impurities in the surrounding liquid. Impurities can diffuse through the surface of the bubble and
combine with the fluid inside. Because the bubble has a low viscosity, the upward motion in a viscous
medium induces a drag on the bubble that is responsible for generating a circulatory motion within it.
This motion can efficiently distribute impurities throughout the whole of the bubble, thereby changing its viscosity and density. Impurities left on the surface of the bubble can form a “skin” that can
significantly affect the rise of the bubble, as the skin layer has its own density and viscosity that are
not included in Equation 46.33. These surface impurities also make a significant contribution to the
inhomogeneous distribution of interfacial tension forces. A balance of these forces is crucial for the
formation of a spherical bubble. The simplest method to minimize the earlier effects is to employ minute
bubbles by introducing a specific volume of fluid (gas or liquid), with a syringe or other similar device,
at the lower end of the cylindrical container. Very small bubbles behave like solid spheres, which make
interfacial tension forces and internal fluid motion negligible.
In all rising bubble viscometers, the bubble is assumed to be spherical. Experimental studies of the
shapes of freely rising gas bubbles in a container of finite extent [20] have shown that (to 1% accuracy) a
bubble will form and retain a spherical shape if the ratio of the radius of the bubble to the radius of the
confining cylindrical container is less than 0.2. These studies have also demonstrated that the effect of
the wall on the terminal velocity of a rising spherical bubble is to cause a large decrease (up to 39%) in
the observed velocity compared to the velocity measured within an unbounded medium. This implies
that the walls of the container influence the velocity of the rising bubble more significantly than its
geometry. In this method, end effects are known to be large. However, a rigorous, analytically or empirically derived correction factor has not yet appeared. To circumvent this, the ratio of container length to
sphere diameter must be in the range of 10–100. As in other Stokesian methods, this allows the bubble’s
velocity to be measured at locations that experience negligible end effects.
Considering all of the earlier complications, the use of minute bubbles is the best approach to ensure a
viscosity measurement that is least affected by both the liquid to be investigated and the container geometry.
46.6 Oscillation Methods
Oscillation methods are based on the measurement of viscous damping of oscillation within a fluid
sample. Either the sample itself can be the wave carrying medium (acoustic method) or a separate
oscillator can be used in forced oscillation or free decay mode (vibrating wire, transducer, or tuning
fork methods). If a liquid is contained within a vessel suspended by some torsional system that is set
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Mechanical Variables
in oscillation about its vertical axis, then the motion of the vessel will experience a gradual damping.
In an ideal situation, the damping of the motion of the vessel arises purely as a result of the viscous
coupling of the liquid to the vessel and the viscous coupling between annular layers in the liquid. In any
practical situation, there are also frictional losses within the mechanical components of the suspension
system that contribute to damping and must be accounted for in the analysis of the measurements.
From observations of the amplitudes and time periods of the resulting oscillations, a viscosity of the
liquid can be calculated. A schematic diagram of the basic setup of the method is shown in Figure
46.10. Following initial oscillatory excitation, a light source (such as a low-intensity laser) can be used
to measure the amplitudes and periods of the resulting oscillations by reflection of the mirror attached
to the suspension rod to give an accurate measure of the logarithmic decrement of the oscillations (δ)
and the periods (T).
Various working formulae have been derived that associate the oscillatory motion of a vessel of
radius r to the absolute viscosity of the liquid. The most reliable formula is the following equation for a
cylindrical vessel [21]:
2
⎡ Id ⎤ ⎡ 1 ⎤
h=⎢ 3
⎥
⎥ ⎢
⎣ (pr HZ ) ⎦ ⎣ prT ⎦
(46.35)
[(3/2) + (4r /pH)]1 [(3/8) + (9r /4H)]a 2
⎛ 1+ r ⎞
Z =⎜
a0 −
+
⎝ 4H ⎟⎠
p
2p2
(46.36)
where
⎛ pr ⎞
p=⎜
⎝ hT ⎟⎠
1/ 2
r
(46.37)
Suspension
fiber
Mirror
ρ
H
2r
FIGURE 46.10 Schematic diagram of the oscillating cup viscometer. Measurement of the logarithmic damping of
the amplitude and period of vessel oscillation is used to determine the absolute shear viscosity of the liquid.
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Fluid Viscosity Measurement
2
⎛ d ⎞ ⎛ 3d ⎞
a0 = 1 − ⎜ ⎟ − ⎜
⎝ 4p ⎠ ⎝ 32p 2 ⎟⎠
(46.38)
2
⎛ d ⎞ ⎛ d ⎞
a2 = 1 + ⎜ ⎟ + ⎜
⎝ 4p ⎠ ⎝ 32p 2 ⎟⎠
(46.39)
where
I is the mass moment of inertia of the suspended system
ρ is the density of the liquid
A more practical expression of Equation 46.35 is obtained by introducing a number of simplifications.
First, it is a reasonable assumption to consider δ to be small (on the order of 10−2 to 10−3). This reduces a0
and a2 to values of 1 and −1, respectively. Second, the effects of friction from the suspension system and
the surrounding atmosphere can be experimentally determined and contained within a single parameter or instrument constant, δ 0. This must then be subtracted from the measured δ. A common method
of obtaining δ 0 is to observe the logarithmic decrement of the system with an empty sample vessel and
subtract that value from the measured value of δ. With these modifications, Equation 46.35 becomes
1/ 2
3 /2
⎛ h⎞
⎛ h⎞ ⎤
(d − d 0 ) ⎡ ⎛ h ⎞
= ⎢A ⎜ ⎟ − B ⎜ ⎟ + C ⎜ ⎟ ⎥
r
⎝ r⎠
⎝ r⎠ ⎥
⎢⎣ ⎝ r ⎠
⎦
(46.40)
where
⎛ p 3 / 2 ⎞ ⎡ ⎛ r ⎞ 3 1/ 2 ⎤
A=⎜
⎟ Hr T ⎥
⎢1 + ⎜
⎝ I ⎟⎠ ⎣ ⎝ 4H ⎠
⎦
(46.41)
⎛ p ⎞ ⎡⎛ 3 ⎞ 4r ⎤ 2
B = ⎜ ⎟ ⎢⎜ ⎟ +
Hr T
⎝ I ⎠ ⎣⎝ 2 ⎠ pH ⎥⎦
(46.42)
⎛ p 1/2 ⎞ ⎡⎛ 3 ⎞ 9r ⎤
3 /2
C =⎜
⎥ HrT
⎢⎜ ⎟ +
⎝ 2I ⎟⎠ ⎣⎝ 8 ⎠ 4H ⎦
(46.43)
It has been noted [22] that the analytical form of Equation 46.40 needs an empirically derived, instrument-constant correction factor (ζ) in order to agree with experimentally measured values of η. The
discrepancy between the analytical form and the measured value arises as a result of the earlier assumptions. However, these assumptions are required as solutions of the differential equations of motion of
this system are nontrivial. The correction factor is dependent on the materials, dimensions, and densities
of each individual system but generally lies between the values of 1.0 and 1.08. The correction factor is
obtained by comparing viscosity values of calibration materials determined by an individual system (with
Equation 46.35) and viscosity values obtained by another reliable method such as the capillary method.
With the earlier considerations taken into account, the final working Roscoe’s formula for the absolute shear viscosity is
3 /2
⎡ ⎛ h ⎞ 1/ 2
⎛ h⎞
⎛ h⎞ ⎤
(d − d 0 )
⎢
= z A⎜ ⎟ −B⎜ ⎟ +C ⎜ ⎟ ⎥
r
⎝ r⎠
⎝ r⎠ ⎥
⎢⎣ ⎝ r ⎠
⎦
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Mechanical Variables
The oscillating cup method has been used and is best suited for use with low values of viscosity within the
range of 10−5 to 10−2 Pa · s. Its simple closed design and use at high temperatures have made this method
very popular for viscosity measurement of liquid metals.
Perhaps the simplest version of the oscillation type of instrument is the vibrating-wire viscometer.
Vibrating-wire viscometers are typically quite small and so require small quantities of fluid. They provide accurate measurements but may require calibration using a known fluid. They can be used in quite
extreme conditions: the vibrating-wire viscometer was first used to measure the viscosity of liquid
helium at temperatures around 2 K [23,24]. Related instruments have been used at pressures in the GPa
range [25]. A viscometer based on a vibrating tuning fork, which is capable of accurate measurements
over a wide viscosity range (A & D Engineering), is available commercially.
The theory of operation of the vibrating-wire viscometer has been described in detail in the literature
[26]. It is based on the phenomenon of resonance. A wire (or more generally a thin beam) fixed at both
ends will have a natural vibration frequency ω 0 determined by its length, physical properties, and tension. The vibration amplitude of the wire will be strongly peaked at that resonance frequency, and the
height and width of the resonant peak depending on the magnitude of the forces that damp the wire’s
motion. For a vibrating wire immersed in a fluid, the width of the peak ∆ω or equivalently the quality
factor Q = ω 0/∆ω thus depends on viscosity, with higher viscosity leading to a smaller, broader peak
and a lower Q. In a typical instrument, the vibrating wire sits in a uniform magnetic field produced by
permanent magnets. A small alternating current of angular frequency ω is passed through the wire, and
the resulting Lorentz forces cause the wire to vibrate at the same frequency. The voltage across the wire
will consist of a constant contribution due to the electrical impedance of the system plus an induced
emf that is proportional to the amplitude of the vibration. Vibrating-wire viscometers can operate in a
forced mode, in which the wire is driven over a range of frequencies and the resonant peak measured
directly, or in a transient mode, in which Q is determined from measurements of the decay rate of the
oscillations following a short pulse of the driving current. In some configurations, the tension on the
vibrating wire is applied by a mass suspended in the fluid. In this case, the instrument can be used to
simultaneously measure fluid density, since the tension and so the resonant frequency will depend on
the buoyant force acting on the mass.
Several recent publications describe miniaturized versions of vibrating-object viscometers. Such
instruments are of interest in applications ranging from the need for simple bedside measurements of
the viscosity of blood, in which the quantity of fluid used must be minimized, to in situ measurement
of viscosity in oil field reservoirs, in which the viscometer must withstand high temperatures and pressures. Very small vibrating beam and vibrating plate viscometers have been fabricated using microelectromechanical system (MEMS) technology [27,28]. Smith et al. [29] describe a MEMS oscillating plate
viscometer for human blood measurements. Dehestru et al. [30] discuss a vibrating-wire viscometer
designed for oil field use. It has an internal volume of a few μL and is accurate to ±10% at pressures up
to 24 MPa and temperatures up to 175°C.
46.7 Acoustic Methods
Viscosity plays an important role in the absorption of energy of an acoustic wave traveling through a liquid.
By using ultrasonic waves (104 Hz < f < 108 Hz), the elastic, viscoelastic, and viscous response of a liquid
can be measured down to times as short as 10 ns. When the viscosity of the fluid is low, the resulting time
scale for structural relaxation is shorter than the ultrasonic wave period, and the fluid is probed in the
relaxed state. High-viscosity fluids subjected to ultrasonic wave trains respond as a stiff fluid because structural equilibration due to the acoustic perturbation does not go to completion before the next wave cycle.
Consequently, the fluid is said to be in an unrelaxed state that is characterized by dispersion (frequencydependent wave velocity) and elastic moduli that reflect a much stiffer liquid. The frequency dependence of
the viscosity relative to some reference viscosity (η0) at low frequency, η/η0, and of the absorption per wavelength, αλ, where α is the absorption coefficient of the liquid and λ is the wavelength of the compressional
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η/η0
Unrelaxed state
αλ
Relaxed state
1
ωτ
FIGURE 46.11 Effects of liquid relaxation (relaxation frequency corresponds to ωτ = 1 where ω = 2πf ) on relative
viscosity (upper) and absorption per wavelength (lower) in the relaxed elastic (ωτ < 1) and unrelaxed viscoelastic
(ωτ > 1) regimes.
Wave
generator
BR-1
Gate
Sample
wave, for a liquid with a single relaxation time, t, is shown in Figure 46.11. The maximum absorption per
wavelength occurs at the relaxation frequency when ωτ = 1 and is accompanied by a step in η/η0, as well as
in other properties such as velocity and compressibility. Depending on the application of the measured
properties, it is important to determine if the liquid is in a relaxed or unrelaxed state.
A schematic diagram of a typical apparatus for measuring viscosity by the ultrasonic method is
shown in Figure 46.12. Mechanical vibrations in a piezoelectric transducer travel down one of the buffer
rods (BR-1 in Figure 46.12) and into the liquid sample and are received by a similar transducer mounted
on the other buffer rod, BR-2. In the fixed buffer rod configuration, once steady-state conditions have
been reached, the applied signal is turned off quickly. The decay rate of the received and amplified signal,
BR-2
Amplifier
a
Amplitude
Amplitude
b
Time
Melt thickness
FIGURE 46.12 Schematic diagram of apparatus for liquid shear and volume viscosity determination by ultrasonic
wave attenuation measurement showing the received signal amplitude through the exit buffer rod (BR-2) using
(a) a fixed buffer rod (BR-2) configuration and (b) an interferometric technique with moveable buffer rod.
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Mechanical Variables
displayed on an oscilloscope on an amplitude versus time plot as shown in Figure 46.12a, gives a measure of α. The received amplitude decays as
A = A0e −(b +ac )t ′
(46.45)
where
A is the received decaying amplitude
A0 is the input amplitude
b is an apparatus constant that depends on other losses in the system such as due to the transducer
and container that can be evaluated by measuring the attenuation in a standard liquid
c is the compressional wave velocity of the liquid
t′ is time
At low frequencies, the absorption coefficient is expressed in terms of volume and shear viscosity:
4h ⎞ arc 3
⎛
h
+
v
⎜⎝
⎟=
3 ⎠ 2p 2 f 2
(46.46)
One of the earliest ultrasonic methods of measuring attenuation in liquids is based on acoustic interferometry [31]. Apart from the instrumentation needed to move and determine the position of one of the buffer
rods accurately, the experimental apparatus is essentially the same as for the fixed buffer rod configuration
[32]. The measurement, however, depends on the continuous acoustic wave interference of transmitted and
reflected waves within the sample melt as one of the buffer rods is moved away from the other rod. The
attenuation is characterized by the decay of the maxima amplitude as a function of melt thickness as shown
on the interferogram in Figure 46.12b. Determining a from the observed amplitude decrement involves
numerical solution to a system of equations characterizing complex wave propagation [33]. The ideal conditions represented in the theory do not account for conditions such as wave front curvature, buffer rod end
nonparallelism, surface roughness, and misalignment. These problems can be addressed in the amplitude
fitting stage, but they can be difficult to overcome. The interested reader is referred to [33] for further details.
Ultrasonic methods have not been and are not likely to become the mainstay of fluid viscosity determination simply because they are more technically complicated than conventional viscometry techniques. And although ultrasonic viscometry supplies additional related elastic property data, its niche
in viscometry is its capability of providing volume viscosity data. Since there is no other viscometer to
measure ηv, ultrasonic absorption measurements play a unique role in the study of volume viscosity.
46.8 Microrheology
On the microscopic scale, viscosity is related to the phenomenon of diffusion. A small (i.e., micron
sized) particle suspended in a viscous fluid such as water undergoes random motion as a result of thermal fluctuations in the forces acting on it. This is the well-known Brownian motion. Einstein showed in
a famous 1905 [34] paper that the mean squared displacement 〈r2〉 of a particle undergoing Brownian
motion is given by
r 2 (t ) = 2dDt,
(46.47)
where
τ is the time
d is the dimensionality of the motion
The angle brackets indicate an average over an ensemble of particles
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Fluid Viscosity Measurement
D is the diffusion constant, which is related to the viscosity η of the fluid by the Stokes–Einstein equation:
D=
k BT
6pha
(46.48)
Here
kB is the Boltzmann constant
T is the absolute temperature
If the radius a of the diffusing particles is known, these equations can be used to determine the viscosity
from measurements of their mean squared displacement. In a viscoelastic fluid, the effects of elasticity
make the mean squared displacement grow more slowly than linearly. In this case, Equations 46.47 and
46.48 can be generalized to relate the frequency-dependent viscous and elastic moduli to 〈r2〉. This is
discussed in detail in Refs. [35–39].
Several techniques collectively referred to as microrheology have been developed to make use of
this relationship to measure viscosity or, more generally, the viscoelastic properties of fluids. The most
commonly used of these will be discussed briefly in the following. The development and application of
microrheological techniques have been discussed in a number of review papers [35,40]. Microrheology
can be used to measure the properties of very small quantities of fluid and so is useful for characterizing expensive or scarce fluids. On the other hand, while at least one commercial instrument is available, microrheology is still largely a research laboratory technique. For quantitative measurements of
the bulk viscous and elastic moduli, the fluids being tested must be homogeneous on the scale of the
particle motion, since, if the fluid has small-scale structure, the local viscoelastic properties felt by the
tracer particles can differ dramatically from the bulk properties. This is not an issue for most Newtonian
fluids, but it can be for many complex fluids such as suspensions and gels.
46.8.1 Particle-Tracking Microrheology
Particle-tracking microrheology is based on direct imaging of small tracer particles undergoing
Brownian motion in a fluid. This limits its application to fluids that are transparent or at least sufficiently so that the positions of the tracer particles can be detected. The maximum viscosity that can be
measured with particle tracking is limited by the requirement that the motion of the tracers must be
large enough to be detectable. This limit depends on the optical resolution of the microscope and image
analysis procedure, as well as on the size of the tracers. For a typical experimental system, this limit is
on the order of tens of Pa . s.
Particle-tracking microrheology is done using a high-quality microscope fitted with a video camera
interfaced to a computer. Tracer particles are suspended in the fluid of interest, which is placed in a
sample holder and mounted on the stage of the microscope. Sample holders can be fabricated in any
volume of fluid, but 10–100 μL is typical. The microscope must be focused on the center of the holder to
avoid any influence of the walls on the motion of the tracer particles. The video camera records images
of the tracer particles at a set frame rate, which are stored on the computer, and image analysis software
is later used to reconstruct the particles’ trajectories and to calculate the mean squared displacement.
To obtain good statistics in the data, the trajectories of many particles are tracked simultaneously, and
the concentration of tracers in the fluid is typically chosen to give about 50 particles in the microscope’s
field of view.
The implementation details are discussed more fully elsewhere [35,40], but several points are worth
mentioning. The tracer particles should be close in density to the fluid of interest so that they do not
settle over the course of the measurement. Polystyrene latex spheres with a density of 1.05 g cm–3
can be obtained in a range of sizes and with a variety of chemical coatings and are typically used for
aqueous solutions. It is important to ensure that the tracer particles do not interact chemically or
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Mechanical Variables
electrostatically with the fluid being tested. Using bright-field microscopy, particles down to 0.5 μm
in diameter can be tracked, while fluorescence microscopy can be used to track fluorescently dyed
particles as small as 0.1 μm in size. The shortest time over which the mean squared displacement
can be measured is limited by the video frame rate and the longest time by computer memory or the
stability of the fluid.
The stored video images of the diffusing particles must be analyzed to extract the trajectory of each
particle and ultimately the mean squared displacement of the ensemble of particles. Several software
packages can be used to do this analysis. Particle-tracking software by Crocker and Grier [41] that
uses the IDL programming language is freely available [42] and has been used extensively. This
package has also been translated into MATLAB ®, and other image analysis software such as ImageJ
could also be used. The image analysis involves filtering the images to remove background intensity
variations, identifying candidate particles by thresholding the images, determining their positions
to subpixel accuracy from the centroid of their brightness distribution, and finally reconstructing
particle trajectories by matching particles from one image to the next. Once the trajectories of all
recorded particles have been determined, their squared displacements are calculated and averaged.
The viscosity, or the viscous and elastic moduli in the case of a non-Newtonian fluid, can then be
calculated. Typically measurement uncertainties limit the observable mean squared displacement to
about 10 −4 μm 2 .
An individual tracer particle moving in a viscoelastic fluid interacts hydrodynamically with other
tracer particles. Two-particle microrheology exploits this fact by using correlations between the motions
of pairs of particles to obtain viscoelastic properties of the fluid on the scale of the separation between
the particles [43]. Two-particle microrheology gives moduli that are independent of the interactions
between the tracer particles and the material under study [43] and so gives the bulk properties even
in microscopically inhomogeneous fluids. Two-particle microrheology requires the same equipment
as described earlier. The difference comes in the data analysis, which, since it involves analyzing the
motion of pairs of particles, requires much more data than in single-particle measurements. This technique has been used to study length-scale-dependent rheology in inhomogeneous materials such as
actin networks [43], but for now it remains a research tool rather than a practical technique for routine
measurements of material properties.
46.8.2 Dynamic Light Scattering
A second microrheological technique uses dynamic light scattering to measure the mean squared displacement of tracer particles in the fluid. In this technique, incident laser light is scattered by particles
suspended in the fluid of interest. A photomultiplier mounted at an angle θ with respect to the incident
beam measures the scattered light intensity, which fluctuates in time due to the random motion of the
scattering particles. A digital correlator calculates the autocorrelation function of the scattered intensity, g2(τ), where τ is the time and g2 is related to g1, the autocorrelation function of the scattered electric
field, which decays in time as
g 1(t ) = e
−(q 2 /6) r 2 (t )
,
(46.49)
where q = (4πn/λ)sin(θ/2), n is the refractive index of the fluid, and λ is the wavelength of the laser
light. The decay rate of g 1(τ) thus gives the mean squared displacement of the tracer particles, from
which the viscosity or viscous and elastic moduli can be calculated using Equations 46.47 and
46.48, as shown earlier. An example of data obtained using this technique is shown in Figure 46.13,
where the mean squared displacement is plotted as a function of time for 110 nm polystyrene
spheres in water.
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Fluid Viscosity Measurement
10–1
<r 2> (μm2)
10–2
10–3
10–4
10–5
10–4
Time (s)
10–3
FIGURE 46.13 Mean squared displacement of 100 nm polystyrene sphere diffusion in water, measured using
dynamic light scattering. These data can be used to obtain the diffusion constant, from which the viscosity of the
fluid can be determined. (From Yang, N. and deBruyn, J.R., unpublished.)
A range of light scattering instruments, including laser, photodetector, and correlator, are available
commercially, although many “black-box” scattering systems are intended for particle size determination assuming a known viscosity. The tracer particles must again be neutrally buoyant in the fluid under
study, and their concentration must be low enough that the laser light is scattered at most once as it
passes through the sample holder. Typically a few milliliters of fluid is required, and the sample must
again be transparent. Light scattering microrheology measures the viscoelastic properties of fluids at
much higher frequencies than particle tracking, which is useful for characterizing complex fluids such
as polymer solutions.
A related microrheological technique uses diffusing-wave spectroscopy. This is also a light scattering
technique but requires the light to be scattered many times in the sample, so that the light effectively diffuses through it. This requires a higher concentration of tracer particles. As shown earlier, the decay rate
of the electric field autocorrelation function is related to the mean squared displacement of the tracer
particles, although the equations involved are slightly more complicated. Details are given in a number
of references. Diffusing-wave spectroscopy allows measurements at even higher frequencies than the
single-scattering technique. An instrument that does microrheological measurements based on this
technique is available commercially.
46.9 High-Pressure Rheology
Measurement of viscosity of fluids at high pressures began with Bridgman [45]. Its primary interest
was to investigate liquids that would remain fluid at high pressures so they could be used as hydrostatic pressure-transmitting media in his experiments. Since that time, continued development of
high-pressure viscometry methods remained in the scientific research realm until recently. Over
the past decade, there has been increasing commercial and industrial interest in the application of
high-pressure technology in the processing of fluids. A wide range of foods, for example, are now
being processed using high-pressure technology principally as a nonthermal alternative for extending
shelf life. The technology also provides the food industry with a variety of new product development
opportunities able to exploit the functional properties of ingredients such as hydrocolloids and proteins. High-pressure technology is proving increasingly attractive and beneficial in the destruction
of microorganisms; activation and deactivation of enzymes; inactivation kinetics of both vegetative
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Mechanical Variables
and pathogenic microorganisms; change of functional properties of biopolymers such as proteins
and polysaccharides used in foams, gels, and emulsions; and the control of phase change such as fat
solidification and ice melting point. The rheology of these fluids and soft solids is important to control
process function and speed. In other applications such as in the automotive industry, understanding
the high-pressure rheology of petroleum-based fuels is critical for the safe and efficient design and
performance of engines. High-pressure rheology has also been recognized as vital information for
studies of planetary interiors, where fluids of mainly silicate or metallic composition play significant
roles in heat and mass flow and therefore fluid rheology is a pivotal parameter in controlling planetary
evolution.
46.9.1 High-Pressure Rheometry
Rheometry methods employed at high pressure are mainly based on traditional methods and can be
grouped into four categories: concentric cylinder, capillary, oscillatory, and falling/rolling body. The
theoretical treatment given earlier for each of these methods applies, but additional corrections may
be required for the container geometry effects due to the limited sample volumes imposed by the highpressure environment. The approaches to measure motion or stress in situ are often applied outside the
high-pressure environment because of the high loads required and can involve detailed experimental
setups. Also, depending on the pressure range, the equipment required to generate high pressure can
be much more elaborate than the equipment required to measure viscosity. Although there are several
high-pressure viscometers commercially available for use below a pressure of 1 GPa and examples are
listed in Table 46.5, measurement of viscosity at pressures above 1 GPa is carried out mainly in the
research laboratory.
Recent reviews of high-pressure viscosity measurement have appeared [46,47] and describe in
detail the pressure devices and methods used up to ∼1 GPa. This appears to be the pressure limit
of the concentric cylinder method because of the engineering challenges presented by the friction
effects in rotating high-pressure seals and the difficulties in precise measurement of torque when
parts are subjected to high mechanical loads. Most high-pressure rotating systems use the Searle
principle with a rotating inner cylinder and stationary outer cylinder. Methods of measuring the
resulting shear stress include elastic torsion tubes, magnetic or inductive coupling, strain gauge
load cell, or variable torque induction motor. The capillary method is based on pressure-driven
flow. The pressure drop along the capillary is normally measured with pressure transducers, which
must detect small pressure differences along the capillary on top of a large pressure signal arising from the static high-pressure environment. High-pressure oscillation methods are based on
the measurement of viscous damping of an oscillator located within a pressurized fluid sample.
Piezoelectric quartz crystals, which can operate at pressures exceeding 1 GPa, show a shift in the
crystal resonant frequency reflecting the mechanical shear impedance of the sample. A second way
to measure fluid viscosity is by observation of the free decay of a disk or wire-shaped oscillator.
TABLE 46.5
High-Pressure Viscometer/Rheometer Manufacturers
Manufacturers
PSL Systemtechnik
Cambridge Viscosity, Inc.
PAC
Chandler Engineering
Stony Brook Scientific, Inc.
F5 Technologie
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Maximum Pressure (Bars)
Method
1200
1379
2040
2721
4082
5000
Rotational
Oscillating piston/EM
Oscillating piston/EM
Rotational
Falling needle/magnetic
Torsional oscillation
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A third way to measure viscosity using the oscillation method at high pressure is by passage of
transducer-generated and transducer-detected ultrasonic waves through the pressurized fluid. The
viscosity is calculated via the wave absorption coefficient of the sample, which is measured by signal attenuation at the receiving transducer.
At pressures above 1 GPa, the falling/rolling body method is the preferred method although at
least one study of a capillary method employed to 6 GPa has been reported [48]. The simplicity and
reliability of construction of components for the Stokesian method make it capable of providing viscosity measurements at the highest pressures, where the sample fluid volume ranges from ∼10 −2 mm 3
to a few mm 3. In principle, viscosity can be measured with this method at pressures up to several 10’s
of GPa and up to an order of magnitude greater than this using the diamond anvil cell. At such high
pressures, high temperature is also required in order to melt the pressure-frozen liquid or because
this range of pressures is mainly of interest to the Earth and planetary sciences and the relevant materials to be studied have melting temperatures in excess of 1500°C. Such high temperatures require
judicious selection of the material for the falling/rolling body to ensure solid-phase stability of the
body at the test conditions, chemical inertness of the body in the fluid, appropriate density contrast
of the body and fluid, and ease of fabrication of the desired geometry. The velocimetry method used
depends on the body and fluid and on the observational access to the pressurized sample region.
Since the diamond anvil cell has the great advantage of optical access through the diamond anvils,
this is the method of choice for velocimetry of the moving body between two fiducial points. In
large-volume presses, which contain optically opaque components (usually ceramics and metals) in
the pressure cell, there are two main approaches to measuring the speed of the moving body. The first
involves allowing the body to move for a measured time in the pressurized fluid and then quenching
the temperature to freeze the fluid, thereby capturing the body in position. Follow-up measurement
of the body position is made by sectioning the recovered sample and measuring the distance the
body has moved. This is an effective method for measuring viscosities in the range of approximately
10 −1–105 Pa s if the density contrast of the body and fluid is chosen appropriately. A technique for
tailoring the density of a probe sphere, by cladding a metallic core sphere with a refractory mantle
to make a composite sphere [49], allows for adjustment of the fall time of the body to within acceptable ranges considering the fluid density and its expected viscosity. However, for measuring much
lower viscosities, the control on quenching the temperature and accurate determination of the body
start/stop position are inadequate, and therefore, in situ velocimetry methods are required. Many
in situ velocimetry methods have been devised based on the following with either in situ or ex situ
components for detection: electromagnetic induction using a two-coil external arrangement and an
electrically conducting body in a nonconducting fluid, detecting passage of an insulating body moving through two electrodes placed in a metallic fluid by measuring electrical resistance of the fluid,
detecting passage of a weakly radioactive body such as a 60Co sphere sensed by an external scintillation counter, and x-ray radiographic imaging of a body with sufficient absorption x-ray contrast
with the fluid. Over the last decade, the last method has become the method of choice for accurate
high-pressure viscometry at pressures above 1 GPa [50] on low-viscosity fluids and is described in
further detail.
Experiments to measure the viscosity of a fluid at high temperatures and at pressures above 1 GPa
are frequently carried out using the x-ray radiography imaging technique in real time at a synchrotron
facility. The high photon flux at a synchrotron is needed in order to produce images of sufficient quality and at high enough frequency (or ideally a movie) at various positions during travel of the body in
order to derive a velocity. The x-ray radiographic technique images the pressurized region according to
the relative x-ray attenuation coefficients of the experimental materials. A diagram of the experimental
imaging setup for a high-pressure Stokesian viscometry measurement using synchrotron x-ray radiography, the resultant sequence of time lapse images, and the sphere displacement and velocity data for a
falling composite sphere is shown in Figure 46.14.
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Mechanical Variables
Second stage anvil (WC)
CCD camera
First stage anvil
Slit
26 mm
White beam
Mirror
Fluorescence screen
High-pressure cell
(a)
5
1.6
4
1.2
3
0.8
2
0.4
Distance (mm)
Velocity (mm s–1)
(b)
1
0.0
0
0.0
(c)
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
FIGURE 46.14 (a) Synchrotron x-ray radiography setup for a high-pressure Stokesian viscometry measurement
[51]. (b) Synchrotron radiographic images of a Pt core—ruby mantle composite sphere falling in liquid Fe–29 wt% S at
2.6 GPa and 1373 K. Images are separated in time by 62 ms [52]. (c) Distance and velocity of falling composite sphere
versus time in Fe–29 wt% S liquid, at 2.6 GPa and 1373 K. Dashed line is terminal velocity [52].
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Further Information
Brooks, R. F., Dinsdale, A. T., and Quested, P. N., The measurement of viscosity of alloys—A review of
methods, data and models, Meas. Sci. Technol., 16, 354–362, 2005, a review of methods used to measure viscosity of liquid metals including methods adaptable to “in plant” measurement of fluid flow
in metallurgical industrial processes.
Hou, Y. Y. and Kassim, H. O., Instrument techniques for rheometry, Rev. Sci. Instrum., 76. 101101, 2005, a
review of some of the latest advances in rheology measuring techniques including new approaches
to conventional velocimetry (e.g., nuclear magnetic resonance, ultrasound pulse Doppler mapping)
as well as other techniques that are important for industrial processes (e.g., extensional rheometry
and interfacial rheometry).
O’Neill, M. E. and Chorlton, F., Viscous and Compressible Fluid Dynamics, Chichester, U.K.: Ellis Horwood,
1989, mathematical methods and techniques and theoretical description of flows of Newtonian
incompressible and ideal compressible fluids.
Ryan, M. P. and Blevins, J. Y. K., The viscosity of synthetic and natural silicate melts and glasses at high
temperatures and 1 bar (105 Pascals) pressure and at higher pressures, U.S. Geological Survey Bulletin
1764, Denver, CO, 1987, p. 563, an extensive compilation of viscosity data in tabular and graphic
format and the main techniques used to measure shear viscosity.
Van Wazer, J. R., Lyons, J. W., Kim, K. Y., and Colwell, R. E., Viscosity and Flow Measurement: A Laboratory
Handbook of Rheology, New York: Interscience Publishers, Division of John Wiley & Sons, 1963,
A comprehensive overview of viscometer types and simple laboratory measurements of viscosity
for liquids.
Author Queries
[AQ1] Please check if the given order of authors is correct.
[AQ2] Please provide the expansion of the acronym “IDL,” if appropriate.
[AQ3] Please update Refs. [13,44].
[AQ4] Please provide in-text citation for Ref. 14.
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