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Falling Object
Rheometry

11.1 INTRODUCTION

In a quest to develop simple and reliable methods for the measurement of
viscosity and other non-Newtonian fluid parameters, considerable effort has
been expended in examining and establishing the suitability of the flows pro-
duced by falling objects (sphere, needle, cylinder), rolling ball and rotating
sphere, vibrating sphere, etc. Over the years, it has been demonstrated convin-
cingly that while most of these devices yield reliable values of shear viscosities
for Newtonian fluids, their applicability to non-Newtonian fluids is severely
limited by the fact that all such flows are nonviscometric. Therefore, neither the
shear stress nor the shear rate are uniform, nor are these known a priori. How-
ever, most non-Newtonian fluids (except for visco-plastic and time-dependent
fluids) approach the Newtonian fluid behavior in the limit of vanishingly small
Reynolds and Deborah (or Weissenberg) numbers. It is thus possible to evaluate
some characteristics of visco-inelastic and visco-elastic fluids from suitable data
obtained from falling object tests under appropriate conditions. This chapter
presents an overview of the developments in this field. In particular, consider-
ation is given to the falling ball method, the falling cylinder method, and the
rolling ball method. Other less common methods such as rotating and vibrating
sphere devices, bubble viscometer, etc. are also mentioned briefly. In order to
maintain the consistency of the style of presentation, a short section on the use
of each device to measure the viscosity of Newtonian fluids is included which
serves as the background for the subsequent discussion for non-Newtonian
fluids.

11.2 FALLING BALL METHOD

11.2.1 N

EWTONIAN

LUIDS

The incompressible and steady flow generated by a sphere translating in an
infinite expanse of a fluid presents a viscosity measuring device which is simple
to fabricate and operate. The principle of the falling sphere viscometry is the

557

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Bubbles, Drops, and Particles in Fluids

well-known Stokes formula, written as

(ρ

− ρ)

18V

(11.1)

It is useful to recall here that the Stokes formula for drag is based on the assump-
tions of the creeping flow (small values of Reynolds number) and unbounded
domain (i.e., no wall and end effects). In practice, however, usually the terminal
settling velocity is measured in a cylindrical fall tube and thus the measured
velocity should be corrected for wall and end effects before calculating the value
of viscosity using Equation 11.1. The creeping flow assumption is verified by
calculating the value of the particle Reynolds number which should not exceed
about 0.1. Beyond this value, a correction for inertial effects is also needed.
While the end effects are known to be negligible (Maude, 1961; Tanner, 1963;
Flude and Daborn, 1982) as long as the terminal velocity is measured three to
four sphere diameters away from the top and bottom ends of the fall tube, the
corrections due to wall and inertial effects have been dealt with in

Chapter 10

and by others such as Sutterby, 1973a, 1973b; Dimova et al., 1999; Ben-Richou
et al., 2003. This technique has been used extensively for absolute measure-
ments for a range of materials including silicate melts (Kahle et al., 2003)
and of suspensions of small spheres and rods by treating them as a pseudo-
homogeneous fluid in comparison with the size of the falling ball (Milliken
et al., 1989a, 1989b; Harlen et al., 1999; Kaiser et al., 2004), though some
unusual phenomena can also be encountered in these systems (Kaiser et al.,
2004).

Naturally when dealing with highly viscous substances like epoxy resins,

greases, pastes, etc., the motive force due to gravity is very small in comparison
with the drag force and therefore the falling velocity of the ball will be extremely
small. This difficulty has been obviated by developing the so-called magnetic
sphere viscometer, in which the translational velocity of the sphere can be
regulated via the strength of the imposed magnetic field (Adam et al., 1984;
Sobczak, 1986; Gahleitner and Sobczak, 1987, 1988; Tran-Son-Tay et al., 1988,
1990; Hermann and Sobczak, 1989; Hilfiker et al., 1989). Linliu et al. (1994)
and Nyrkova et al. (1997) have also described the design and operation of a
centrifuge ball viscometer and tested its reliability for scores of Newtonian
liquids and polymer melts. On the other hand, Ringhofer and Sobczak (1997)
have attempted to relate the values of viscosity from a magneto-viscometer to
that of the melt indexer.

The aforementioned corrections due to inertial, wall and end effects are

necessary only when the absolute values of viscosity are needed. When only
a relative value is desired (or the falling ball device has been calibrated) for

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559

quality control purposes, Equation 11.1 can be simplified as

= k(ρ)t

(11.2)

where k is the system constant and t is the time of fall between two marks in
the viscometer tube. Obviously, under these conditions, it is expected that
the resulting values of the Reynolds number do not vary too much in the
range of operation, else the constant k will also depend upon the value of
the Reynolds number. The commercially available falling ball viscometers have
been described in detail by van Wazer et al. (1963) and Heywood (1985).

11.2.2 N

-N

EWTONIAN

LUIDS

As mentioned earlier, most time-independent (excluding visco-plastic) and
visco-elastic substances exhibit Newtonian flow behavior at very low shear
rates. The limiting value of the apparent viscosity in the low shear rate region
is termed the zero-shear viscosity, denoted by

. The range of shear rates over

which the transition from the zero-shear viscosity to the so-called power-law
region occurs is quite narrow for polymers of narrow molecular weight distri-
bution and vice versa, and it is also a function of the polymer concentration,
temperature and the type of solvent, etc. It is, however, not yet possible to predict
a priori the range of shear rate (or shear stress) for the onset of shear-thinning
behavior in a new application.

The determination of the zero-shear viscosity is of both fundamental and

pragmatic importance. Most rheological equations of state include this limiting
behavior as parameter and thus its determination by an independent method is
desirable rather than treating it as an adjustable parameter. From an engineering
standpoint, a knowledge of the zero-shear viscosity is also useful in the ana-
lysis of flow configurations which entail low shear rates and involve stagnation
points such as the flow around an isolated sphere or a cylinder and flow past
an assemblage of particles, etc. Unfortunately, it is generally not possible to
achieve sufficiently low shear rates in the conventional rotational and capillary
viscometers to measure the zero-shear viscosity directly, especially for dilute
polymer solutions. Admittedly, although some modern instruments are capable
of producing extremely low shear rates, the corresponding shear stress produced
is too small to be measured with the desirable level of accuracy. In this context,
the historic success of the falling ball method for measuring the viscosity of
Newtonian fluids has motivated several workers to explore the possibility of
extending this method to non-Newtonian systems. Likewise, one of the main
characteristics of visco-plastic media is the existence of a yield stress, and the
fluid deformation occurs only when the applied shear stress (due to the buoyant
weight) exceeds this value. Intuitively it appears that the falling ball method can

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Bubbles, Drops, and Particles in Fluids

be employed to measure the yield stress for a given system by continually chan-
ging the size/density of a sphere until the cessation or commencement of sphere
motion. Thus, the falling ball method has been employed to extract a range of
quantitative rheological information (including zero-shear and shear-dependent
viscosities, fluid characteristics time for visco-elastic fluids, yield stress, etc.)
as well as for qualitative quality control purposes in a spectrum of industrial
settings.

11.2.2.1 Zero-Shear Viscosity

While the value of

obtained from Equation 11.1 is the true viscosity of a

Newtonian fluid, the significance and meaning of this quantity (

) for a non-

Newtonian fluid is far from clear, as due to its shear-dependence, the viscosity
of the liquid is not uniform around the sphere surface. Strictly speaking, the
measurements carried out in the so-called second-order region only would yield
the true value of the zero-shear viscosity (Bird et al., 1987a). For a given liquid,
two or more spheres of different size and density can be used to establish whether
the measurements relate to the constant viscosity region or not. Admittedly, it is
possible, in principle at least, to achieve arbitrarily small values of shear rate by
varying the size and density of the sphere for a given liquid. However, owing to
the practical considerations, such as too small a size and impracticable values of
sphere density, it is not always possible to perform experiments in the constant
viscosity region. In view of these difficulties, it is not at all surprising to find an
abundance of extrapolation procedures for the evaluation of zero-shear viscosity
using falling sphere data. In essence, all such methods involve extrapolating the
value of viscosity, calculated using Equation 11.1, to zero-shear rate or zero-
shear stress conditions. Some of the procedures are based on sound theoretical
considerations while others are simply empirical in nature. Extensive studies
contrasting the performance and reliability of the different methods are available
in the literature (Subbaraman et al., 1971; Chhabra and Uhlherr, 1979; Gottlieb,
1979). Here, only a selection of the widely used extrapolation procedures is
included, and the resulting values of zero-shear viscosity,

are compared

with the directly measured values for six polymer solutions. The corresponding
falling sphere data obtained in these polymer solutions are free from the end,
wall and inertial effects.

11.2.2.1.1 Theoretical Extrapolation Methods
As discussed in

Chapter 3

and

Chapter 5,

numerous theoretical studies of

the slow non-Newtonian flow around a sphere are available. Some of these
(Caswell, 1962, 1970; Caswell and Schwarz, 1962; Giesekus, 1963) provide a
sort of theoretical framework for analyzing the falling sphere data with a view to
evaluate the zero-shear viscosity. However, the theory only identifies as to what

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Falling Object Rheometry

561

variables should be used for extrapolation to zero-shear rate or stress conditions
and does not provide any theoretical basis for the extrapolation per se.

1. Caswell method (1962): Based on a perturbation analysis for the creeping

motion of a third-order Rivlin–Ericksen fluid past a sphere, Caswell (1962)
presented the following expression:

= µ

(11.3)

In this equation, W (dimensionless) is the combined correction factor for the
wall and end effects and

is a combination of Rivlin–Ericksen fluid model

parameters. Evidently, this method involves plotting

against

(V/d)

and

the zero-shear viscosity,

is obtained as the intercept on the y-axis. How-

ever, experimental data often deviate from the linear dependence suggested
by Equation 11.3.

Figure 11.1

shows data for three polymer solutions plotted

in accordance with Equation 11.3; similar curvatures have been observed by
others (Subbaraman et al., 1971; Gottlieb, 1979). Obviously, the extrapolation
to

(V/d)

= 0 is not very meaningful in such instances. However, the data in

the region

(V/d)

< 0.1 s

−2

conform to the linear variation and the resulting

values of

are summarized in

Table 11.1.

2. Giesekus method (1963): In a fundamental study, Giesekus theoretically

investigated the simultaneous translation and rotation of a sphere in a four-
constant Oldroyd model fluid in the absence of inertial effects. For the case
of pure translation motion, his expression, in its corrected form (Bird et al.,
1987a), can be rearranged as

= µ

− {µ

2
1

φ(λ

)}

+ · · ·

(11.4)

where

, etc. are the Oldroyd model parameters and this analysis is applic-

able only for

(2λ

/d) 1. From the extrapolation view point, this method

also involves plotting

against

(V/d)

(if the higher-order terms are neglected)

whence it is equivalent to the Caswell’s method in this regard.

3. Modified Caswell–Schwarz method (1962): Based on the analysis of the

creeping flow of a Rivlin–Ericksen fluid, Caswell and Schwarz (1962) obtained
the following expression for viscosity

= µ

ρdV + C

+ · · ·

(11.5)

where C (dimensional) is a model parameter.

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Bubbles, Drops, and Particles in Fluids

0.02

0.04

0.1

0.2

0.3

a s)

0.5

(V /d )

–2

)

1.0

1.5

FIGURE 11.1 Determination of zero-shear viscosity using the method due to Caswell
(1962).

Turian (1967) simply rearranged Equation 11.5 and used

(µ

−

ρVd)

vs.

(V/d)

plots for extrapolating the falling sphere data to the zero-shear rate

condition. Though some minor numerical errors in Equation 11.5 have been
pointed out, the functional form is however, correct (Verma and Sacheti, 1975).

Under most conditions, the term

ρVd is negligible in comparison with

and thus this method is identical to method (1) above. For instance, the

largest value of

ρdV is only about 0.02% of µ

for the six polymer solutions

examined herein.

4. Caswell’s second method (1970): In a subsequent paper, Caswell (1970)

elucidated the extent of retardation exerted by cylindrical boundaries on the
creeping sphere motion in Rivlin–Ericksen model fluids. His final result can be
rearranged as

− C

+ O

(11.6)

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Falling Object Rheometry

563

TABLE 11.1

Comparison of True Zero-Shear Viscosities With the Values
Obtained by Extrapolation Procedures

Test ﬂuid number

Values of

(Pa s)

Actual value (Chhabra, 1980)

3.30

8.20

10.50

19.90

28.80

59.00

Caswell method (1962)

3.20

9.10

14.00

17.30

33.00

64.00

Caswell method (1970)

3.18

8.40

10.40

21.40

31.10

70.30

Williams method (1965)

3.46

11.00

12.75

25.00

44.00

90.00

Sutterby method (1966)

3.39

9.40

10.26

21.50

34.00

71.50

Cygan and Caswell method

(1971)

3.39

13.20

13.70

62.50

66.70

105.00

(1/µ

) vs. (V/d) method

3.23

9.40

12.50

23.80

37.00

83.30

Symbols used in

Figure 11.1

through

Figure 11.7

•

Note: All in water.

Measured using Weissenberg Rheogoniometer; (1) 1.46% Methocel, (2) 0.85% Separan,

(3) 0.75% Separan/0.75% CMC, (4) 1.25% Separan, (5) 1.63% Separan, (6) 2% Separan.

Source: From Chhabra, R.P. and Uhlherr, P.H.T., Rheol. Acta 18, 593 (1979).

where C

is a constant involving the model parameters. The data shown in

Figure 11.2

confirms the validity of Equation 11.6 provided the higher order

terms are neglected. Similar linear dependence has been reported by other
investigators also (Subbaraman et al., 1971; Broadbent and Mena, 1974). The
resulting values of the zero-shear viscosity (corresponding to

(µ

/d)

= 0)

are also listed in Table 11.1.

11.2.2.1.2 Empirical Extrapolation Methods
Most of the procedures in this category entail the extrapolation of either
the average Newtonian shear stress

(∼dρg) or shear rate (∼V/d) to

zero. A variety of combinations and coordinates have been employed for
extrapolating the falling sphere data to evaluate zero-shear viscosity.

1. Williams method (1965): This method involves plotting log

vs.

the maximum shear stress for a Newtonian fluid, that is,

(ρ

− ρ)g

and

= µ

(11.7)

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Bubbles, Drops, and Particles in Fluids

0.1

0.2

0.3

0.4

–

(Pa

)

x10

–2

x10

–2

x10

–2

x10

–3

x10

–3

FIGURE 11.2 Evaluation of zero-shear viscosity using the second method due to
Caswell (1970).

where f

, f

, respectively, are the correction factors for the end, wall, and

inertial effects, all of which are, however, redundant in the context of the
present results for the six polymer solutions and therefore

= µ

Figure 11.3

shows representative plots of

vs.

on semilogarithmic coordinates. Similar

pronounced curvatures in such plots have also been reported by Subbaraman
et al. (1971). However, the results for

< 35 Pa conform to Equation 11.7

and the resulting values of the zero-shear viscosity are included in

Table 11.1.

2. Sutterby method (1966): There is no reason why shear stress should be

used as the independent variable rather than the shear rate. Sutterby (1966) sug-
gested plotting

vs. (3V

/d) and the intercept on ordinate yields the value of the

zero-shear viscosity. This method has also been used by numerous other work-
ers (Torrest, 1982, 1983; Gahleitner and Sobczak, 1987, 1988). Though there
is no question that the average shear rate around a sphere may be expressed as
(

αV/d), considerable uncertainty exists regarding the exact value of α. Strictly

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Falling Object Rheometry

565

a s)

–1

(

× 10

–1

)

(Pa)

FIGURE 11.3 Evaluation of zero-shear viscosity by Williams method (1965).

speaking,

α is a function of the rheological parameters itself, for example, the

power-law flow behavior index (Hirota and Takada, 1959; Sato et al., 1966;
Uhlherr et al., 1976; Cho et al., 1984; Butcher and Irvine, 1990). Based on
the limited experiments in carboxymethyl cellulose and hydroxyethyl cellulose
solutions, Sato et al. (1966) proposed

α = 0.6 whereas some data (Gahleitner

and Sobczak, 1987) obtained for polypropylenes seems to suggest

α = 1; the

latter value is also in line with the theoretical estimates available in the literature
(EI Kayloubi et al., 1987). However, such an uncertainty in

α will only shift

the x-axis without affecting the intercept on the ordinate. In this method then,
the zero-shear viscosity is evaluated from log

vs.

(V/d) plots.

Figure 11.4

shows the representative data for five polymer solutions plotted in this manner;
appreciable curvatures are seen to be present. The linear variation of log

with

/d) is obtained only in the range (V/d) < 0.2 s

−1

and the resulting values

of the zero-shear viscosity are included in

Table 11.1.

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Bubbles, Drops, and Particles in Fluids

100

(V /d ) (s

–1

)

s (P

a s)

(

× 10)

–1

FIGURE 11.4 Evaluation of zero-shear viscosity using the extrapolation method due
to Sutterby (1966).

3. Cygan and Caswell method (1971): These investigators carried out a

systematic experimental study with a view to verify the validity of Equation 11.6
by gleaning falling sphere data for polyisobutylene (PIB) in toluene solutions.
Their data, however, did not conform to Equation 11.6. Instead, data resulted in
straight lines when the fluidity is plotted against

(µ

/d) rather than (µ

/d)

as predicted by the theory (Caswell, 1970).

Figure 11.5

supports this assertion

and the corresponding values of the zero-shear viscosity are listed in

Table 11.1.

In this case also, one can conceivably replace the apparent shear stress

(µ

/d)

by the apparent shear rate (V

/d). This is demonstrated in

Figure 11.6

and the

values of the zero-shear viscosity so obtained are summarized in Table 11.1.
Another method (Turian, 1967) uses extrapolating log

vs.

(µ

/d) plots,

but this method is equivalent to that of Williams.

An inspection of Table 11.1 clearly reveals that most extrapolation proced-

ures generally overestimate the value of the zero-shear viscosity by varying
amounts; the predictions of the Caswell method (1970) being the closest to
actual values. However, it is difficult to delineate a priori the most appropriate
range of conditions over which falling sphere data must be gleaned to obtain
reliable values of the zero-shear viscosity, as this will vary from one poly-
mer solution to another. Notwithstanding these limitations, this configuration

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Falling Object Rheometry

567

0.1

0.2

0.3

2.5

5.0

7.5

–

msV

(Pa)

FIGURE 11.5 Determination of zero-shear viscosity using the method of Cygan and
Caswell (1971).

0.1

0.2

0.3

0.4

(V/d ) (s

–1

)

(1/

) (Pa. s)

–

FIGURE 11.6 Evaluation of zero-shear viscosity by extrapolating (1/

) vs.

/d) data.

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Bubbles, Drops, and Particles in Fluids

does offer a simple device to estimate the value of the zero-shear viscosity for
analytical as well as comparative purposes.

The falling ball method has been used to study the pressure dependence of

the zero-shear viscosity of polypropylene melts (Foltz et al., 1978; Linliu et al.,
1994; Mattischek and Sobczak, 1997), evaluation of the leveling characteristics
of latex paints (Quach and Hansen, 1974) and for measuring the viscosity of
blood (Doffin et al., 1984) and other biological systems (Munzel and Schaub,
1961). In a series of papers, Sobczak and coworkers (Sobczak, 1986; Hermann
and Sobczak, 1989) and others (Gauthier-Manuel et al., 1984; Hilfiker et al.,
1989) have presented detailed descriptions of the various detection methods
used for timing the sphere descent as well as a new design based on the sphere
motion in magnetic fields to facilitate the measurement of unusually high vis-
cosities. On the other hand, Maclean-Fletcher and Pollard (1980) have used the
falling ball method to monitor the kinetics of the gelation of Acanthamoeba.

11.2.2.2 Shear-Dependent Viscosity

Few investigators (Hirota and Takada, 1959; Uhlherr et al., 1976; Cho et al.,
1984; Butcher and Irvine, 1990) have attempted to extend the range of utility
of the falling sphere configuration to determine shear stress–shear rate data for
purely viscous fluids. Owing to the nonviscometric nature of the flow around a
sphere, both shear stress and shear rate vary from point to point on the surface of
the sphere and therefore all such attempts rely on the use of the surface averaged
values of shear stress and shear rate. The first attempt at obtaining the shear
stress–shear rate data for pseudoplastic polymer solutions using the falling
ball method is due to Hirota and Takada (1959) who presented the following
definitions of the shear stress and shear rate, respectively.

1
6

(ρ

− ρ)g

(11.8)

˙γ

[1 + 2.5(δ − 2) − 0.75(δ − 2)

]

(11.9)

where

δ = (d log V)/(d log R).

Note that Equation 11.8 gives the maximum value of the shear stress pre-

vailing on the surface of a sphere falling in Newtonian media in the creeping
regime. Without giving sufficient details for their results to be recalculated,
Hirota and Takada (1959) concluded that the predictions of Equation 11.8 and
Equation 11.9 agreed very well with the corresponding capillary viscometer
data for six polymer solutions. Subsequently, both Uhlherr et al. (1976) and
Cho et al. (1984) have outlined schemes for the evaluation of shear stress and

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569

shear rate data for power-law model fluids using falling sphere data. The method
of Uhlherr et al. (1976) assumes that the surface average shear stress for a sphere
is the same as that in a Newtonian fluid, that is,

1
9

(ρ

− ρ)

(11.10)

and the corresponding average shear rate is obtained, using the approximate
flow field used by Slattery (1962) to evaluate the drag of a sphere, as

˙γ

(1 + 4A)

(11.11)

where the values of A, a function of the flow behavior index n, are available
in their original paper (Slattery, 1962). Satisfactory agreement was reported
between the viscometric and falling ball results for two weakly pseudoplastic
(n

= 0.825 and n = 0.865) polymer solutions. However, not only is this method

restricted to a rather narrow range of the power-law index (1

≥ n ≥ 0.763) but

was also found to be unsuitable for visco-elastic fluids. Cho et al. (1984), on the
other hand, proposed corrections to both the average shear stress and shear rate
relevant for a sphere in Newtonian media. Based on the assumed velocity and
stress profiles (Wasserman and Slattery, 1964; Cho and Hartnett, 1983a,b), the
following semiempirical expressions for the upper and lower bounds for shear
stress and shear rate were derived as

= F

(n)

(ρ

− ρ)

(11.12)

where

(n)

= 0.2827 + 0.8744n + 0.4562n

− 0.7486n

(11.13a)

and

(n)

= 0.6388 + 0.6418n − 0.4344n

+ 0.1560n

(11.13b)

˙γ

= F

(n)

(11.14)

where

(n)

= −1.731 + 41.28n − 116n

+ 123.9n

− 46.72n

(11.15a)

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Bubbles, Drops, and Particles in Fluids

and

(n)

= −2.482 + 54.35n − 160.1n

+ 178.2n

− 69.04n

(11.15b)

and

µ =

˙γ

(11.16)

While in actual calculations, one should use the mean values of F

(n) and

(n), these expressions are retained here in terms of the upper and lower

bounds to give an idea about the variability of the predictions of this method.
Equation 11.12 to Equation 11.16 are applicable in the complete range of
power-law index, that is, 1

≥ n ≥ 0.1. However, their use necessitates

a prior knowledge of the value of the power-law index, n. Cho et al. (1984)
suggested that it can be evaluated from the slope of a log–log plot as
d log

{R(ρ

− ρ)}/d log(V/R).

It is well known that the power-law model does not predict the transition

from the zero-shear viscosity region to power-law region. In a subsequent
numerical study (Butcher and Irvine, 1990), this difficulty was obviated by
employing a three parameter truncated form of the power-law model. Again,
the average shear stress and shear rate were obtained in the form of corrections
to the corresponding Newtonian values. The correction factors, which depend
only upon the value of n, are available both in tabular and graphical forms in
the original paper (Butcher and Irvine, 1990).

Figure 11.7

shows a typical com-

parison between the predictions (Cho et al., 1984; Butcher and Irvine, 1990)
and viscometric measurements for three polymer solutions. An examination of
this figure shows that the method of Cho et al. (1984) always overpredicts the
values of viscosity, and the deviations between the falling sphere results and
viscometric measurements increase, as the shear rate decreases. This is not at all
surprising as the predictions assume a constant value of the power-law index, n,
to be applicable over the whole range, whereas strictly speaking it varies some-
what with shear rate. Despite these limitations, the predictions rarely seem to
differ from the viscometric data by more than 50% and hence this method does
offer an attractive alternative for measuring shear-dependent viscosities, at least
for qualitative comparison purposes. This approach also seems to work moder-
ately well for weakly visco-elastic polymer solutions (Kanchanalakshana and
Ghajar, 1986).

11.2.2.3 Yield Stress

As noted in

Chapter 4,

by virtue of its yield stress, a visco-plastic substance

in an unsheared state has the capacity to support the weight of an embedded

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571

0.1

Shear rate (s

–1

)

100

Viscometric data

100

Apparent viscosity (P

a s)

0.1

FIGURE 11.7 Typical comparison between the predicted and measured apparent vis-
cosity in the power-law region.

, ,

•

— Predictions of Butcher and Irvine (1990);

I — Predictions of Equation 11.12 through Equation 11.16.

particle for a sufficiently long time. This idea has been used to evaluate the
yield stress of visco-plastic media. This method hinges on the fact that for
a given medium, it is possible to find the heaviest ball which will not move
under its own weight. Obviously, the criterion to delineate whether a ball is
falling or not implicitly assumes a timescale of observation (Barnes and Walters,
1985). Notwithstanding this arbitrariness and subjectivity, since the sphere is
not moving, the only relevant forces are that due to its weight and due to the
yield stress of the medium. Many investigators (Maclean-Fletcher and Pollard,
1980; Hartnett and Hu, 1989; Schurz, 1990; Wunsch, 1990) have used this
configuration to evaluate the value of yield stress. While Wunsch (1990) made
use of the numerical results of Beris et al. (1985) to estimate the value of

, all

other methods are based on heuristic considerations.

However, much confusion exists regarding the calculation of the force due

to the yield stress, F

, acting on a sphere. For instance, Maclean-Fletcher and

Pollard (1980) and many others (such as Johnson, 1970) have approximated
it as

= C

πd

(ρ

− ρ) g

(11.17)

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572

Bubbles, Drops, and Particles in Fluids

where C

is a constant, the values ranging from 1.75 to 3.14 (Maclean-Fletcher

and Pollard, 1980; Chhabra and Uhlherr, 1988a) have been used in the liter-
ature. This uncertainty coupled with the practical difficulties of finding a ball
which will just remain suspended has led to the adaptations of the falling ball
method. For instance, Wunsch (1992, 1994) developed a method based on a
oscillating sphere and outlined a scheme which allows the evaluation of the
yield stress. Unlike Equation 11.17, the method of Wunsch (1994) is obviously
based on the choice of a specific fluid model. However, Wunsch (1994) reported
good agreement between the predicted and measured values for a few carbopol
solutions. Similarly, Singh et al. (1991) have outlined a method of evaluating
the fluid parameters of a Herschel–Bulkley fluid from the behavior of a sphere
moving in visco-plastic media subject to vibrations.

Uhlherr and coworkers (Uhlherr, 1986; Guo and Uhlherr, 1996) have

developed the so-called pendulum methods which can be used to measure
the yield stress simply from the equilibrium position of the pendulum (see
Figure 11.8). This method does not require a priori knowledge of the fluid
model. While initially Uhlherr (1986) introduced the sphere–pendulum method,
subsequently he argued that the deformation is more uniform around a cylinder
with hemispherical ends than that around a sphere (Guo and Uhlherr, 1996). By
writing a moment balance at equilibrium, it can easily be shown that for a given

Liquid

Air

Fg2

Fg1

FIGURE 11.8 Schematics of a cylinder pendulum. (Redrawn from Guo, J. and
Uhlherr, P.H.T., Proceedings of the XII International Congress on Rheology, p. 73,
Quebec City, PQ, Canada, 1996.)

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Falling Object Rheometry

573

system, the equilibrium angle (

θ) depends upon the depth of immersion, l, as

sin

θ = −A

+ B

+ C

(11.18)

From a knowledge of the intercept C

,one can readily extract the value of the

yield stress

+ m

(H/L) − ((π/4)d

+ (πd

/6))ρ]g sin θ

min

πdb + ((πd)/2)

(11.19)

where m

and m

are the masses of the cylinder and the plate-hook assembly,

respectively. Guo and Uhlherr (1996) reported a good match between the values
of

obtained using the pendulum and the well-established vane technique for

scores of polymer solutions and silica suspensions. Likewise, a plate pendulum
was successfully used for measuring the yield stress of fire fighting foams
(Gardiner et al., 1998).

In summary, this method does yield values of the yield stress (static) com-

parable to the vane technique, provided due care is taken to eliminate the wall
effects (large container), the end effects (by using several values of l), and the
slip effects. Notwithstanding the theoretical pitfalls, this method offers a simple
and convenient tool for comparative and quality control purposes when yield
stress is an important characteristic of the fluid.

11.2.2.4 Characteristic Time for Visco-Elastic Fluids

Some anomalous effects associated with the motion of spheres in concen-
trated polymer solutions have been observed which are attributed to the
visco-elasticity of polymer solutions. For instance, Walters and Tanner (1992)
presented photographic evidence of the initial oscillatory motion of a sphere
released from rest into a visco-elastic fluid before attaining its terminal fall-
ing velocity. Likewise, Cho and Hartnett (1979) and Cho et al. (1984) found
the terminal velocity of spheres to depend upon the time elapsed between two
successive experiments. In some cases, up to 30 min were required before the
original value of terminal velocity could be reproduced. The time-dependent
terminal velocity measurements were explained qualitatively by hypothesiz-
ing that the elastic energy produced by the sphere motion must be gradually
relaxed for the polymer solution to return to its undeformed virgin state. Though
these authors also suggested that this information can be used to evaluate the
fluid characteristic time, the available limited experimental results, however,
do not seem to support this contention (Tanner, 1964; Ambeskar and Mashelkar,
1990). Adam et al. (1984) have also outlined a method of evaluating the longest
relaxation time using a magnetic sphere rheometer.

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Bubbles, Drops, and Particles in Fluids

11.3 ROLLING BALL METHOD

11.3.1 N

EWTONIAN

LUIDS

Similar to the falling ball method, the velocity of a sphere rolling along the
wall of a cylindrical tube filled with a liquid and inclined at a fixed angle to
the horizontal gives a measure of the viscosity (Flowers, 1914). While the flow
field produced is quite complex (especially when the ball diameter is almost
equal to the tube diameter, as is common for this device), Hubbard and Brown
(1943a) used dimensional arguments to show that for a given arrangement, the
rolling velocity and the viscosity of a Newtonian fluid are linked as

µ =

(ρ

− ρ)

(11.20)

where V

is the rolling velocity of the ball. This expression has been used

successfully to measure the pressure and temperature dependent viscosity of
scores of Newtonian fluids (Hubbard and Brown, 1943b; Harrison and Gosser,
1965; Geils and Keezer, 1977; Medani and Hasan, 1977; Hasan, 1983). While
Equation 11.20 is really applicable when the streamline (laminar) flow condi-
tions exist, Hasan (1983) has provided the necessary correction for inertial
effects. Obviously, the instrument constant k

includes the system specif-

ics such as tube and ball diameter, acceleration due to gravity, the angle of
inclination, etc.

Lewis (1953) improved upon the treatment of Hubbard and Brown (1943a)

by considering it as one-dimensional pressure-driven flow in the constriction
formed between the tube wall and the surface of the ball. He presented the
following expression for viscosity:

µ =

πJ

g sin

(ρ

− ρ)

− d

2.5

(11.21)

where J

is a constant which equals 0.398. In contrast to Equation 11.20,

Equation 11.21 yields absolute values of shear viscosity and no calibration
is needed (Bagchi and Chhabra, 1991b). Commercially available rolling ball
viscometers have been discussed by van Wazer et al. (1963) and by Sherman
(1970).

11.3.2 N

-N

EWTONIAN

LUIDS

HEAR

-D

EPENDENT

ISCOSITY

)

In view of the exceedingly complex nature of the flow field in a rolling ball
device, little effort has been devoted to the extension of this method to measure

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Falling Object Rheometry

575

non-Newtonian characteristics. Bird and Turian (1964) extended the approach
of Lewis (1953) for power-law fluids. Subsequently, Sestak and Ambros (1973)
refined the analysis of Bird and Turian (1964) and also presented limited data to
demonstrate the applicability of their analysis for a few power-law fluids. Their
analysis is however restricted to

[(D − d)/d] 1, that is, for a closely fitting

ball. Without any justification, Stastna and Dekee (1987) argued that only

5
7

the buoyant weight of the sphere balances the resistance force. According to
Stastna and Dekee (1987), the power-law index n of a fluid can be evaluated
using the rolling velocity as a function of the angle of inclination as

log

(sin β

/ sin β

)

log

)

(11.22)

and the power-law consistency, m, is given by the following expression due to
Sestak and Ambros (1973)

(ρ

− ρ) ((D − d)/D)

(4n+1)/2

sin

[{(2n + 1)/n}πV

]

(11.23)

The values of J

for a range of values of n are available in the paper of Sestak

and Ambros (1973). In the limit of d

≈ D and n = 1, Equation 11.23 does

reduce approximately to Equation 11.21. Bagchi and Chhabra (1991b) reported
moderate agreement between the values of m and n obtained using this method
and those from a rotational viscometer for a few polymer solutions. Finally,
much confusion exists in the literature regarding the calculation of the mean
shear rate in the rolling ball device. For instance, Sestak and Ambros (1973)
derived the following expression for the average shear rate at the surface of
the ball

˙γ

2DV

(D − d)

(n + 1)(2n + 1)

(2 + n)(3n + 2)

(11.24)

whereas the corresponding expression due to Stastna and Dekee (1987) is
slightly different as

˙γ

πd

(D − d)

(n + 1)(2n + 1)

(n + 2)(3n + 2)

(11.25)

In the limit of n

= 1, Equation 11.24 yields for a Newtonian fluid, ˙γ

/(D − d)

whereas Equation 11.24 leads to a different result, as it does

not impose any restriction on the value of

[(D − d)/d]. Bryan and Silman

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576

Bubbles, Drops, and Particles in Fluids

(1990), on the other hand, simply postulated the shear rate to be propor-
tional to

∼(V

/d) and reported good agreement between the rolling ball

viscometer and Brookfield viscometer data for a few fermentation broths. Sim-
ilarly, following the early work of Scott Blair and Oosthuizen (1960), Briscoe
et al. (1992b, 1994) have postulated the mean shear rate to be given by

˙γ

4DV

− d

(11.26)

and they also reported good agreement between the rolling ball data and the
viscometric data for a few bentonite suspensions and polyox polymer solu-
tions. In essence, the approaches of Bryan and Silman (1990) and Briscoe
et al. (1992b, 1994) hinge on identifying appropriate shift factors for shear
stress–shear rate data from the rolling ball device to superimpose on to the
corresponding data from a viscometer. Undoubtedly, this simple approach is
probably quite adequate for comparative purposes, but is difficult to justify on
theoretical grounds for the shear rate itself is expected to be a function of the
power-law index, n.

11.3.3 Y

IELD

TRESS

By analogy with the falling ball and the pendulum techniques presented in
the preceding section, some attempts have been made for using the rolling
ball method to evaluate the yield stress of visco-plastic substances. Gruber et
al. (1973) and Schurz (1990) outlined the design and operation of a rolling
ball device in which a sphere rolled down a curved groove (a circular arc, see

Figure 11.9).

The underlying principle is that if the material under study is

indeed visco-plastic, the rolling ball should come to an equilibrium position
when the buoyant weight equals the force due to the yield stress acting on
the ball, a kin to the sphere pendulum method. Therefore, the yield stress
can be shown to be proportional to sin

θ, θ being the angle of equilibrium

from the vertical direction. Some data was presented supporting the validity
of this method. Similarly, Briscoe et al. (1992b, 1994) have also outlined an
empirical scheme to extract the values of the Bingham plastic model parameters
by extrapolating the rolling ball data.

11.4 ROTATING SPHERE VISCOMETER

Some attempts have also been made to use a rotating sphere to evaluate
zero-shear viscosity and characteristic relaxation time for visco-elastic fluids
(Walters and Waters, 1963; Bourne, 1965; Walters and Savins, 1965; Hermes,
1966; Mashelkar et al., 1972; Kelkar et al., 1973; Acharya and Maaskant,
1978). In the absence of secondary flows and inertial effects, all one needs to

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Falling Object Rheometry

577

Angular scale

Sphere

Sphere release
mechanism

Groove (8 mm)

100

10 mm

irc

a = 15°

30°

FIGURE 11.9 Rolling ball device of Schurz (1990).

measure is the torque as a function of the speed of rotation to evaluate viscous
and elastic characteristics, albeit these results are not only model dependent but
are also restricted to the vanishingly small values of the Reynolds and Deborah
(Weissenberg) numbers. For a power-law fluid, Bourne (1965) solved the gov-
erning equations for a sphere of diameter d rotating in a power-law fluid filled
in a sphere of diameter (D). In the absence of wall effects, that is, D

d, the

shear rate at the surface of the sphere is given by

˙γ =

sin θ

(11.27)

which is seen to vary from 0 at the poles (

θ = 0, π) to a maximum at the equator

(θ = π/2). The value of power-law index n is obtained as the slope of torque vs.
speed plots (for the same sphere) on log–log coordinates. Acharya and Maaskant
(1978) and Cairncross and Hansford (1978) have presented some experimental
results on the feasibility of this method for visco-elastic fluids. More recently,
Schatzmann et al. (2003) have developed a ball measuring system to measure
the flow curves for debris and muds which yielded shear stress–stress rate data
comparable to those obtained from a rotational viscometer.

Another variant of the ball devices is the so-called oscillating sphere

configuration which has also received some attention in the literature (Sellers
et al., 1987; Tran-Son-Tay et al., 1988, 1990). There is no doubt that in prin-
ciple all these devices can yield reliable values of viscosity for Newtonian fluids,
especially at high pressures and temperatures, but the interpretation of results

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578

Bubbles, Drops, and Particles in Fluids

obtained for non-Newtonian fluids is far from straightforward and must there-
fore be treated with caution. Also, the simple design and operation of the falling
ball and the rolling ball configurations makes them much more attractive than
the rotating and oscillating ball methods.

11.5 FALLING CYLINDER VISCOMETER

11.5.1 N

EWTONIAN

LUIDS

Considerable attention has been devoted to the development of suitable ana-
lytical frameworks to evaluate the viscosity of Newtonian fluids from a falling
cylinder configuration. While detailed descriptions of various designs of this
kind of viscometer are available in the literature (Sherman, 1970; Irving and
Barlow, 1971; Mclachlan, l976; Claesson et al., 1983; Park and Irvine, 1984,
1988; Uchida et al., 1984),

Figure 11.10

shows the basics of this device. The

earliest attempts to develop an analytical expression relating the viscosity to
the other pertinent variables are due to Smith (1957) and Lohrenz et al. (1960).
For a circular cylinder of radius R

falling vertically in a fluid at the axis of a

tube of radius R

, the viscosity is given by

µ =

2
2

{(1 + k

) ln k

− (k

− 1)}

(1 + k

)

(11.28)

where k

= (R

The assumptions underlying the development of Equation 11.28 include

no end effects, no eccentricity, long cylinder (large length-to-diameter ratio),
and small Reynolds numbers. Obviously, for given values of R

and R

, the

viscosity is given by

µ =

(ρ

− ρ)

(11.29)

where k

is a system constant.

In practice, however, end effects are always present and as is the degree of

eccentricity, howsoever small. Therefore, the use of Equation 11.29 requires
calibration of the falling cylinder viscometer. Also, the constant k

not only

represents the system geometry, but also indirectly accounts for the end effects,
and for the finite size of the cylinder. It is tacitly assumed that the value of k

constant in the narrow range of conditions of interest.

In order to minimize the possibility of eccentricity, some novel designs of

cylinders (or slugs) including one with hemispherical ends (Jackson and Bed-
borough, 1978; Chan and Jackson, 1985; Chu and Hilfiker, 1989), a cylinder

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Falling Object Rheometry

579

Fluid of density

and viscosity

Cylinder

Cylindrical tube
closed at both ends

FIGURE 11.10 Schematics of a falling cylinder viscometer.

with stabilizers on both ends (Lohrenz and Kurata, 1962), conical bottom sim-
ilar to of rotameter floats (Dandridge and Jackson, 1981), cylinders with central
holes or hollow cylinders (Irving and Barlow, 1971; Mclachlan, 1976), with
cone at one end and a propeller at the other end (Claesson et al., 1983), roun-
ded corners (Lescarboura and Swift, 1968), etc. have been developed and used
to examine the temperature and pressure (up to

∼350 MPa) dependence of the

Newtonian viscosities up to about 10

Pa s. While Equation 11.29 does apply to

all such variants of the falling cylinder (also called sinker or slug) with an appro-
priate value of k

determined via calibration, rigorous analytical treatments

are clearly not possible. However, for the case of a hollow cylinder of inner
radius R

, Irving and Barlow (1971) presented the following expression for k

2
2

(1 − k

){(k

− 1 + k

) ln k

− (k

− 1)

}

− 1 + k

)

(11.30)

Clearly, in the limit of k

(=R

) → 0, Equation 11.30 reduces to the limiting

behavior for a solid cylinder given by Equation 11.28.

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580

Bubbles, Drops, and Particles in Fluids

The applicability of the aforementioned expressions has been amply demon-

strated among others by McDuffie and Barr (1969), Irving and Barlow (1971),
Claesson et al. (1983), Chan and Jackson (1985), Dandridge and Jackson
(1981), and Mclachlan (1976).

While Equation 11.28 and Equation 11.30 are valid over the entire range of

, Park and Irvine (1984, 1988) have advocated the use of the so-called falling

needle viscometer, that is, k

→ 0. In this case, the system constant k

(for

< 0.033) is given by (Park and Irvine, 1984) as

2
2

{−(1 + ln k

)}

(11.31)

The experimental verification of Equation 11.31 has been provided by Park and
Irvine (1984) who measured the viscosity of aqueous glycerol solutions using
such a falling needle viscometer. Subsequently, a theoretical justification has
been provided by Davis and Brenner (2001) which also eliminates the necessity
of experimental determination of the system constant k

for the falling needle

viscometers.

Some effort has also been directed at ascertaining the role of eccentricity

(Heyda, 1959; Chen et al., 1968; Lescarboura and Swift, 1968; Liu et al., 2004),
end effects (near the top and bottom of the falling cylinder) (Chen and Swift,
1972; Park and Irvine, 1988; Wehbeh et al., 1993) and the effects arising from
the bottom end of the outer cylinder (Wehbeh et al., 1993; Cristescu et al.,
2002). For a needle with hemispherical ends, Park and Irvine (1988) presented
the following end correction factor (ECF) for the settling velocity to be used in
Equation 11.28:

ECF

= 1 +

(1 − ln k

) − (1 + ln k

)

+ k

−1

(11.32)

where the wall correction factor, f

, arises from the hemispherical ends and

is the same as that given by Faxen (1923) and discussed in

Chapter 10.

It is a

function of k

(0.1) as

= 1 − 2.104k

+ 2.09k

+ · · ·

(11.33)

The dimensionless length

− 1

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Falling Object Rheometry

581

Subsequently, Kim et al. (1994) reported a flow visualization study and found
the flow disturbance due to the end effects to persist only up to about 6% of
the needle length. This finding is also consistent with the theoretical analysis
of Chen and Swift (1972) and experiments of Wehbeh et al. (1993). Thus, by
an appropriate choice of (L

), one can minimize the end effects. Chen et

al. (1968) have also carried out a theoretical analysis to elucidate the role of
eccentricity on the fall velocity and hence on the measured viscosity. They
presented their results in the form of a eccentricity ratio, ER, defined as

(1 − k

)

(11.34)

where

ε is the distance between the axes of the falling cylinder and the tube.

While they have tabulated extensive results as functions of eccentricity ratio
(0.05 ≤ ER ≤ 0.95) and 0.80 ≤ k

≤ 0.99, the influence of eccentricity is

negligible up to about ER

∼ 0.20. In fact, the effect becomes increasingly

significant with the increasing value of k

. Conversely, the role of eccentricity

is expected to be minimal in a falling needle viscometer. These predictions are
supported by the limited experimental results of Lescarboura and Swift (1968).
The role of the end effects due to the finite length (closed end) of the tube has
been assessed by Wehbeh et al. (1993) and Cristescu et al. (2002).

11.5.2 N

-N

EWTONIAN

LUIDS

11.5.2.1 Shear-Dependent Viscosity

Some analytical attempts have also been made to explore the use of a falling
cylinder viscometer to obtain shear stress–shear rate curves for power law flu-
ids (Ashare et al., 1965; Eichstadt and Swift, 1966; Park and Irvine, 1988;
Park et al., 1990; Phan-Thien et al., 1993; Zheng et al., 1994; Yamamoto and
Shibata, 1999), for Bingham plastic fluids (Eichstadt and Swift, 1966) and
for visco-elastic fluids (Phan-Thien et al., 1993; Zheng et al., 1994; Tigoiu,
2004). The early studies of Ashare et al. (1965) and Eichstadt and Swift (1966)
are straightforward extensions of the analysis of Lohrenz et al. (1960). For
large values of k

, the narrow annular gap between the falling cylinder and the

tube wall is treated as a planar slit and then the Newtonian corrections for the
curvature and the end effects are used for power-law and Bingham plastic flu-
ids. Inspite of these developments, the interpretation of falling cylinder data for
non-Newtonian fluids is far from straightforward for large values of k

. On the

other hand, the falling needle configuration has been found to be promising. Park
and Irvine (1988) and Park et al. (1990) have outlined a theoretical procedure
that allows the determination of the shear stress–shear rate curves for power-law
fluids. Subsequently, some weaknesses of this analysis have been pointed out

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582

Bubbles, Drops, and Particles in Fluids

by Phan-Thien et al. (1993) and Zheng et al. (1994). The latter authors have also
studied this configuration numerically in detail and their results are presented
here. The average shear rate at the surface of the needle is given by

˙γ

= α

(11.35)

where

α is a function of both n and k

log

α = α

− α

log n

(11.36)

= 0.4579 + 3.2305(k

− 0.1)

(11.37a)

= 0.07044(k

)

−0.585

(11.37b)

Equation 11.37 correlates their numerical results with an average error of less
than 2% over wide ranges of n (0.1–0.9) and k

(0.05–0.3).

The corresponding shear stress is estimated from a knowledge of the drag

force on a needle falling at the same velocity in a power law-fluid and in an equi-
valent Newtonian fluid (of viscosity

). The drag force F

in a Newtonian

fluid is given by the formula

πµ

(1 + k

)

(1 − k

) + (1 + k

) ln k

(11.38)

The drag on the needle in a power-law fluid F

is given as

−1

(11.39)

Note that at the terminal fall condition, the drag force F

is simply equal to the

buoyant weight of the needle and is thus known for a needle–liquid combination.
Thus, the slope of log

/(F

/µ

)) vs. log(V/R

) plot will yield the value

of n. This, in turn, will allow the evaluation of

α and hence the shear rate

via Equation 11.35 to Equation 11.37. A comparison of Equation 11.39 with
the power-law model suggests the corresponding shear stress to be given by
(F

) ˙γ

and thereby yielding the desired shear stress–shear rate curve

for a fluid. This analysis, however, assumes the needle to be infinitely long and
Zheng et al. (1994) recommend the value of (L

) > 40 for the end effects to

be negligible.

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Falling Object Rheometry

583

11.5.2.2 Yield Stress

The falling cylinder configuration (also known as a penetrometer) has also
been exploited to evaluate the static yield stress of visco-plastic materials. Park
et al. (1988) outlined an extrapolation procedure to evaluate the value of the
yield stress from a series of falling velocity data for needles of the same size,
but with different values of

ρ. Obviously, this method is equivalent to the

extrapolation of viscometric shear stress–shear rate data to zero-shear rate. On
the other hand, Uhlherr et al. (2002) have proposed a novel design of a hollow
penetrometer whose mass can be changed by addition of tungsten powder or
mercury to locate the motion/no motion point, that is, when the penetrometer
does not move. Under such static equilibrium conditions, the buoyant weight
must equal the vertical component of the force due to the yield stress of the
medium acting on the penetrometer (shown schematically in Figure 11.11). The
static yield stress is calculated from the expression

− ρπd

((h/4) + (d/12))

πd (h + (πd/8))

(11.40)

Uhlherr et al. (2002) reported good agreement between the penetrometer and
the vane values of the yield stress for a few carbopol solutions and titania
suspensions.

Air

Liquid

FIGURE 11.11 Schematics of a penetrometer. (From Uhlherr, P.H.T., Guo, J.,
Fang, T.-N., and Tiu, C., Korea–Australia Rheol. J., 14, 17, 2002.)

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Bubbles, Drops, and Particles in Fluids

Before concluding this chapter, it is appropriate to mention here that many

other devices based on the principles presented herein are available including
the so-called air bubble viscometer (Barr, 1926), diamond anvil cell viscometer
(Piermarini et al., 1978; King et al., 1992; Herbst et al., 1993; Cook et al., 1993).
However, while their utility for Newtonian fluids is well established, none of
these have yet been extended to the viscometry of non-Newtonian fluids.

11.6 CONCLUSIONS

In this chapter, consideration has been given to the use of falling, rolling, oscil-
lating and rotating sphere viscometry and of falling cylinder, and falling needle
viscometery for Newtonian and non-Newtonian media. While their utility for
the measurement of viscosity of Newtonian liquids at high temperatures and
pressures is established beyond doubt, their application to the measurement
of non-Newtonian fluid characteristics is severely limited by the fact that the
flow is nonviscometric in each case. It is, however, possible to extract some
information under appropriate experimental conditions. For instance, falling
ball method can be used to obtain reasonable estimates of the zero-shear viscos-
ity. On the other hand, this method yields less reliable values of shear-dependent
viscosity. The falling needle method offers an attractive alternative for shear-
thinning fluids simply because the shear rate is more uniform in a falling needle
device than that in a falling ball rheometer. The rolling ball method can certainly
be used for quality control purposes. The pendulum and penetrometer methods
offer simple and reliable means of evaluating static yield stress of visco-plastic
materials. Finally, it is not at all easy to obtain visco-elastic characteristics from
such devices.

NOMENCLATURE

Function of n, Equation 11.11 (-)

Length of cylinder in pendulum, Equation 11.19 (m)

Constant, Equation 11.5 (kg s m

−1

)

Constant, Equation 11.7 (-)

Constant, Equation 11.6 (kg m

−1

−3

)

Cylinder or sphere diameter (m)

Tube diameter (m)

ECF

End correction factor, Equation 11.32 (-)

Eccentricity ratio, Equation 11.34 (-)

Correction due to bottom end (-)

Correction due to inertia ( - )

Wall correction factor (-)

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Falling Object Rheometry

585

Drag force on a needle (N)

Drag force on a needle in a Newtonian medium (N)

Force on a sphere due to yield stress (N)

(n), F

(n)

Functions of n, Equation 11.12 and Equation 11.14 (-)

Acceleration due to gravity (m s

−2

)

Center of mass,

Figure 11.8

(m)

Depth of immersion of penetrometer (m)

Constant, Equation 11.2 (m

−2

)

Constant, Equation 11.20 (m

−2

)

Ratio of radii (

) (-)

Ratio of radii (

) (-)

System constant, Equation 11.29 (m

−2

)

Immersed length, Equation 11.18 (m)

Length of needle or cylinder (m)

Power-law consistency index (Pa s

)

Mass of cylinder, Figure 11.8 (kg)

Mass of plate-hook assembly, Equation 11.19 (kg)

Mass of penetrometer, Equation 11.40 (kg)

Power-law index (-)

Inner radius of hollow cylinder (m)

Outer radius of hollow cylinder (m)

Radius of cylindrical tube (m)

Fall time (s)

Falling velocity free from wall, end and eccentricity

effects (m s

−1

)

Measured fall velocity (m s

−1

)

Rolling velocity of ball (m s

−1

)

REEK

YMBOLS

Constant, Equation 11.36 (-)

Constants, Equation 11.37 (-)

Angle of inclination from horizontal (-)

˙γ

Surface average shear rate (s

−1

)

True zero-shear viscosity (Pa s)

Viscosity of a Newtonian fluid (Pa s)

Viscosity defined by Equation 11.1 (Pa s)

Rotational velocity (rad s

−1

)

Fluid density (kg m

−3

)

Density of falling ball or cylinder (kg m

−3

)

Density difference (

− ρ) (kg m

−3

)

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586

Bubbles, Drops, and Particles in Fluids

Inclination of pendulum from vertical or spherical coordinate (-)

Surface average shear stress (Pa)

Maximum shear stress (Pa)

Yield stress (Pa)

UBSCRIPTS

Newtonian

Lower bound

Upper bound

Document Outline

Contents
Chapter 11 Falling Object Rheometry

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