Advances in Nanoparticles, 2013, 2, 266-279
http://dx.doi.org/10.4236/anp.2013.23037 Published Online August 2013 (http://www.scirp.org/journal/anp)
Viscosity of Nanofluids—Why It Is Not Described by the
Classical Theories*
Valery Ya. Rudyak
Department of Theoretical Mechanics, Novosibirsk State University of
Architecture and Civil Engineering, Novosibirsk, Russian Federation
Received May 8, 2013; revised June 8, 2013; accepted June 15, 2013
Copyright © 2013 Valery Ya. Rudyak. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Transport properties of nanofluids are extensively studied last decade. This has been motivated by the use of nanosized
systems in various applications. The viscosity of nanofluids is of great significance as the application of nanofluids is
always associated with their flow. However, despite the fairly large amount of available experimental information, there
is a lack of systematic data on this issue and experimental results are often contradictory. The purpose of this review is
to identify the typical parameters determining the viscosity of nanofluids. The dependence of the nanofluid viscosity on
the particles concentration, their size and temperature is analyzed. It is explained why the viscosity of nanofluid does
not described by the classical theories. It was shown that size of nanoparticles is the key characteristics of nanofluids. In
addition the nanofluid is more structural liquid than the base one.
Keywords: Nanofluids; Gas Nanosuspensions; Nanoparticles; Viscosity; Kinetic Theory; Molecular Dynamics;
Rheology of Nanofluids; Effect of Nanoparticles Size
1. Introduction
Nanofluid is a two-phase system consisting of a carrier
medium (liquid or gas) and nanoparticles. The term nan-
ofluid was first used by Choi [1] to describe a suspension
consisting of carrier liquid and solid nanoparticles. In our
papers [2-5], nanoparticle suspensions in gases (gas nano-
suspensions) are also called nanofluids. This is done for
several reasons: 1) both gas and liquid nanosuspensions
have extensive practical applications; 2) many properties
of nanofluids and gas nanosuspensions are very similar,
especially if the carrier gas is dense; 3) in some cases,
transport processes in gas and liquid suspensions of
nanoparticles are studied using the same methods or
models. For example, the procedure of molecular dy-
namics simulation is the same in both cases.
Typical carrier fluids are water, organic liquids (eth-
ylene glycol, oil, biological liquids, etc.), and polymer
solutions. The dispersed solid phase is usually nanoparti-
cles of chemically stable metals and their oxides. Fullere-
nes with a diameter of about 1 nm can be considered the
smallest nanoparticles. Viruses are intermediate in size:
their sizes are of the order of tens of nanometers. On the
other hand, nanofluids based on carbon nanotubes have
also been widely studied, but these liquids will not be
considered in this paper.
Investigation of the physics of nanofluids and their
transport properties began relatively recently. This has
been motivated by the use of nanosized systems in vari-
ous applications. The special properties of nanoparticles
are due to their small size. Nanofluids have peculiar
transport properties. In contrast to large dispersed parti-
cles, nanoparticles do not sediment and erode the chan-
nels in which they move. For these and some other rea-
sons, nanofluids have already been successfully used, or
are proposed for use, in chemical processes, including
catalysis, for cooling devices, in various bio-, MEMS-,
and nanotechnologies, in the development of thermal en-
rgy production and transport systems, in pharmaceutical
and cosmetic products, for pollution detection and air and
water purification systems, as lubrication materials and
drug delivery systems. This list can be extended, but the
transport properties and flow patterns of nanofluids play
a key role in these applications.
The wide potential application of nanofluids has stimu-
lated the development of their production processes and
extensive research activity. In particular, numerous ex-
*This work was supported in part by the Russian Foundation for Basic
Research (grant No. 13-01-00052-а).
C
opyright © 2013 SciRes. ANP

V. Ya. RUDYAK 267
perimental and theoretical studies have focused on the
thermal conductivity and viscosity of nanofluids. The vis-
cosity of nanofluids is of great significance as the appli-
cation of nanofluids is always associated with their flow.
However, despite the fairly large amount of available
experimental information, there is a lack of systematic
data on this issue and experimental results are often con-
tradictory. Two good reviews published recently [6,7]
have not improved the situation. The importance of ob-
taining adequate data on the viscosity coefficients has
motivated a series of special measurements simultane-
ously in more than thirty laboratories around the world.
However, the results of these measurements have not
clarified the situation [8]. This is due to the fact that the
measurements were carried out without adequate tem-
perature control, in a narrow range of volume concentra-
tions of nanoparticles, with a wide spread of nanoparticle
sizes, and for different carrier liquids. Therefore, a fur-
ther critical review of the results is needed to make pro-
gress in the understanding of what nanofluid viscosity is.
Such a review should not be limited to listing available
data, but should identify the methodological shortcom-
ings of the procedures used to obtain these data and indi-
cate ways to overcome them. The present paper is written
with exactly this goal in mind. The purpose of the review
is to identify the typical parameters determining the vis-
cosity of nanofluids. Below methods for producing nan-
ofluids will not be discussed. We note only that in almost
all studies, nanofluids are prepared using the so-called
two-step method [9,10], in which a nanopowder contain-
ing particles of a given size is added in a certain ratio to
the carrier (base) fluid.
2. Viscosity of Molecular Mixtures and Gas
Nanosuspensions
Nanoparticle size varies within two orders of magnitude.
At the lower limit are nanoparticles with characteristic
sizes of one to a few nanometers. These particles are only
a few times larger than conventional inorganic molecules
and are similar in size to some organic molecules. At the
upper limit are nanoparticles with sizes of 80 - 100 nm.
In fact, they are close in size to usual Brownian particles
(in Perrin’s experiments [11], the size of Brownian parti-
cles was 10−4 - 105 cm). For this reason, it is clear that the
mechanisms of transport of small nanoparticles should be
close to those of conventional molecules, the mecha-
nisms of transport of large nanoparticles should be simi-
lar to those of Brownian particles. However, the indicated
mechanisms of transport processes are significantly dif-
ferent. Therefore, to clarify the momentum transfer char-
acteristics of nanofluids that determine their viscosity, it
is necessary to analyze both molecular systems and dis-
perse systems with large macroscopic particles (Brownian,
in particular). This will be done in the next two sections.
However, before doing so, we note that even a cursory
analysis of the size range of nanoparticles shows that the
transport properties of nanofluids should depend on the
size of the particles. This conclusion is unusual for the
classical physics of transport processes. The classical
relations for the Einstein viscosity [12] and Maxwell
thermal conductivity [13] are independent of the size of
the dispersed particles.
Transport processes in gases at normal pressures are
described by the kinetic Boltzmann equations, whose
solution for the viscosity coefficient of binary gas mix-
tures can be expressed as [14]

 
1
Z
XY 

, (1)
where
22
112
112
22
2
x
xx x
X


 ,

2
22
*112
121 2
112
11
0.6 0.52
2
x
x
YA xx








 





,

22
*2 2
1211 2
1
0.6 21
4
x
ZAxxx










   








,


*
12
2,2
2
π
5
16 π
i
i
ii i
mkT
 ,


*
12
12
12 2,2
2
12 12
π
5
16 π
mkT
 .
Here




12121 2

  
, x1 and x2 are the mole
fractions of components 1 and 2, 12 mm

, where m1
and m2 are the masses of the molecules of the first and
second components, T is the temperature.
 
*
2,2 1,1
*
12 1212
A * , where are reduced Ω-inte-
grals, and ii

*
,
12
lm


and ij

are parameters of the pair inter-
action potential (for example, the Lennard-Jones potential)
which characterize the effective scattering cross sections
of molecules of the same kind and the interaction between
different kinds of molecules, respectively.
If there is a mixture of gases with a small addition of
one of them (say, the second) and this addition contains
molecules with a large mass (analog of a gas suspension),
small parameters appear in formulas (1) and these for-
mulas are simplified to



*
*
2,2
2
12
12
1,1
12
11.22 1
s
xO  





 





. (2)
Thus, the addition of a small amount of a heavy gas to a
lighter gas will lead to an increase in the viscosity of the
mixture compared to the viscosity of the light component.
Generally, however, this increase does not depend mono-
tonically on the concentration of the heavy component and
changes significantly with increasing temperature [15].

V. Ya. RUDYAK
268
Kinetic theory of gas nanosuspensions has been de-
veloped (see papers [2,5,16-21]). In particular, it has been
shown that the viscosity of rarefied gas nanosuspensions
can be described by expression (1), where the subscript 2
refers to nanoparticles and 22R

 (R is the radius of
the nanoparticle). Thus, as for molecular mixtures of
gases, calculating the viscosity coefficient for gas nano-
suspensions reduces to calculating the corresponding Ω-
integrals for the RK molecule-nanoparticle interaction
potential [19,22].
The viscosity coefficient of the gas nanosuspensions (1)
is a multiparameter function and varies considerably with
nanoparticle size and concentration and with the tem-
perature of the gas suspension. For small mole fractions
of the dispersed phase, , the coefficient (1) be-
comes
21x

2
1*
12
2
** 12 1
12 12
1
110.6
1
0.32 1.22
x
A
AA
 

 







1 2



 

 


 

.
(3)
Function (3) depends significantly on the mass ratio μ,
nanoparticle radius, temperature, and interaction potential
parameters. In particular, for certain values of these quan-
tities, function (3) can change sign. This implies that the
addition of small volume fractions of dispersed solid
particles to a gas can both increase or decrease the effect-
tive viscosity of the medium.
Calculations have shown that this is indeed true. As an
example, Figure 1 shows the dependence of the viscosity
of a gas nanosuspension

2
H
U on the volume frac-
tion of 1 nm diameter particles at different temperatures: T
Figure 1. Effective viscosity (poise) of a gas suspensions of U
nanoparticles in H2 (a) and Zn particles in Ne (b) versus
volume concentration φ of nanoparticles.
= 200 K (1), T = 300 K (2), T = 400 K (3), T = 500 K (4),
T = 600 K (5), T = 800 K (6), and T = 1000 K (7) [5,23].
At concentrations of the order of 2 × 10−4 and room tem-
peratures, the viscosity of the gas nanosuspension is 90%
higher than the viscosity of the carrier gas. This effect
depends significantly on the temperature, and at T = 1000
K, the ratio η/η1 ~2.3 at the same concentrations.
On the other hand, the viscosity of a Zn-Ne gas sus-
pension with nanoparticles of the same size decreases, as
shown in Figure 2 (where the different curves correspond
to the same temperatures as in Figure 1). At room tem-
perature and a volume concentration of particles of 2 ×
10−4, the effective viscosity of this gas suspension is about
15% lower than the viscosity of pure neon, and this effect
increases with increasing temperature.
The results derived from kinetic theory have been con-
firmed experimentally by studies of the diffusion of na-
noparticles [24-27]. The main conclusion that can be
drawn is that the viscosity of gas nanosuspensions of
small particles, first, differs significantly from the vis-
cosity of ordinary molecular mixtures, and second, it is
not described by the classical Einstein theory [12], ac-
cording to which the effective viscosity of a disperse fluid
depends only on the concentration of dispersed particles


012.5

   , (4)
and is always greater than the viscosity of the carrier fluid
0

, where φ is the volume fraction of dispersed particles.
3. Viscosity of Coarse Suspensions
The effect of dispersed particles on the viscosity of sus-
pensions was first studied in Einstein’s classical work
cited above. Considering the motion of a small particle in
a fluid, he determined the flow field perturbations caused
by it, then calculated the effective stress tensor, and, as a
result, obtained formula (4) for the effective viscosity
coefficient. The volume concentration of particles was
Figure 2. Normalized viscosity of a nanfluid versus volume
concentration of particles for different ratios of the radius
of the nanoparticle to that of the molecule.

V. Ya. RUDYAK 269
assumed to be small enough to take into account the per-
turbations caused by an isolated particle. Thus, the addi-
tional momentum transfer in the liquid due to the presence
of dispersed particles was determined only by hydrody-
namic processes. Comparison of formula (4) with ex-
periments has shown that it adequately describes the vis-
cosity of suspensions at volume concentrations of parti-
cles [28]. In this paper, we consider systems in
which the particle concentrations are relatively low, up to
about 10% - 15%. For this, it is necessary to find a cor-
rection of order
2
10 


2

to formula (4). To extend formula (4)
to such particle concentrations within Einstein’s concept,
it is necessary to take into account the hydrodynamic
perturbations due to the mutual influence of closely
spaced dispersed particles. This hydrodynamic interaction
leads to an increase in the energy dissipation of viscous
friction. As a result, the viscosity increases linearly with
increasing particle concentration. Theory incorporating
these effects was developed by Batchelor [29]. In addition,
he took into account the contribution of Brownian motion
to the average stress and obtained the following formula
for the effective viscosity, accurate to the second order in
concentration [30]
2
012.5 b



 
, (5)
where the coefficient b = 6.2. Similar ideas were deve-
loped in [31] and some other papers.
A fundamentally different approach to calculating the
viscosity of suspensions was proposed by Mooney [32],
who considered possible structuring of disperse systems.
Later, this approach was developed in a number of papers.
In particular, Krieger and Dougherty [33] obtained the
formula
01
m








, (6)
where m

is the volume fraction of particles at their
maximum packing density and α is a parameter that de-
pends on the properties of the medium. In particular, for a
dilute suspension in which the particles do not interact
with each other, α = 2.5. Later, formula (6) was modified
in several papers (see [34,35]). A comprehensive list of
the formulas used can be found in reviews [6,7], but it is
important to emphasize that at not too high concentrations,
almost all formulas lead to relation (5), in which the co-
efficient b usually varies from 4.3 to 7.6. Thus, at a 10%
concentration of dispersed particles, formulas of the type
(5) gives an about 25% - 30% increase in the viscosity.
It is worth noting that almost all classical theories of
viscosity describe the dependence of the effective viscos-
ity coefficient only on the volume concentration of parti-
cles. However, the dependence of the viscosity of even a
homogeneous liquid varies greatly with temperature. So
far as is known, there is no consistent theory of the vis-
cosity of liquids that would give universal formulas for
fluid viscosity. In the absence of such a theory, almost all
empirical or semi-empirical formulas for the viscosity
coefficient actually describe its dependence on tempera-
ture. A typical dependence was proposed by Andrade
[36,37]
B
T
A
e  , (7)
where A and B are some constants determined experi-
mentally for a particular fluid. The exponential depend-
ence of viscosity on temperature is often violated near
the liquid-solid phase transition.
In practice, liquid viscosity is often calculated using the
correlation proposed by Orrick and Erbar [15],
1
ln ln
A
BT M



 , (8)
here ρ is the density of the liquid, M is its molecular
weight, and the constants A and B for a number of liquids
can be found in [15]. Some other semi-empirical correla-
tions (see, for example, [15]) are also used, but the tem-
perature dependence of viscosity the remains qualita-
tively the same as in (7). Formula (8) also takes into ac-
count the physical properties of the liquid: its molecular
weight and density. According to this formula, the liquid
viscosity increases with increasing its density.
Density, however, is not the only physical characteris-
tic that affects liquid viscosity. In liquids, as opposed to
gases, the viscosity is known to decrease with increasing
temperature. In contrast, in gases it increases with in-
creasing temperature. This is related to the different
mechanisms of momentum transfer. In gases, smoothing
of momentum is due to the kinetic mechanisms: the mo-
mentum is transferred by molecules in motion and colli-
sion. It is for this reason that in not too dense gases, the
viscosity coefficient is proportional to the mean free path
of the molecules l: η ~ l. As the density increases, the
viscosity coefficient becomes a nonlinear function of den-
sity and increases with increasing density. Increasing tem-
perature increases the velocity of molecules, and, hence,
the momentum transfer rate per unit time. This implies
an increase in the viscosity with increasing gas tempera-
ture.
In liquids, the kinetic mechanism of momentum trans-
fer also takes place. However, in liquids, the molecules
are packed closely enough, so that it is difficult to speak
about the mean free path of the molecules. Here, the main
kinetic mechanism of momentum transfer is associated
with collisions of molecules. On the other hand, although
such collisions redistribute the momentum in the system,
they do it locally, near a distinguished molecule. This is
not sufficient to smooth out large-scale momentum fluc-
tuations in a liquid. Liquids have short-range order and
momentum transfer by molecules is accompanied by its

V. Ya. RUDYAK
270
destruction. Thus, the viscosity of a liquid depends sig-
nificantly on its structure. Attempts to take into account
the liquid structure in calculations of liquid viscosity have
been reported in the literature. This approach has been
successfully applied to model the viscosity of non-New-
tonian fluids (for example, see [38]). However, similar
approaches are known for modeling the viscosity of sim-
ple liquids. An example is the correlation proposed by
Van Velzen, Cardozo, and Langenkamp [15]
measurements (International Nanofluid Properties Bench-
mark Exercise) also examined the dependence of viscosity
only on the volume concentration of particles [40]. What
has been established? At low concentrations, the viscosity
of all measured nanofluids is several times higher than
predicted by Einstein’s theory (4). However, the coeffi-
cient of increase a was different in different studies. In
paper [40] the data for three nanofluids with large (100 nm)
Al2O3 nanoparticles in poly-alpha olefin (PAO) were
presented. The coefficient a was found to be an order of
magnitude larger than predicted by Einstein’s theory (4).
Investigating an ethylene glycol based nanofluid with
TiO2 particles Chen et al. [41,42] obtained a correlation
coefficient a = 10.6. In these experiments, nanoparticles
with an average size of 25 nm formed aggregates in the
nanofluid with an average size of 140 nm. Colla et al. [43]
proposed a correlation with a coefficient a = 18.64 based
on the results of measurements for water-based nanofluids
containing Fe2O3 particles with an average size of 67 nm.
For an ethylene glycol based nanofluid containing 200 nm
Cu particles, Garg et al., [44] obtained a correlation with a
coefficient a ≈ 11. In a study [45] of the viscosity of a
water-based nanofluid with Al2O3 nanoparticles, this factor
equaled 7.3. The correlation used in this study was derived
from experimental data [46-48]. It is worth to say that
these experiments were performed with different nan-
ofluids.


*1 1
0
ln BT T 

, (9)
where the parameters and 0 T are determined by the
structure of the liquid molecules.
*
B
Short-range order in liquids occurs at scales ranging
from one to a few nanometers. Thus, whereas in gases,
viscosity forms at scales of the order of the mean free
path, in liquids it occurs at mesoscales ms much larger
than the characteristic size of the molecules1 r0: rms > r0.
r
Thus, the dependence of the viscosity coefficient on
the concentration of dispersed particles and temperature
has been noted in all major studies (both experimental
and theoretical). At low particle concentrations, the ef-
fective viscosity coefficient of suspensions increased in
proportion to φ compared to the viscosity coefficient of
the carrier liquid. With further increase in the concentra-
tion, the viscosity coefficient was described by formula
(5). The dependence of viscosity on temperature gener-
ally fits the classical relation (7). In experiments, correla-
tions between viscosity and the size of dispersed particles
were not found. Nevertheless, an attempt to take into
account the effect of particle size on viscosity was made
in studies [39] using a linear approximation for particle
concentration. In accordance with the proposed formula,
the effective viscosity coefficient increases with increas-
ing particle size.
Some more data can be given, but even those cited
above are enough to see that there is no universality in the
obtained correlations. In contrast, the Einstein formula for
conventional suspensions is universal for all fluids and
depends only on the volume concentration of particles.
What are the reasons for the lack of universality for nan-
ofluids? There may be two possible reasons. The coeffi-
cient a may depend on: 1) the size of nanoparticles; 2) the
density of nanoparticles (i.e., the nanoparticle material).
4. Dependence of Nanofluid Viscosity on
Nanoparticle Concentration In all cases, as the volume (or mass) concentration of
nanoparticles increases, a quadratic dependence of the
viscosity on

is obtained
The viscosity of nanofluids has been persistently inves-
tigated over about fifteen years in more than thirty groups
throughout the world. However, a universal formula that
would describe the viscosity coefficient of any nanofluid
has not been derived. Moreover, measurements often lead
to diametrically opposite results. Why does this occur?
2
01ab
 




, (10)
However, as in the case of the coefficient a, the coef-
ficient b is not universal, but far exceeds that for conven-
tional suspensions. Several correlations obtained at dif-
ferent times are given below. One of the first correlations
was obtained for a nanofluid with TiO2 particles [49]
It was long thought, and is still being argued by many,
that similarly to the viscosity of conventional suspensions,
the viscosity of nanosuspensions is determined only by
the mass concentration of particles. It is noteworthy that in
special benchmark measurements made as part of an in-
ternational project on viscosity and thermal conductivity
2
01 5.45108.2
 


 

. (11)
A year later, the following experimental correlation was
proposed for a water-based nanofluid containing Al2O3
nanoparticles [50]
1and larger than the mean free path of molecules, which can be calculated,
e.g., for hard-sphere molecules and it is even smaller than the typical
size of molecules.
2
017.3 123
 




. (12)

V. Ya. RUDYAK 271
It is worth noting that in the same paper, a different
correlation was proposed for a suspension of the same
nanoparticles in ethylene glycol
2
010.19 306
 

 

. (13)
This correlation, however, is certainly not universal
because at low concentrations, it gives a reduction in the
viscosity, which is physically unreasonable. Several more
similar correlations have been proposed by different au-
thors, but, for the mentioned reason, they will not be
discussed in this paper.
Chen et al. [41] proposed the following correlations for
two different nanofluids: for an ethylene glycol based
nanofluid with TiO2 particles,

2
01 10.610.6
 

 

, (14)
and for a water-based nanofluid with Cu particles,
2
00.995 3.645468.72
 


 
. (15)
The second correlation, however, is not entirely satis-
factory in the limit 0

.
From a study of nanofluids containing Al2O3 and CuO
particles, Nguyen et al. [48] derived the correlation
23
01.475 0.3190.0510.009
 


 
. (16)
However, it is also inadequate in the limit 0

 and
at low concentrations of nanoparticles.
For water-based nanofluids with Fe2O3 particles, in
paper [45] the following correlation based on experi-
mental data [46-48] was proposed
2
01 18.64248.3
 

 
 
. (17)
Summing up this brief overview, we note again that all
formulas are dissimilar and not universal. The possible
reasons are the same as before: the possible dependence of
viscosity on nanoparticle size and material.
5. Dependence of the Viscosity of Nanofluids
on the Size of Nanoparticles
A fundamental disadvantage of all the correlations dis-
cussed in the previous section is not the lack of univer-
sality, but the fact that in the limit, none of them reduce to
the corresponding formulas for the viscosity of conven-
tional suspensions. In the first part of the paper, it was
noted that nanoparticles are intermediate in size d between
conventional molecules and macroscopic dispersed parti-
cle. Moreover, the largest nanoparticles are close in size to
usual Brownian particles. Therefore, the properties of nan-
ofluids with such particles should be similar to the prop-
erties of suspensions of submicron particles. In this regard,
it is reasonable to expect that the formula for the viscosity
of nanofluids with large particles reduces to the corre-
sponding formula for conventional suspensions when
, where D is the diameter of the conventional
dispersed particle. It is therefore reasonable to assume that
the viscosity of nanofluids should be a function of nanopar-
ticle size. It was shown in Section 2 that the viscosity of
gas nanosuspensions depends significantly on the size of
the nanoparticles.
dD
Experimentalists came to the understanding that nano-
particle size can affect the viscosity of nanofluids almost a
decade ago. Nevertheless, the dependence of the viscosity
on nanoparticle size has been the subject of very few
studies. According to [7], they are only about one-quarter
of the total number of papers devoted to the viscosity of
nanofluids. This is not surprising since determining the
dependence of the viscosity on nanoparticle size is much
more difficult than determining viscosity for a particular
size. First, measurements should be performed simulta-
neously for, at least, three or four sizes of nanoparticles in
the same base fluid. Second, one has to carefully monitor
the particle size distribution in the nanofluid. Third, it is
possible that many have been discouraged by conflicting
information on the influence of nanoparticle size. Thus,
He with colleagues [51] asserts that the viscosity increases
with increasing size of nanoparticles. A similar assertion
is made by Nguyen et al. [52]. They state that at low
concentrations of nanoparticles, their size has virtually no
effect on the viscosity of nanofluids. On the other hand, as
the particle concentration increases, the viscosity coeffi-
cient depends strongly on nanoparticle size and is higher
for nanofluids with larger particles. In paper [53], it is
noted that the viscosity of nanofluids is virtually inde-
pendent of the size of nanoparticles.
In contrast, studies [46,54,55] showed a reduction in the
viscosity of nanofluids with increasing particle size. This
is supported by molecular dynamics simulations [56,57].
Before analyzing these experimental data, we present the
main result of these molecular dynamics simulations. In
studies [56,57], nanofluids were simulated using the hard
sphere interparticles interaction potential. In the calcula-
tions, the dimensionless radius of nanoparticles Rr
(where r is the radius of the carrier-fluid molecule) was 2,
3, and 4. The dependence of the viscosity coefficient of
this nanofluid on the volume concentration of nanoparti-
cles is shown in Figure 2. Here curves 1 - 3 correspond to
R/r = 2, 3, 4.
Here the dependence of nanofluid viscosity on nanopar-
ticle size is quite obvious: the viscosity coefficient in-
creases with decreasing particle size. These data, however,
have a significant drawback. They were obtained for rather
small nanoparticles. In addition, molecular dynamics simu-
lation is an ideal experiment. In real experiments, nano-
suspensions are never monodisperse. A nanofluid always
contains nanoparticles of different sizes, and the viscosity
of the nanofluid depends on the size distribution of the

V. Ya. RUDYAK
272
nanoparticles. Thus, in experiments, one should carefully
monitor the nanoparticle size distribution. At the same
time, the molecular dynamics method is, in fact, the only
method that can be used to model an “ideal nanofluid” and
rigorously examine the effect exerted on the viscosity and
other transport coefficients by not only the concentration
of nanoparticles, but also by their material and size. It is
necessary, however, to clearly distinguish between the
effects of the volume concentration of nanoparticles and
their size. Of course, for a fixed number of nanoparticles
in a fluid, the volume fraction of the particles will increase
with their size. However, nanofluids with particles of
different sizes are different. Therefore, the results of mo-
lecular dynamics simulations of water-based nanofluids
[58] should be treated with caution.
Let us now analyze the data of experimental studies of
the effect of nanoparticle size on the effective viscosity of
nanofluids. The suggestion that the viscosity of nanofluids
can increase with increasing size of nanoparticles can be
found in two papers we are aware of [51,52]. He with
colleagues [51] studied the viscosity of a water-based
nanofluid with relatively large particles of TiO2 (95, 132,
and 230 nm). The volume concentrations of nanoparticles
were rather small (less or equal than 1.2%). At such low
concentrations, the viscosity coefficient should still in-
crease almost linearly with increasing particle concentra-
tion. In the graph given in [51] this is not so. On the other
hand the measurement accuracy, apparently, does not
exceed 2% - 3%, but an increase in the viscosity with
increasing nanoparticle size is observed in exactly this
range. In addition, in these experiments, the size distribu-
tion of particles and their possible agglomeration were not
monitored.
Prasher, Song and Wang [53] studied a propylene glycol
based nanofluid with Al2O3 nanoparticles. Nanoparticles
of three sizes, 27, 40, and 50 nm were used, but it is not
clear what as the distribution of nanoparticles in each of
the nanofluids. The accuracy of viscosity measurements in
these experiments is not high: about 3% at 20˚C and about
11% at 40˚C. The examined concentrations of nanoparti-
cles were not great: 0.5%, 2%, and 3%. The independence
of the viscosity on particle size stated in [53] can well be
explained by the inaccuracy of the measurements. This is
suggested, in particular, by the nonsystematic temperature
dependence of the viscosity of the nanofluid.
Namburu et al. [46] were among the first to show that
nanofluid viscosity increases with decreasing particle size.
They studied the viscosity of a suspension of Al2O3 na-
noparticles of three sizes, 20, 50, and 100 nm, in an
aqueous solution of ethylene glycol. The effect depended
significantly on temperature and was greater at lower
temperature of the nanofluid.
The dependence of the viscosity of water-based na-
nofluids with SiC particles was studied in sufficient detail
by Timofeeva et al. [55]. They considered nanofluids with
four particle sizes: 16, 29, 66, and 90 nm. Maximum
viscosity was observed for nanofluids with the smallest
particles, and the lowest viscosity for nanofluids with the
largest particles. Furthermore, the viscosity of the nan-
ofluid (4.1% concentration of nanoparticles) with 90 nm
particles was 30% higher and that of the nanofluid with 16
nm particles was about 85% higher than the viscosity of
the base fluid.
Finally, recently a special series of experiments was
performed to measure the dependence of the viscosity of
an ethylene glycol based nanofluid with SiO2 particles on
the size of nanoparticles [59]. The mass concentration of
nanoparticles in ethylene glycol was varied from 0.5% to
14%, which was equivalent to a volume concentration of
0.25% to 7%. Viscosity measurements were performed on
a Brook field LVDV-II+Pro rotational viscometer equipped
with a small sample adapter (cup diameter 20 mm) using
an SC4-18 spindle with an outer diameter of 17.5 mm.
The measurement accuracy was not less than one percent.
A VPZh 2 viscometer with a capillary diameter of 1.31
mm was used for reference measurements. The experi-
ments were conducted using a thermostat to provide meas-
urements at a given temperature. Test measurements of
the viscosity of pure ethylene glycol were performed. The
resulting viscosity 17.1 sP at a temperature Т = 25˚C
agrees to within one percent with available experimental
data [42]. The data obtained by the capillary and rotational
viscometers were identical within the measurement error.
Since the main objective of this work was to determine
the dependence of the viscosity of the nanofluid on na-
noparticle size, it was necessary to accurately determine
the average size of the particles and their size distribution.
A typical electronic photograph of nanoparticles with an
average size of 28.3 nm is shown in Figure 3. The dif-
ferential particle size distributions f obtained after proc-
essing of an ensemble of such photograph are presented in
Figure 3. Electronic photograph of silica nanoparticles with
an average size of 28.3 nm.

V. Ya. RUDYAK 273
Figure 4. Here rhombuses correspond to an average par-
ticle size of 18.1 nm, triangles to 28.3 nm, and squares to
45.6 nm. In all cases, these distributions were found to be
lognormal.
dependence of the viscosity of a nanofluid is its most
important thermophysical characteristic. In liquids, unlike
in gases, the viscosity coefficient decreases with increas-
ing temperature. From physical considerations, similar
behavior should be expected for nanofluids. In nearly all
known works where this dependence was determined, the
viscosity of nanofluids indeed decreased with increasing
temperature. An exception is a paper [53], where contra-
dictory data were presented. The general bibliography of
papers dealing with the temperature dependence of nan-
ofluid viscosity contains about 50 references, some of
which can be found in reviews [6,7]. The temperature
dependences of viscosity obtained in all studies are fairly
typical. Naturally, the viscosity depends on the volume
content of nanoparticles. As an example, Figure 6 gives
data on the temperature dependence of viscosity obtained
for an ethylene glycol nanofluid [60].
The resulting dependences of the increase in the vis-
cosity 01

  on the volume concentration of
nanoparticles are presented in Figure 5. These measure-
ments were performed at 25˚C. Here, as above, rhombuses
correspond to particles with an average size of 1.18 nm,
triangles to 28.3 nm, and squares to 45.6 nm. The solid
line corresponds to the viscosity predicted by Einstein's
theory: 2.5

  . The viscosities of the three fluids
considered are different and increase with decreasing
nanoparticle size. The viscosity increases significantly
with increasing particle concentration, and at a volume
concentration of nanoparticles equal to seven percent, the
viscosity of the nanofluid with the largest particles in-
creased by 40%, and that of the nanofluid with the smallest
particles increased by almost 80%.
Many different correlations have been proposed to de-
scribe the temperature dependence of the viscosity on
nanofluids, but they are all not universal and vary widely,
depending on the concentration of nanoparticles, their
material and size, and the viscosity of the base fluid. For
this reason, it is useful to understand the temperature
dependence of the relative viscosity of nanofluids ηr =
η/η0. For the nanofluid shown in Figure 6, the temperature
dependence of ηr for different concentrations of nanopar-
ticles is shown in Figure 7 [60]. At low to moderate
concentrations of nanoparticles, the relative viscosity co-
efficient does not change with temperature, and decreases
somewhat (by about 3%) only at a concentration of 8.2%.
A similar dependence was observed in studies [42,46,48]
dealing with nanofluids based on ethylene glycol and TiO2
particles, an aqueous solution of ethylene glycol and SiO2
particles, and water and Al2O3 and CuO particles, respec-
tively.
6. Temperature Dependence of Nanofluid
Viscosity
In Section 3, we have already noted that the temperature
7. Rheology of Nanofluids
Figure 4. Differential distributions of SiO2 nanoparticles by
size. Until now, there have been very few systematic studies of
the rheology of nanofluids. This is not surprising since the
rheological behavior of nanofluids depends on many fac-
0
10
20
30
20 3040 50 60
T,
o
C


0%
2.70%
4.80%
Figure 6. Temperature dependence of the viscosity of eth-
ylene glycol base d nanofluid containing 28.3 nm SiO2 nano-
particles versus temperature for different volume concen-
trations.
Figure 5. Viscosity of an ethylene glycol nanofluid with SiO2
nanoparticles versus their volume concentration.

V. Ya. RUDYAK
274
0.8
1
1.2
1.4
1.6
1.8
2
20 30 40 50 60
T,
o
C

r
0.61%
1.30%
2.70%
3.99%
4.80%
8.20%
Figure 7. Temperature dependence of the relative viscosity
coefficient o f eth ylene gl ycol based nanofluid containing 28.3
nm SiO2 particles for different volume conce ntr ations.
tors: the concentration of nanoparticles, their size, mate-
rial, and temperature. It is nevertheless useful to give a
brief overview of typical results.
The rheology of a water-based nanofluid containing 2
to 4% Fe2O3 particles has been studied by Phuoc and
Massoudi [61], who reported non-Newtonian behavior
this nanofluid. However, the nanofluid was stabilized using
polymer dispersants (polyvinylpyrrolidone or polyethyl-
ene oxide) which themselves could change the rheology
of water. Nevertheless, it was found that at a nanoparticle
concentration less than 0.02%, the nanofluid remained
Newtonian. With further increase in the concentration of
nanoparticles, non-Newtonian behavior of the nanofluid
was observed. It is important to note that the presence of
yield stress was recorded, and the nanofluid behaved as a
shear-thinning non-Newtonian fluid.
Water-based nanofluids with the same particles were
later studied in paper [43] without using dispersants and at
much higher concentrations of nanoparticles: from 5 to
20% by weight. Nevertheless, in all cases, the nanofluid
behaved as a Newtonian one. Yet, it is worth noting that
the particles used were quite large, with an average size of
67 nm.
Newtonian behavior of ethylene glycol based nanoflu-
ids containing copper particles was also observed in [44].
Here large particles (average size 200 nm) were used, and
their concentrations did not exceed 2%.
In general, as shown in [47], the rheological behavior of
nanofluids depends on temperature. This, of course, is not
surprising. Namburu with colleagues [47] studied the
rheological behavior of an aqueous solution of ethylene
glycol containing SiO2 nanoparticles of three sizes: 20, 50,
and 100 nm. At a 6% volume concentration of nanoparti-
cles, nanofluids with 50 nm particles exhibited Newtonian
behavior at temperatures above −10˚C, but became non-
Newtonian at lower temperatures.
Thus, it can be argued that almost all nanofluids begin
to show non-Newtonian properties as their concentration
increases. Furthermore, the rheology of nanofluids gen-
erally depends on the nanoparticle size. Newtonian be-
havior of nanofluids at lower volume concentrations of
particles appears the sooner the smaller the nanoparticle
size. This is confirmed by our experiments with ethylene
glycol nanofluids containing SiO2 particles [59,60]. Re-
lated to this is the fact that in a study [62] of the rheology
of a propanol nanofluid with 10 nm Al2O3 particles, non-
Newtonian behavior was observed already at a 0.5% vol-
ume content of the nanoparticles.
8. Discussion and Concluding Remarks
The experimental data and molecular dynamics simula-
tions of nanofluid viscosity discussed in the previous sec-
tions of this paper allow us to make definite conclusions
and formulate the problems that should be solved in the
near future. First, we can speak of Newtonian behavior of
nanofluids only in the case where the base fluid is New-
tonian and the nanoparticle concentration is not too high.
Apparently, the volume concentration of nanoparticles in
this case does not exceed 10% - 15%. The effective vis-
cosity coefficient at such concentrations can be repre-
sented in the form (10). Now, however, the coefficients in
this equation should be functions of the nanoparticle size d


2
01 2
1kd kd
 


 

. (18)
All experimental data and molecular dynamics simula-
tions suggest that at a fixed volume concentration of na-
noparticles, the nanofluid viscosity is significantly higher
than the viscosity of conventional suspensions. Why? In
fluids in which there is a short-range order and the short-
range molecules are in quasi-bound states, one of the main
mechanisms of momentum transfer, as already noted, is
associated with the destruction of the short-range order
and has characteristic scales of the order of nanometers.
How does the presence of a nanoparticle affect the short-
range order in a fluid? Figure 8 presents the results of
molecular dynamics calculations of nanoparticle-mole-
cule radial distribution functions

2
g
r for an argon
based nanofluid containing 2 nm diameter Li nanoparti-
Figure 8. Nanoparticle-molecule pair distribution functions
at the same pressure.

V. Ya. RUDYAK 275
cles at a temperature of 150 K. The radial distribution
function of the base fluid is shown by the solid curve, and
the radial distribution functions of the nanofluids with 5%
volume concentrations of nanoparticles are shown by the
points line. The pressure and temperature in this two sys-
tems are the same. The nanofluids are found to be more
ordered than the base fluid. The degree of order of the
nanofluid is increased if the particles concentration grows.
Note that the ordering is due not only to an increase in the
first maximum of the radial distribution function corre-
sponding to the molecules of the immediate environment
of the nanoparticle. The second and third maxima in the
suspension also increase severalfold. In addition, the fifth,
sixth, and seventh maxima appear, which are virtually
absent in the base fluid. The characteristic linear scale of
the short-range order of molecules near the nanoparticle is
approximately twice that in the base fluid. Increasing the
degree of ordering of the fluid increases its effective vis-
cosity.
It is interaction with these microfluctuations that de-
termines the main mechanism of relaxation of the nanopar-
ticle velocity [3-5]. Energy is spent in producing such
fluctuations, which accelerates the relaxation of the par-
ticle velocity and increases the viscosity of the medium. In
fact, the production of density and velocity microfluctua-
tions is similar to the generation of perturbations of the
fluid flow field by the motion of conventional dispersed
particles considered in Einstein's theory [12]. These last
perturbations, however, do not depend on the size of the
dispersed particles. Why, then, does the nanoparticle size
have such a significant effect on the viscosity of nan-
ofluids? A simple explanation of this may be found in the
following consideration. In a suspension with macroscopic
particles at volume concentrations less than or of the order
of φ ~ 10−3, the distances between the particles are large
enough, so that their interaction can be neglected. Indeed,
suppose that the dispersed particle size is d ~ 10−4 cm.
Then, at the specified volume concentration of these par-
ticles, the average distance between them is


3
~
p
mp
ln

,
where


3
6π
p
n  
At low flow velocities, especially typical of microflows,
the Brownian motion of particles plays a significant role
in the viscosity of suspensions. A moving nanoparticle
induces density and velocity microfluctuations in the car-
rier medium. These microfluctuations have been modeled
by the molecular dynamics methods for motion of an
isolated nanoparticle in a molecular fluid [3,5,63]. A
fragment of the carrier fluid velocity field around a na-
noparticle at some time is presented in Figure 9. The
arrows indicate the directions and magnitudes of the ve-
locity of the medium. The velocity of the nanoparticle is
directed to the right, and a part of the boundary of the
nanoparticle is depicted by the arc. A vortex structure can
be seen near the surface of the particle. It has a toroidal
shape and is in the plane passing through the center of the
particle and perpendicular to the direction of its velocity.
The diameter of the formed vortex is of order of the na-
noparticles size.
d is the number density of the par-
ticles. For these data, np ~ 2 × 109 and 3 ~10
p
m
l

cm. At
the same time, the mean free path of a pseudo-gas of these
particles is


22
1~510d 
pp
ln cm, i.e., p
mp
ll

.
The van der Waals parameter of this gas is equal to
3
~2 10
p
nd 3

. In the kinetic theory of dilute gases, it has
been shown that in this case, pair collisions of particles
should be taken into account. However the mutual influ-
ence of particles on the perturbations induced by these
particles in the carrier fluid velocity field can be neglected.
In this case, the viscosity coefficient of the suspension is
described by Einstein’s formula (4).
In nanofluids, the particle size is of the order of σ × 10−7
cm, where the parameter σ takes values of 1 to 102. At the
same volume concentration (φ ~ 10−3), the average dis-
tance between nanoparticles in the fluid is


3
~
np
mnp
ln

.
The number density of nanoparticles in this case is of the
order of 19 3
~10
np
n


and varies from 2 × 1019 to 2 ×
1013, depending on the size of the particles. Accordingly,
the distance between the nanoparticles

3
n p
n~
np
m
l

varies
from 4 × 10−7 to 4 × 10−5. In all cases the distance between
the nanoparticles are of the order of their size. On the
other hand the physically infinitesimal scale for the con-
tinuous medium (fluid) is of the order 0 ~
h [64],
where 0 and L are the characteristic diameter of the
molecule and the characteristic linear size of the flow,
respectively. It is easy to see that in the metric of the
continuous medium, the distances between nanoparticles
are almost always infinitesimal and their hydrodynamic
interaction should be considered even at these low con-
centrations. In formula (10) for conventional suspensions,
it is the coefficient b that mainly takes into account the
mutual influence of nanoparticles on the perturbations
they induce in the velocity field of the suspension. There-
rd L
d
Figure 9. Ve loc ity f ield of the ca rri er f lui d i n the vic in it y of a
nanoparticle.

V. Ya. RUDYAK
276
fore, for nanofluids with low concentrations (in the lin-
earized approximation for the concentration) of nanopar-
ticles viscosity, the viscosity coefficient should be repre-
sented as:

01ab
 




. Batchelor [29,30], took
into account the contribution of the hydrodynamic inter-
action and Brownian motion of dispersed particles and
obtained a coefficient b = 6.2. If this value is substituted
into the above formula, the viscosity coefficient in the
linear approximation for the concentration is equal to


018.7

   . This value would provide a good fit to
most of the available experimental data for nanofluids
with small volume concentration of nanoparticles. Figure 10. Viscosity of an ethylene glycol nanofluid con-
taining SiO2 particles versus nanopartic le size .
There is also a second important point. The van der
Waals parameter of a pseudo-gas of nanoparticles is of the
order . It is known from kinetic theory
that this corresponds to a fairly dense gas in which not
only pair collisions of particles but also multiparticle
(three-particle, etc.) collisions should be taken into ac-
count. A nanoparticle interaction potential that takes into
account the forces of both attraction and repulsion was
constructed in [65]. The scale of the attractive forces is of
the order of several sizes of nanoparticles. It follows that,
even at such low concentrations, the contribution of the
direct interaction of nanoparticles is also appreciable and
should be considered when developing a coherent theory
of nanofluid viscosity.
3
~2 10
np
nd  
2

molecule. Thus, correlation (21) allows for the possible
dependence of nanofluid viscosity on the physical char-
acteristics of the base fluid. Moreover it may hope that
correlation in form (21) will be applied for description of
the viscosity coefficient at any temperature. Certainly, the
volume fraction of the nanoparticles should be rather low.
However, it does not take into account the possible
dependence of nanofluid viscosity on nanoparticle mate-
rial. Unfortunately, as yet, there have been no experiments
giving a clear answer to the question of whether such
dependence exists or not. Molecular dynamics simula-
tions [56,59,66] indicate that for sufficiently small parti-
cles, this dependence exists. However, the results of the
kinetic theory of gas nanosuspensions [5,21] indicate that
starting from certain sizes of nanoparticles (about 10 nm),
this dependence should disappear.
In constructing a correlation for nanofluid viscosity, it
is necessary to proceed from formula (18). It is important,
however, that in the limit of macroscopic fluids, it would
reduce to the classical relations. Therefore, (18) needs to
be represented as With further increase in the concentration of nanopar-
ticles, it is necessary to take into account the possible
structuring of the pseudo-gas of nanoparticles. In addition,
due to the Van der Waals forces between nanoparticles,
they can coagulate to form agglomerates. In this case the
structure of nanofluids may resemble the structure of
polymer solutions. From this point of view, it is not sur-
prising that at high concentrations of nanoparticles, nan-
ofluid become non-Newtonian. The study of the rheologi-
cal properties of liquids requires systematic experiments
with varying concentration, material, and shape of nanopar-
ticles and varying properties of the base fluid. Interesting
results can also be expected in the case where the base
fluid is itself non-Newtonian.
 

2
01 1
12.5 6.2mD mD
 
 

, (19)
Here, the coefficient b is taken to be the value obtained
by Batchelor [29,30], which provides a good fit to many
experimental data. To specify the dependence of the co-
efficients mi on the particle size, it is necessary to take into
account two factors. First, it is desirable that in the limit of
large particles, this correlation reduce to the classical one.
Second, the number of the parameters in the correlation
should not be too large. Figure 10 shows the dependence
of nanofluid viscosity on nanoparticle size obtained in
experiments [59,60]. This dependence is well enough
described by an exponential. Therefore, correlation (19)
can be written as

2
01 2
12.5 6.2
Dd Dd
nen e
 



 
. (20) REFERENCES
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

0
0
0.013
0
0.013 2
12.5 13.43
7.35 38.33
dd
dd
e
e






 
 , (21)
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