1. Introduction
1.3. Viscosity
The evaluation of rheological properties of suspension are essential for various applications.
Viscosity is one of the most important rheological property from practical point of interest.
Viscosity of a suspension depends on the structural organization and interaction of constituents within the suspensions. Viscosity of suspension increases with the increase in solid volume fraction. There has been significant effort in the literature to characterize the viscosity of different types of suspension, both from theoretical point of view and from experimental focusing on dilute, semi dilute and concentrated systems. Assuming the absence of interactions between the suspended particles in a dilute suspension of spherical particles in Newtonian fluid, Einstein (1906) predicted the bulk viscosity of the dilute suspension. He showed that the effective suspension viscosity linearly increases with the spheres volume fraction and derived the following formula:
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Chapter 1 (1 2.5 )
η η= o + φ (1.3)
Equation (1.3) is valid for ߶ << 1. In the above equation, η is the effective viscosity of the suspension ηo is the viscosity of the suspending fluid and φ is particle volume fraction.
For more concentrated suspensions it is necessary to consider corrections to the viscosity that are of higher order in the volume fraction. When the flow field around a sphere is influenced by the presence of neighboring spheres, the hydrodynamic interactions cannot be neglected and they could be treated as a contribution toη that is proportional to
φ
2 for two bodies, toφ
3for three bodies and so on. Batchelor and Green (1972a, 1972b) expressed bulk suspension viscosity in terms of higher order of particle volume fractions as follows.(1 2)
o a b
η =η + φ+ φ (1.4)
In the above equation, the parameters are defined as a=2.5and second order coefficient 6
. 7
=
b and it becomes equal to 6.2 when Brownian motion is considered (Batchelor, 1977).
Ball and Richmond (1982) assumed that in a concentrated suspension the effect of all the particles is additive and proposed the following correlation:
η ηo C 2φm
5
) 1 ( − −
= (1.5)
Here, C accounts for the so-called “crowding” effect. Ball and Richmond's expression is effectively identical to that of Krieger and Dougherty (1959) whose theory also states that, in the general case, the 5
2factor should be replaced by the intrinsic viscosity. The value of 5 2is the intrinsic viscosity for an ideal dilute suspension of spherical particles. To simplify a little, from now onwards, we will be denoting,
8
o
s η
η = η (1.6)
as the normalized viscosity (relative viscosity), where η is the effective suspension viscosity.
Many rheologists measured and modeled the experimental data via various phenomenological equations. In almost all of these models the maximum possible packing fraction of particles
φ
m is taken into account in the expression of effective suspension viscosity.A variety of empirical equations have considered dependence of effective viscosity (ߟ௦) on particle concentration(߶). Some of these equations have been proposed by Maron and
Pierce (1956), Krieger and Dougherty (1959), Kriger,(1972), Leighton and Acrivos (1986) and Morris and Boulay (1999) to determine the dependence of effective viscosity on particle volume concentration. We have shown only a few of these and one can chose the equation which best represents the system. The maximum particle volume fraction is one of the important parameter in these equations. According to the Dorr et al. (2013), the majority of the models are derived either analytically or by the fitting experimental data. Evidently each of the approaches has its limitations. Krieger and Dougherty (1959) formula is one of the most used one. Maron and Piece (1956) model is valid for low concentrations and all other models are applicable to wide range of concentrations (φ to φm). In this work we have considered the correlation proposed by Krieger (1972) which is found to work well for suspension of mono- dispersed non-colloidal particles. The absolute value for the maximum packing fraction (φm) depends on the particle shape and particle size distribution. In the Kreiger model, we have used
0.68.
φm= However, φm could be considered an adjustable parameter. The models mentioned above are given below as follows.
9
Chapter 1
Maron and Pierce (1956):
2 s 1
m
η φ
φ
−
= −
(1.7)
Krieger and Dougherty (1959):
2.5
1
m
s
m
φ φ
η φ
−
= −
(1.8)
Leighton and Acrivos (1986):
2
m s
1 5 . 1 1
− +
=
φ φ
η φ (1.9)
Morris and Boulay (1999):
1 2 2
1 2.5 1 0.1 1
s
m m m
φ φ φ
η ϕ
φ φ φ
− −
= + − + −
(1.10)
Zarraga et al. (2000): 3
m 34 . 2 s
1 e
−
=
−
φ φ η
φ
(1.11)
Krieger (1972) has given the following empirical correlation for suspension viscosity that could be fit for concentrated suspensions:
1.82
( ) o 1
m
η φ η φ φ
−
= −
(1.12)
Zarraga et al. (2000) measured ߶ experimentally by using fluid displacement method and found it to be 0.62. The value of φm is highly sensitive especially when the particle concentration approaches the level close to it (φ ∼φm). It implies that there is no longer sufficient fluid to lubricate the relative motion of particles and the viscosity rises to infinity. In the present study, the maximum packing fraction is set to the value of 0.68 in computations which is reasonable only for rigid mono-dispersed spherical particles and this was also considered by Phillips et al. (1992) in their simulation of circular Couette flow. If one were to
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predict effective viscosity with good precision then the value of φm must be close to the realistic value suitable for that particular type of suspension. φm has also been determined by numerical simulations of random close-packing of monodisperse spheres. Rintoul and Torquato (1996) obtained a value of φm ≈0.62. and Kitano et al. (1981) found that φm ≈0.68.
Experiments of Chong et al. (1971) gave the value of maximum packing fractions as 0.605 whereas it was between 0.55 - 0.71 in the experiments of Shapiro et al. (1992).