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Differential multiphase models for polydispersed suspensions and particulate solids
J,m~d ELSEVIER
J. Non-Newtonian Fluid Mech., 72 (1997) 305-318
Differential multiphase models for polydispersed suspensions and particulate solids N. P h a n - T h i e n a,,, D.C. P h a m b a
Department of Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia b Institute of Mechanics, 224 Doi Can, Hanoi, Vietnam Received 17 March 1997
Abstract
We report some novel applications of the differential scheme to construct the effective viscosity and moduli of suspensions of spheres of diverse sizes. Exact closed form expression for the effective viscosity of droplets of diverse sizes is derived, which includes the generalised Einstein formula of Brinkman and Roscoe. Exact closed form solutions for the effective moduli of a particulate solid consisting of rigid spheres of diverse sizes embedded in an elastic solid shows that the Poisson's ratio of the composite should approach the value 1/5 at high volume fraction. The effect of a limited range of sizes of the included phase and locally-inhomogeneous distribution of inclusions is taken into account by considering a three-component system, with a fictitious inclusion phase made up of the matrix (solvent) material, which limits the space available for the real inclusions. © 1997 Elsevier Science B.V.
Keywords: Multiphase models; Suspensions; Particulate solids
1. Introduction One speaks of a suspension or a particulate solid when a typical dimension of the embedded particles (the microscale) is considerably greater than that of the solvent or matrix molecules, but is considerably less than a typical length scale of the deformation field (the macroscale). For suspensions, there will be a range of sizes of particles above which Brownian motion is unimportant. This is the particles' size range that we are concerned with in this paper. There are several approaches to set up a constitutive equation for suspensions and particulate solids. Bounds of effective properties have been derived based on variational principles and geometric information [1-9]. Asymptotic solutions focused on pairwise interaction of two generic particles have been derived, allowing the effective viscosity and moduli of the material * Corresponding author. 0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 5 7 ( 9 7 ) 0 0 0 2 8 - 1
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N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
to be evaluated at high volume fractions [10-14]. In addition, these solutions allow computation techniques to be developed, capturing the evolution of the suspensions [15,16]. Other exact models of suspensions and particulate solids have also been derived, for example the periodic materials [17], and the differential models scheme originally due to [18]. In particular, although the differential scheme has been developed successfully by many authors to estimate the overall properties of suspensions and particulate solids consisting of two phases [19,20,7,21-25], it is not well-known in the fluid mechanics literature, and to our knowledge, has not been generalised to multiphase materials. The basic idea in the differential scheme is quite aesthetiscally simple: the composite material is constructed from an initial material through a series of incremental additions of compounds materials until their final volume fractions are reached. At each stage of the process, the relevant solution of a dilute suspension problem is used to construct the effective properties of the composite, to be employed as the base properties of the next incremental step. The scheme is idealistic, but its simplicity, exactness, explicitness and flexibility makes it attractive for applications in suspensions and particulate solids. Several interesting results have been obtained by the scheme: Brinkman and Roscoe [19,20] used the scheme to generalise the well-known Einstein formula for the effective viscosity of suspensions of rigid particles at finite concentrations. The method predicts the empirical Archie's law for the effective conductivity of rocks [21]; it has also been used to construct geometric models of two-phase composites with maximal and minimal elastic moduli [7,25]. The scheme can be used to model not only asymmetric matrix composites, but also the quasi-symmetric ones [5]. In this paper, we report some interesting applications of the method. Models of suspensions of rigid particles and droplets and of particulate solids consisting of several elastic phases embedded in an elastic matrix are constructed, yielding exact close-form expressions of the effective properties. These models will be useful in correlating experimental data, and in testing constitutive equations and numerical solutions.
2. Two-phase materials To illustrate the method, we first construct the effective properties of two-phase materials. In essence, the construction process starts with a base material (a Newtonian fluid or a Hookean solid). Into this base material, a small volume fraction of spheres (a Newtonian fluid or a Hookean solid) is added. The effective medium is either a Newtonian fluid, or a Hookean solid. Into this effective medium, a small volume fraction of spheres, of considerably larger sizes than previously-added spheres, is introduced. The new set of spheres will see an effective medium, with previously calculated effective properties. The properties of the new effective medium can be calculated, and serves as the base properties for the next step. The construction process continues until a volume fraction of the added phase is reached, where one has a suspension of spheres of diverse sizes in the original solvent or matrix. 2.1. Suspensions o f droplets
Let us now consider a suspension of droplets at a volume fraction 0 < v < 1. The construction process starts by adding an infinitesimal volume fraction c~v of a droplet phase, a Newtonian
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
307
fluid of viscosity q~, in a Newtonian solvent, of viscosity r/s. The effective medium is Newtonian, with a viscosity given by [26,27]. r / ' = q s ( 1 + 6vqS + (5/2) + th th)
(2.1)
This viscosity plays the role of q~ in the next step of introducing yet another infinitesimal volume fraction of droplets, of considerably larger sizes than those of the previous step, so that the new droplets will see an effective medium at all steps. At an intermediate step of droplet volume fraction v, let us denote the effective viscosity by ~/. To a normalised unit volume of this effective Newtonian fluid, we add yet another infinitesimal volume Av of droplets. The new mixture is a suspension of droplets, of volume fraction Av 1 +Av
in a Newtonian fluid of viscosity r/. Its effective viscosity increases by an infinitesimal amount d~/, where Av r/+ (5/2) ~/~ q + d~/= 1/+ - q. l+Av I/+ql
(2.2)
Since the real volume fraction of droplets increases by
v + Av dv - A 1 +~
Av v - A-----~ 1+ (1 - v),
(2.3)
we find the following differential equation (hence the name of the scheme) for the effective viscosity of our droplet model dq dv
1 r/+ (5/2)/11 1 - v r/+ rh ~/'
0 < v < 1,
r/(0)
t/s.
(2.4)
which has the solution
U= 1 __(~)2/5 (21~s-~-51~1~3/'5 \2-qq ~ 5--~-~/
'
(2.5)
When the droplets are rigid, r/1 ~ ~ , this yields the effective viscosity t//~/s = (1 - v) -s/2,
(2.6)
which has been noted by Roscoe [20]. In the limit of small volume fractions, this further reduces to Einstein's relation ~I/qs = 1 + 5v/2. Relation (2.6) has been evaluated by Roscoe [20] against available data on suspensions of rigid spheres. He concluded that it works well for a suspension of spheres of diverse sizes at low volume fractions. For a suspension of spheres at high volume fractions and limited dispersity, he recommended replacing the volume fraction v with 1.35v, so that the maximum allowable volume fraction is 1.35- ~ --- 0.74. The rationale behind this substitution is that a certain amount of fluid is effectively frozen between the spheres at high volume fractions, causing the effective volume fraction of the rigid phase to be higher than it actually is.
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
308
When the viscosity of the droplets is nil (bubbles), (2.5) yields q/~=(1-v)-',
(2.7)
with the corresponding low-volume fraction limit ~l/qs = 1 + v [27]. We expect that Eq. (2.7) will be a good approximation to a suspension of droplets of diverse sizes. Fig. 1 shows the reduced viscosity r//r/s for a number of values of ~h/qs, including some experimental data on water/oil emulsion, but with a monodispersed droplets [28]; the bars represent the variation in the diameter of the droplets, from 1.4-3.3 pm (the smaller the droplets, the higher is the effective viscosity). It is clear that the experimental results follow reasonably well the prediction for bubbles. The more viscous the droplets, the higher is the effective viscosity. In practice, since particles can only have a limited range of sizes, it is expected that the maximum volume fraction cannot approach unity and a physically reasonable model must reflect this feature. We will return to this issue in a later section. 2.2. Composite materials
We now turn our attention to a particulate solid consisting of elastic spheres (shear modulus kt~ and bulk modulus Xl) of diverse sizes embedded in an elastic matrix of shear and bulk moduli /'/m and Xm, respectively. The construction process for the effective moduli follows the same
1000 Rigid
t~
100 111 /'112 =
o
>
"ql/q2~ ®
rh ~2 =.o.5 /
10
Bubble
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Volume fraction Fig. 1. Reduced viscosity of a suspension of droplets of diverse sizes.
0.9
1.0
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
309
procedure as discussed above. At the intermediate step, where the volume fraction of the inclusion phase is v, and the effective shear and bulk moduli are a and x, respectively, we added an infinitesimal volume Av in a unit volume of the homogenised material. The new mixture is a two-phase composite, with matrix m o d u l i / ~ and K, and inclusion's volume fraction Av/(1 + Av). F r o m the classical inclusion problems for spheres [29,24], we find that the effective moduli increase by lc+dK=K4
Av K1 - x 1 + Av 1 + (~cl - tc)/(x + ~c,)'
Av /q - / z /t + d l t =/~ 4 1 + Av 1 + (,u~ -/~)/(/t + / t , ) '
(2.8)
where K, =4/~,
/**
9x + 8/~ 6 x + 12/~/~"
(2.9)
Since the increase in the volume fraction of the inclusion phase is still given by Eq. (2.3), we obtain the following coupled differential equations for the effective moduli of the composite dK dv
1 xl - x 1 - v 1 + (Xl -- K)/(t¢ + tO,)'
d/* dv
1 ¢q --/~ 1 - v 1 + (/t x - / 0 / ( / 2 + / t , ) '
K(0) = gin,
¢t(0) = /~m,
0 < V < 1.
(2.10)
AS before, for the scheme to be exact the sizes of the inclusions added at every step must be considerably larger than all of the previous ones, so that one ends up with a particulate of greatly diverse sizes, allowing the volume fraction to approach one. Numerical solutions to Eq. (2.10) are generally required. However, in the very special case where b o t h the matrix and the inclusion phase have a Poisson's ratio of 1/5, i.e., K~ = (4/3)fll and Xm = (4/3)/Zm, then the Poisson's ratio of the composite material always remains at 1/5. In this case, Eq. (2.10) are uncoupled to yield v=l--(ltm'~X/2(Itl--fl'~, k ll / \ l h -- l*m/
K=4/t.
(2.11)
Exact solutions to inclusions problems are rare, and should serve as models for testing constitutive equations. Fig. 2 shows the reduced shear modulus #//~m according to Eq. (2.11) for the case where /Xl//~m = 10 (the curve labelled 'exact'), together with the numerical solutions to Eq. (2.10) for a variety of parameter values, ]-/rn= 1 , ~/1 = 10, Km = 0.1 - 10, ~:l = 10 - 20. The numerical solutions are sufficiently close to the exact solution Eq. (2.11) to r e c o m m e n d its use in general In the limiting case of rigid inclusions, both Xl,/q---, oo, and Eq. (2.10) become d~ --= dv
, ( 4) x+ /l 1- v 3
'
310
N. Phan-Thien, D.C. Pham / J . Non-Newtonian Fluid Mech. 72 (1997) 3 0 5 - 3 1 8
10
/
9
rigid Nn = 10 ,"
8
~:"
o,'
7 "o 0
/
F__
6
Ill ¢-
5
"o ® o "o
/
1
rigid r m = O. 1 /
t ,ss°~
exact
4 3
W,m= 10 I,tm= 1 Ic1 = 2 0 p . 1 = 10
oos"~
er
2 r-~a= 1 gin= 1 K1 = 11.tl = 10
1 0
---------, 0.0
0.2
0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Volume fraction Fig. 2. The reduced shear modulus of a two-phase composite material for values.
d/~ dv K (0)
a variety of parameter
1 15~c+ 20/z 1 - v 6x+12/z /~' =
Km,
fl (0) =/~m,
For the special case
~m
~/l/~/rn= 10 and
( l - - v ) 2'
Km =
0 < V < 1.
(2.12)
(4/3)/~n,, these equations are uncoupled as before to yield
4
K=gfl.
(2.13)
This case is also plotted in Fig. 2 (the curve labelled 'rigid'). It is to be compared with the numerical solution of Eq. (2.12) where Krn/~lm = 0 . 1 , 10. (The reduced bulk modulus behaves in a similar manner and is not shown here for brevity.) Clearly, the reduced moduli for a composite of rigid inclusions increase much faster with the volume fraction than those of elastic inclusions. At high volume fractions, it is interesting to find that the ratio of bulk and shear moduli converges to 4/3 independent of original ratio of Krn/fl m. Thus, the Poisson's ratio of a composite of diverse sized rigid inclusions approaches 0.2, independent of the values of the matrix moduli. The limit of K1, /~1 --' 0 corresponds to a matrix filled with voids, which is also of practical interest. In this case, Eq. (2.10) become
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
311
,
dK
dv -
--K 1- v
1+
d/~ dv
1 --l, 1- v
14
/z(0) : / z ~ ,
K(O) = ~m,
0 ~ v ~ 1.
(2.14)
For the special case Xm = (4/3)/tm these equations are uncoupled, yielding the solution - -/~ -~-
(1
-- /.))2
(2.15)
X = ~4 ~/.
]/m
In general, Eq. (2.14) have to be solved numerically. We remark here that if the inclusion phase has the same moduli as the matrix phase, then the effective moduli remain unchanged. This fact will be used later to construct models where the volume fraction of the included phase is not allowed to approach unity, as must be the case of a limited diverse inclusion sizes.
3. Multiphase materials 3. I. Composite materials
We now consider a multiphase composite material, consisting of n different inclusion phases of shear and bulk moduli/z~, x~, respectively, and volume fraction v~ (~ = 1,..., n) in a matrix of shear modulus /Zm and bulk modulus Xm. The construction of the multiphase composite material follows the same procedure described in the previous section, starting with the base matrix. At each step of the procedure, we add proportionally infinitesimal volume amounts v, At, At << 1, 0~= 1,..., n of elastic spheres into an already constructed composite of the previous step, which contains volume fractions vat, of the inclusion phases (the parameter t increases from 0 to 1, as the differential scheme proceeds). The particles added at this step must be considerably greater in sizes to those that have been added previously. The new mixture is considered as a dilute suspension of spherical particles from phase ~, of volume fraction v~ At 1+ ~
v~At
~=1
in an elastic matrix of shear and bulk moduli/~ and to, respectively. The effective moduli of the new composite material is v~ At
K+d~c=K+ ct=
1
1+
I) I
v~ At
/~ +d/~ = / t + c~=l
x~ - ~c
At 1 + (K~ - K)/(K + x , ) ' /z~ - lz
1 + '/)1At 1 + (/z~ ~-~-3~-/~ + ~ , ) '
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
312
where Vl ~" 2,,a Uc~ a=l
is the volume fraction of the included phases, a n d / t , and x , are still given by Eq. (2.9). Since the volume fraction of the included phase 0~ increases by v j + v= A t l+vtAt
v=dt-
vJ
v= A t l+vzAt(1--Vlt)'
We obtain the following differential equations for the effective moduli of the composite dx dt
1 --
dl~
~ 2,~
K= - K
1 - Vzt ~ = 1
Vc~
1 + (K= - x ) / ( K + K , ) '
1
~ lt= -- t~ V= L 1 - vzt ~=, 1 + (It= - tt)/(/~ + / ~ , ) '
dt
tc(0) = xm,
/z (0) =/~m,
0 ~ t < 1.
(3.1)
The numerical solution of system of Eq. (3.1) is straightforward. As in the two-component case, when K m = (4/3)/tm and x~ = (4/3)/t~, 7 = 1,..., n, then the system is decoupled, giving a composite material of Poisson's ratio of 1/5 at all volume fractions. 3.2. T w o - p h a s e materials with m a x i m u m
v< 1
In two-phase materials with limited inclusions' sizes, the volume fraction of the included phase cannot approach one arbitrarily, since there are interstitial spaces that cannot be filled with inclusions. In fact, the maximum volume fraction of randomly packed monodispersed spheres is about 0.6 [30]. For a multimodal packing of spheres, the maximum volume fraction of the included phase can be higher, since more spheres can be packed in a given volume. This maximum allowable volume fraction can be incorporated into a differential model by considering a special three-component material, where one inclusion phase is actually made up from the matrix material (which does not contribute to the increase of the effective moduli). More specifically, we consider a composite material consisting of a matrix phase, of volume fraction Vm (shear and bulk moduli Jim and Km, respectively), and two inclusion phases. One inclusion phase is a fictitious one having a volume fraction of Vo and the matrix moduli ~tm, Xm. The second inclusion phase is rigid, of volume fraction vl and moduli ~q, ~---, ~ - - w e will be interested in the limit of infinite inclusion moduli. Thus, since v0 + Vl + vm = 1, the maximum allowable volume fraction for the rigid inclusion phase is 1 - v0 = Vm,x. In this manner, the effects of a limited size range, which limits the maximum allowable volume fraction of the inclusions can be simulated. We expect that Vmax- 0.6 for a random distribution of rigid particles. Now, under the conditions stipulated here, Eq. (3.1) become dx dt
to+K, { --
_
_
1 - vlt
UI +
V0
Km_ZX__.~ K m 71- K , )
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
313
d/t _ # + #* {v, + v0 ~__~_-~-.1 dt
1
- - VI t
¢tm + I t * )
~c(0) = Xm,
/~(0) =/zm,
'
0 _< t _ 1.
(3.2)
These two equations become uncoupled in the special case where x~ = (4/3)ktm in which the solution is given implicitly by 1 + ~
-1
=(l_v0_v0
-2,
x=4/t.
(3.3)
As /)l approaches the maximum allowable volume fraction /)max, /~//~m>> 1, and we find ~ (Vn~ax-- V0- 2(2,'r,ax- 1)
(3.4)
/tm The case of Xm, x--' or, which corresponds to an incompressible matrix, is also of great interest, since it would be physically identical to the case of a suspension of rigid particles in a viscous solvent. In this case # can also be regarded as the effective viscosity of a suspension of rigid spheres in a viscous solvent of viscosity /Zm. The differential equation for ~t becomes
_d/~ dt
5
/~
2 1 - vtt
{ /)~ +v0 /tin + (3/2)/t
/~/(0) = ]Am,
0 ~ t ~ l,
(3.5)
which yields the implicit solution ¢z
]/m
[
1 +
3vl - 2Vo
-
1
5~o~(3,,,- 2~,0)
= (1 - v 0 -
Vl)-S/2
(3.6)
51)1
At high volume fractions, vl ~ Vmax, /~/Ftr~>> 1, and we find I/ ~ (/)max -- /)1)- 5(5Vmax 2)/6
(3.7)
/Zm With /)max~ 0.6, this singular behaviour scales as (/)max- Vl) -°83, which is less singular than the empirical Krieger's formula (/)max- V0-~.8 [31] for monodispersed spheres. In Fig. 3, we show the prediction of the differential model for the case where 1 - v0 = Vmax= 0.58, together with the experimental data of Ilic and Phan-Thien [32] for spheres, Gardala-Maria and Acrivos [33] for spheres and coal, and the experimental bounds of Thomas [34]. The bounds represent the range of the scatter in the data obtained from different sources. The prediction of the differential model (solid line) is closer to Thomas' lower bound, presumably due to the diverse size range of the inclusion phase. Near the maximum volume fraction, the differential model displays the correct asymptotic behaviour. It is interesting to plot there also Roscoe's formula /t//z m = (1 - 1.35v0 - 5/2
(3.8)
Eq. (3.6) and Eq. (3.8) represent two competing processes in suspensions: while a certain amount of fluid is frozen between the inclusions and behaves like the rigid phase, some amount of fluid tends to occupy some space equal in geometric sense to that of inclusions (the fictitious
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997)305-318
314
inclusions) giving rise to a locally-inhomogeneous distribution of inclusions. One feels that Eq. (3.8) overestimates the amount of frozen fluid, while Eq. (3.6) treating the solvent only as the lubricant plus fictitious inclusions underestimates this effect. Perhaps a combined formula might better fit experimental data. It is interesting to note that Eq. (3.6) is weakly dependent on Vo, at low volume fractions of the inclusion phase (Vl < 0.3, v0 < 0.5). The main effect of the fictitious phase is to limit the space available for the inclusion phase. The case of a suspension of droplets of limited size range can be treated in the same manner. Here if V/s is the viscosity of the solvent, r/1 is the viscosity of the droplets (of volume fraction vl), and Vo is the volume fraction of the fictitious solvent phase, to account for the maximum allowable volume fraction Vmax= 1 - V0, then the effective viscosity is given by solving the differential equation J" (5/2)t/(r/s- I/) r/+ (5/2)q] "~ Vo + v~ t/ , 1 - vlt ~ ~s + (3/2) v/ v/+ rh
dr/
1
dt
v/(0) = r/s,
0 < t < 1.
(3.9)
The limit v/] = 0 corresponds to a suspension of bubbles. In this limit
dq dt
~ {5Vo v&- q + v,} , 1 - vzt 2Vls + 3~ 100
. . . .
~
. . . .
~
'
Differential ......
'
'
'
i
,
,
,
.
,
. . . .
,
. . . .
o o
. . . .
model
Thomas'
bounds
.' o°
• spheres, Ilic & Phan-Thien O coal,Gardala-Maria& Aerivos • spheres,Gardala-Maria&Aerivos
.~_
,
. ./.." . / / / "
_
10 ~) ¢.) -I
n-
.....----" .
.
i
.
I
0.0
'
'
'
'
!
0.1
'
'
'
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!
0.2
'
•
'
'
!
'
0.3
'
'
i
!
0.4
i
.
i
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0.5
.
•
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.
0.6
Volume fraction Fig. 3. Reduced viscosity for a suspension of rigid spheres of diverse sizes. The experimental data of Gardala-Maria and Acrivos, Ilic and Phan-Thien and the experimental bounds of Thoas are also plotted for comparison.
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
I/(0) = q s,
315
(3.1o)
0 < t < 1,
which has the implicit solution q-- 1 +
571
- 1
= (1 - Vo - / ) 1 )
-{5vO+2t'l)/2(v0+vl)
(3.11)
At high volume fractions where v~ ~ Vm,x, we find L
, ~ O ) m a x __ i ) l ) --{8 . . . . . --
5)/3,
(3.12)
which is m u c h weaker than a suspension of rigid particles.
3.3. Three-phasematerials It is clear that a suspension of combined rigid spheres and bubbles of the same size range can be treated in the same manner. In particular, let v~, r/l be the volume fraction and viscosity of rigid phase, v2, r/2--the volume fraction and viscosity of the bubble phase. We have the differential equation:
vzt
d5r / -vq ~1° - a t {
2rlsqS-*l+ 3q ~-~vl+v2},
r/(0) = r/s,
0 1< , _t _<
(3.13)
which has the implicit solution
=
(/')max - - Vl - - U2) -- (Sv0 + 5vl + 2v2)/2(vo + vl + v2)
(3.14)
In case vo = 0, (3.14) reduces to q-- = (1 - v, - v2) (s,,,+2~2)/20~+,~2)
(3.15)
It is interesting to note that a small a m o u n t of bubbles would significantly increase the effective viscosity of a rigid sphere suspension. In Fig. 4, the reduced viscosity of a suspension of rigid spheres, with 10% bubbles, is c o m p a r e d to those of a rigid sphere suspension, a bubble suspension, and a bubble suspension with 10% rigid spheres. The increase in the effective viscosity of a rigid sphere suspension with a small a m o u n t of bubble can be significant, especially at high volume fractions of the rigid phase. It is of interest to compare Eq. (3.15) with the respective formulae for the cases where the rigid and bubble phases are considerably different in sizes. Let us assume that the rigid phase is c o m p o s e d of spherical particles of a scale smaller than that of those of the bubble phase. Then the suspension of rigid particles in the solvent should be treated first as an effective m e d i u m with the viscosity (with the actual inclusion volume fraction of Vl/(1-V2) ) --= r/~
1
vl 1 ----v2
(3.16)
N. Phan-Thien, D.C. Pham / J . Non-Newtonian Fluid Mech. 72 (1997) 305-318
316
1000
rigid spheres . . . . . . rigid spheres & 10% of bubbles ,P
- - bubbles & 10% of rigid spheres
lOO
,t
---.
0
bubbles
._~
..-"
10
n"
s ~'" °
/
//
.
°°°w
..-S> 1 t f _ . j / / / 0 0
0.0
J
0.2
0.4
0.6
0.8
1.0
V o l u m e fraction
Fig. 4. Reduced viscosity of a suspension of rigid spheres with 10% volume fraction of bubbles, and of a suspension of bubbles, with 10% of rigid spheres of the same sizes.
In the next step we consider the suspension of bubbles in the solvent with a viscosity given by Eq. (3.16) and obtain the effective viscosity of the overall suspension r/
(1
-- =
t/s
UI
1 --
~ -5/2 (1 -
v2)-'
/)2//
(3.17)
"
In the reverse case when the rigid particles are of a scale larger than that of the bubbles we get -I1~
1
/)2 1 -
(1
-- I)1) -5/2
(3.18)
v]/
A comparison between Eq. (3.15) and Eqs. (3.17) and (3.18) is given in Fig. 5, where the volume fraction of the bubbles is kept fixed at 0.2, and the reduced viscosity is plotted as a function of the volume fraction of the rigid phase. The increase in the effective viscosity is largest when the rigid spheres are of a smaller size compared with the bubbles, and smallest when the bubbles are of a smaller size.
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
1000
,
.
*
,
I
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,
i
,
I
,
*
*
.
I
,
*
.
*
I
*
.
J
,
I
,
,
i
,
I
,
J
,
.
I
Bubbles & rigid spheres of same sizes
.
.
*
'
'
/
...... Small rigid spheres, large bubbles
"'/
i
,..'/
- - - - Small bubbles, large rigid spheres
1O0
,
317
//
8 > "o ° m
~
10
•
0.0
'
'
'
I
0.1
,
'
'
'
I
0.2
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"
!
'
0.3
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•
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0.4
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0.5
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0.6
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"
'
'
I
0.7
'
'
0.8
Volume fraction Fig. 5. Reduced viscosity of suspensions of rigid spheres and bubbles, of the same and different sizes.
Acknowledgements We thank the University of Sydney for providing support to PDC through an International Development Link Program grant.
References [1] R. Burridge, S. Childress, G. Papanicolaou, Macroscopic Properties of Disordered Media: Lecture Notes in Physics, Vol. 154, Springer-Verlag, Berlin, 1982. [2] Z. Hashin, IUTAM Symposium on Non-homogeneity in Elasticity and Plasticity, Warsaw, Poland, Pergamon, New York, 1959. [3] G.W. Milton, N. Phan-Thien, New bounds on effective elastic moduli of two-component materials, Prec. Roy. Soc. Lend. A 380 (1982) 305-331. [4] D.C. Pham. Bounds on the effective shear modulus of multiphase materials, Int. J. Eng. Sci. 31 (1993) 11-17; Bounds for the effective conductivity and elastic moduli of fully-disordered multicomponent materials, Arch. Rational Mech. Anal. 127 (1994) 192-198; Improved bounds for the elastic moduli of perfectly-random composites, J. Elast. 41 (1995) 1-12; On macroscopic elastic and conductivity properties of perfectly-random cell composite. Int. J. Sol. Struct. 33 (1996) 1745-1755 [5] D.C. Pham, N. Phan-Thien, On the optimal bounds for the effective conductivity of isotropic quasi-symmetric multiphase media (1996) submitted.
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[6] N. Phan-Thien, G. Milton, New bounds on the effective thermal conductivity of N-phase materials, Proc. R. Soc. Lond. A 380 (1982) 333-348; New third-order bounds on the effective moduli of N-phase composites, Quart. Appl. Math. XLI (1983) 59-74. [7] R. Roscoe, Isotropic composites with elastic or viscoelastic phases: general bounds for the moduli and solutions for special geometry, Rheol. Acta 12 (1973) 404-411. [8] K.C. Le, D.C. Pham, On bounding the effective conductivity of isotropic composite materials, Z. angew. Math. Phys. 42 (1991) 614-622. [9] J.R. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites, J. Mech. Phys. Solids 25 (1977) 185-202. [10] F. Gardala-Maria, The Rheology of Concentrated Suspensions, PhD. Dissertation, Stanford University, 1979. [11] N.A. Frankel, A. Acrivos, On the viscosity of a concentrated suspension of solid spheres, Chem. Eng. Sci. 22 (1967) 847-853. [12] N. Phan-Thien, Lubrication approximation in particulate solids, ZAMM 75 (1995) 563-565. [13] N. Phan-Thien, S. Kim, Microstructure in Elastic Media: Principles and Computational Methods, Oxford University Press, New York, 1994. [14] B.H.A.A. van den Brule, R.J.J. Jongschaap, Modeling of concentrated suspensions, J. Stat. Phys. 62 (1991) 1225-1237. [15] J.F. Brady, G. Bossis, Stokesian Dynamics, Annu. Rev. Fluid Mech. 20 (1988) 111-157. [16] I.L.A.M. Claeys, J.F. Brady, Suspensions of prolate spheroids in Stokes Flow, Part 1, Dynamics of a finite number of particles in an unbounded fluid, J. Fluid Mech. 251 (1993) 411-442; Part 2, Statistically homogeneous dispersion, J. Fluid Mech. 251 (1993) 443-477; Part 3, Hydrodynamic transport properties of crystalline dispersions, J. Fluid Mech. 251 (1993) 479-500. [17] K.C. Nunan, J.B. Keller, Effective elasticity tensor of a periodic composite, J. Mech. Phys. Solids 32 (1984) 259-280; Effective viscosity of a periodic suspension, J. Fluid Mech. 142 (1984) 269-287. [18] D.A.G. Bruggeman, Berechnung verschiedener physkilisher konstante von heterogene substazen, Ann. Physik 24 (1935) 636. [19] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571. [20] R. Roscoe, The viscosity of suspensions of rigid spheres, Br. J. Appl. Phys. 3 (1952) 267-269. [21] P.N. Sen, C. Scala, M.H. Cohen, A self-similar model for sedimentation rocks with application to the dielectric constant of fused glass beads, Geophysics 46 (1981) 781-795. [22] R. McLaughlin, A study of the differential scheme for composite materials, Int. J. Eng. Sci. 15 (1977) 237-244. [23] S. Boucher, Modules effectifs de materiaux quasi homogenes et quasi isotropes, constitues d'une matrice elastique et d'inlusions elastiques: II Cas des concentrations finies en enclusions, Revue M 22 (1976) 1. [24] A.N. Norris, A differential scheme for the effective moduli of composites, Mech. Mat. 4 (1985) 1-16. [25] A.N. Norris, A.J. Callegari, P. Sheng, A generalized differential effective medium theory, J. Mech. Phys. Solids 33 (1985) 525-543. [26] G.I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. Roy. Soc. Lond. A 146 (1934) 501-523. [27] G.K. Batchelor, An Introduction to Fluid Dynamics, CUP, Cambridge, 1970. [28] C.D. Han, R.G. King, Measurement of the rheological properties of concentrated emulsions, J. Rheol. 24 (1980) 213-237. [29] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. Roy. Soc. Lond. A 241 (1957) 376-396. [30] J.G. Berryman, Definition of dense random packing, in: M. Shahnipoor (Ed.), Advances in the Mechanics and the Flow of Granular Materials, Vol. I, Trans. Tech. Publications, 1983. [31] I.M. Krieger, Rheology of monodispersed lattices, Adv. Coll. Int. Sci. 3 (1972) 111-136. [32] V. Ilic, N. Phan-Thien, Viscosity of concentrated suspensions of spheres, Rheol. Acta 33 (1994) 283-291. [33] F. Gadala-Maria, A. Acrivos, Shear-induced structure in a concentrated suspensions of solid spheres, J. Rheol. 24 (1980) 799-814. [34] D.G. Thomas, Transport characteristics of suspensions: VIII, A note on the viscosity of Newtonian suspensions of uniform spherical particles, J. Colloid Sci. 20 (1965) 267-277.
J. Non-Newtonian Fluid Mech., 72 (1997) 305-318
Differential multiphase models for polydispersed suspensions and particulate solids N. P h a n - T h i e n a,,, D.C. P h a m b a
Department of Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia b Institute of Mechanics, 224 Doi Can, Hanoi, Vietnam Received 17 March 1997
Abstract
We report some novel applications of the differential scheme to construct the effective viscosity and moduli of suspensions of spheres of diverse sizes. Exact closed form expression for the effective viscosity of droplets of diverse sizes is derived, which includes the generalised Einstein formula of Brinkman and Roscoe. Exact closed form solutions for the effective moduli of a particulate solid consisting of rigid spheres of diverse sizes embedded in an elastic solid shows that the Poisson's ratio of the composite should approach the value 1/5 at high volume fraction. The effect of a limited range of sizes of the included phase and locally-inhomogeneous distribution of inclusions is taken into account by considering a three-component system, with a fictitious inclusion phase made up of the matrix (solvent) material, which limits the space available for the real inclusions. © 1997 Elsevier Science B.V.
Keywords: Multiphase models; Suspensions; Particulate solids
1. Introduction One speaks of a suspension or a particulate solid when a typical dimension of the embedded particles (the microscale) is considerably greater than that of the solvent or matrix molecules, but is considerably less than a typical length scale of the deformation field (the macroscale). For suspensions, there will be a range of sizes of particles above which Brownian motion is unimportant. This is the particles' size range that we are concerned with in this paper. There are several approaches to set up a constitutive equation for suspensions and particulate solids. Bounds of effective properties have been derived based on variational principles and geometric information [1-9]. Asymptotic solutions focused on pairwise interaction of two generic particles have been derived, allowing the effective viscosity and moduli of the material * Corresponding author. 0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 5 7 ( 9 7 ) 0 0 0 2 8 - 1
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N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
to be evaluated at high volume fractions [10-14]. In addition, these solutions allow computation techniques to be developed, capturing the evolution of the suspensions [15,16]. Other exact models of suspensions and particulate solids have also been derived, for example the periodic materials [17], and the differential models scheme originally due to [18]. In particular, although the differential scheme has been developed successfully by many authors to estimate the overall properties of suspensions and particulate solids consisting of two phases [19,20,7,21-25], it is not well-known in the fluid mechanics literature, and to our knowledge, has not been generalised to multiphase materials. The basic idea in the differential scheme is quite aesthetiscally simple: the composite material is constructed from an initial material through a series of incremental additions of compounds materials until their final volume fractions are reached. At each stage of the process, the relevant solution of a dilute suspension problem is used to construct the effective properties of the composite, to be employed as the base properties of the next incremental step. The scheme is idealistic, but its simplicity, exactness, explicitness and flexibility makes it attractive for applications in suspensions and particulate solids. Several interesting results have been obtained by the scheme: Brinkman and Roscoe [19,20] used the scheme to generalise the well-known Einstein formula for the effective viscosity of suspensions of rigid particles at finite concentrations. The method predicts the empirical Archie's law for the effective conductivity of rocks [21]; it has also been used to construct geometric models of two-phase composites with maximal and minimal elastic moduli [7,25]. The scheme can be used to model not only asymmetric matrix composites, but also the quasi-symmetric ones [5]. In this paper, we report some interesting applications of the method. Models of suspensions of rigid particles and droplets and of particulate solids consisting of several elastic phases embedded in an elastic matrix are constructed, yielding exact close-form expressions of the effective properties. These models will be useful in correlating experimental data, and in testing constitutive equations and numerical solutions.
2. Two-phase materials To illustrate the method, we first construct the effective properties of two-phase materials. In essence, the construction process starts with a base material (a Newtonian fluid or a Hookean solid). Into this base material, a small volume fraction of spheres (a Newtonian fluid or a Hookean solid) is added. The effective medium is either a Newtonian fluid, or a Hookean solid. Into this effective medium, a small volume fraction of spheres, of considerably larger sizes than previously-added spheres, is introduced. The new set of spheres will see an effective medium, with previously calculated effective properties. The properties of the new effective medium can be calculated, and serves as the base properties for the next step. The construction process continues until a volume fraction of the added phase is reached, where one has a suspension of spheres of diverse sizes in the original solvent or matrix. 2.1. Suspensions o f droplets
Let us now consider a suspension of droplets at a volume fraction 0 < v < 1. The construction process starts by adding an infinitesimal volume fraction c~v of a droplet phase, a Newtonian
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
307
fluid of viscosity q~, in a Newtonian solvent, of viscosity r/s. The effective medium is Newtonian, with a viscosity given by [26,27]. r / ' = q s ( 1 + 6vqS + (5/2) + th th)
(2.1)
This viscosity plays the role of q~ in the next step of introducing yet another infinitesimal volume fraction of droplets, of considerably larger sizes than those of the previous step, so that the new droplets will see an effective medium at all steps. At an intermediate step of droplet volume fraction v, let us denote the effective viscosity by ~/. To a normalised unit volume of this effective Newtonian fluid, we add yet another infinitesimal volume Av of droplets. The new mixture is a suspension of droplets, of volume fraction Av 1 +Av
in a Newtonian fluid of viscosity r/. Its effective viscosity increases by an infinitesimal amount d~/, where Av r/+ (5/2) ~/~ q + d~/= 1/+ - q. l+Av I/+ql
(2.2)
Since the real volume fraction of droplets increases by
v + Av dv - A 1 +~
Av v - A-----~ 1+ (1 - v),
(2.3)
we find the following differential equation (hence the name of the scheme) for the effective viscosity of our droplet model dq dv
1 r/+ (5/2)/11 1 - v r/+ rh ~/'
0 < v < 1,
r/(0)
t/s.
(2.4)
which has the solution
U= 1 __(~)2/5 (21~s-~-51~1~3/'5 \2-qq ~ 5--~-~/
'
(2.5)
When the droplets are rigid, r/1 ~ ~ , this yields the effective viscosity t//~/s = (1 - v) -s/2,
(2.6)
which has been noted by Roscoe [20]. In the limit of small volume fractions, this further reduces to Einstein's relation ~I/qs = 1 + 5v/2. Relation (2.6) has been evaluated by Roscoe [20] against available data on suspensions of rigid spheres. He concluded that it works well for a suspension of spheres of diverse sizes at low volume fractions. For a suspension of spheres at high volume fractions and limited dispersity, he recommended replacing the volume fraction v with 1.35v, so that the maximum allowable volume fraction is 1.35- ~ --- 0.74. The rationale behind this substitution is that a certain amount of fluid is effectively frozen between the spheres at high volume fractions, causing the effective volume fraction of the rigid phase to be higher than it actually is.
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
308
When the viscosity of the droplets is nil (bubbles), (2.5) yields q/~=(1-v)-',
(2.7)
with the corresponding low-volume fraction limit ~l/qs = 1 + v [27]. We expect that Eq. (2.7) will be a good approximation to a suspension of droplets of diverse sizes. Fig. 1 shows the reduced viscosity r//r/s for a number of values of ~h/qs, including some experimental data on water/oil emulsion, but with a monodispersed droplets [28]; the bars represent the variation in the diameter of the droplets, from 1.4-3.3 pm (the smaller the droplets, the higher is the effective viscosity). It is clear that the experimental results follow reasonably well the prediction for bubbles. The more viscous the droplets, the higher is the effective viscosity. In practice, since particles can only have a limited range of sizes, it is expected that the maximum volume fraction cannot approach unity and a physically reasonable model must reflect this feature. We will return to this issue in a later section. 2.2. Composite materials
We now turn our attention to a particulate solid consisting of elastic spheres (shear modulus kt~ and bulk modulus Xl) of diverse sizes embedded in an elastic matrix of shear and bulk moduli /'/m and Xm, respectively. The construction process for the effective moduli follows the same
1000 Rigid
t~
100 111 /'112 =
o
>
"ql/q2~ ®
rh ~2 =.o.5 /
10
Bubble
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Volume fraction Fig. 1. Reduced viscosity of a suspension of droplets of diverse sizes.
0.9
1.0
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
309
procedure as discussed above. At the intermediate step, where the volume fraction of the inclusion phase is v, and the effective shear and bulk moduli are a and x, respectively, we added an infinitesimal volume Av in a unit volume of the homogenised material. The new mixture is a two-phase composite, with matrix m o d u l i / ~ and K, and inclusion's volume fraction Av/(1 + Av). F r o m the classical inclusion problems for spheres [29,24], we find that the effective moduli increase by lc+dK=K4
Av K1 - x 1 + Av 1 + (~cl - tc)/(x + ~c,)'
Av /q - / z /t + d l t =/~ 4 1 + Av 1 + (,u~ -/~)/(/t + / t , ) '
(2.8)
where K, =4/~,
/**
9x + 8/~ 6 x + 12/~/~"
(2.9)
Since the increase in the volume fraction of the inclusion phase is still given by Eq. (2.3), we obtain the following coupled differential equations for the effective moduli of the composite dK dv
1 xl - x 1 - v 1 + (Xl -- K)/(t¢ + tO,)'
d/* dv
1 ¢q --/~ 1 - v 1 + (/t x - / 0 / ( / 2 + / t , ) '
K(0) = gin,
¢t(0) = /~m,
0 < V < 1.
(2.10)
AS before, for the scheme to be exact the sizes of the inclusions added at every step must be considerably larger than all of the previous ones, so that one ends up with a particulate of greatly diverse sizes, allowing the volume fraction to approach one. Numerical solutions to Eq. (2.10) are generally required. However, in the very special case where b o t h the matrix and the inclusion phase have a Poisson's ratio of 1/5, i.e., K~ = (4/3)fll and Xm = (4/3)/Zm, then the Poisson's ratio of the composite material always remains at 1/5. In this case, Eq. (2.10) are uncoupled to yield v=l--(ltm'~X/2(Itl--fl'~, k ll / \ l h -- l*m/
K=4/t.
(2.11)
Exact solutions to inclusions problems are rare, and should serve as models for testing constitutive equations. Fig. 2 shows the reduced shear modulus #//~m according to Eq. (2.11) for the case where /Xl//~m = 10 (the curve labelled 'exact'), together with the numerical solutions to Eq. (2.10) for a variety of parameter values, ]-/rn= 1 , ~/1 = 10, Km = 0.1 - 10, ~:l = 10 - 20. The numerical solutions are sufficiently close to the exact solution Eq. (2.11) to r e c o m m e n d its use in general In the limiting case of rigid inclusions, both Xl,/q---, oo, and Eq. (2.10) become d~ --= dv
, ( 4) x+ /l 1- v 3
'
310
N. Phan-Thien, D.C. Pham / J . Non-Newtonian Fluid Mech. 72 (1997) 3 0 5 - 3 1 8
10
/
9
rigid Nn = 10 ,"
8
~:"
o,'
7 "o 0
/
F__
6
Ill ¢-
5
"o ® o "o
/
1
rigid r m = O. 1 /
t ,ss°~
exact
4 3
W,m= 10 I,tm= 1 Ic1 = 2 0 p . 1 = 10
oos"~
er
2 r-~a= 1 gin= 1 K1 = 11.tl = 10
1 0
---------, 0.0
0.2
0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Volume fraction Fig. 2. The reduced shear modulus of a two-phase composite material for values.
d/~ dv K (0)
a variety of parameter
1 15~c+ 20/z 1 - v 6x+12/z /~' =
Km,
fl (0) =/~m,
For the special case
~m
~/l/~/rn= 10 and
( l - - v ) 2'
Km =
0 < V < 1.
(2.12)
(4/3)/~n,, these equations are uncoupled as before to yield
4
K=gfl.
(2.13)
This case is also plotted in Fig. 2 (the curve labelled 'rigid'). It is to be compared with the numerical solution of Eq. (2.12) where Krn/~lm = 0 . 1 , 10. (The reduced bulk modulus behaves in a similar manner and is not shown here for brevity.) Clearly, the reduced moduli for a composite of rigid inclusions increase much faster with the volume fraction than those of elastic inclusions. At high volume fractions, it is interesting to find that the ratio of bulk and shear moduli converges to 4/3 independent of original ratio of Krn/fl m. Thus, the Poisson's ratio of a composite of diverse sized rigid inclusions approaches 0.2, independent of the values of the matrix moduli. The limit of K1, /~1 --' 0 corresponds to a matrix filled with voids, which is also of practical interest. In this case, Eq. (2.10) become
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
311
,
dK
dv -
--K 1- v
1+
d/~ dv
1 --l, 1- v
14
/z(0) : / z ~ ,
K(O) = ~m,
0 ~ v ~ 1.
(2.14)
For the special case Xm = (4/3)/tm these equations are uncoupled, yielding the solution - -/~ -~-
(1
-- /.))2
(2.15)
X = ~4 ~/.
]/m
In general, Eq. (2.14) have to be solved numerically. We remark here that if the inclusion phase has the same moduli as the matrix phase, then the effective moduli remain unchanged. This fact will be used later to construct models where the volume fraction of the included phase is not allowed to approach unity, as must be the case of a limited diverse inclusion sizes.
3. Multiphase materials 3. I. Composite materials
We now consider a multiphase composite material, consisting of n different inclusion phases of shear and bulk moduli/z~, x~, respectively, and volume fraction v~ (~ = 1,..., n) in a matrix of shear modulus /Zm and bulk modulus Xm. The construction of the multiphase composite material follows the same procedure described in the previous section, starting with the base matrix. At each step of the procedure, we add proportionally infinitesimal volume amounts v, At, At << 1, 0~= 1,..., n of elastic spheres into an already constructed composite of the previous step, which contains volume fractions vat, of the inclusion phases (the parameter t increases from 0 to 1, as the differential scheme proceeds). The particles added at this step must be considerably greater in sizes to those that have been added previously. The new mixture is considered as a dilute suspension of spherical particles from phase ~, of volume fraction v~ At 1+ ~
v~At
~=1
in an elastic matrix of shear and bulk moduli/~ and to, respectively. The effective moduli of the new composite material is v~ At
K+d~c=K+ ct=
1
1+
I) I
v~ At
/~ +d/~ = / t + c~=l
x~ - ~c
At 1 + (K~ - K)/(K + x , ) ' /z~ - lz
1 + '/)1At 1 + (/z~ ~-~-3~-/~ + ~ , ) '
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
312
where Vl ~" 2,,a Uc~ a=l
is the volume fraction of the included phases, a n d / t , and x , are still given by Eq. (2.9). Since the volume fraction of the included phase 0~ increases by v j + v= A t l+vtAt
v=dt-
vJ
v= A t l+vzAt(1--Vlt)'
We obtain the following differential equations for the effective moduli of the composite dx dt
1 --
dl~
~ 2,~
K= - K
1 - Vzt ~ = 1
Vc~
1 + (K= - x ) / ( K + K , ) '
1
~ lt= -- t~ V= L 1 - vzt ~=, 1 + (It= - tt)/(/~ + / ~ , ) '
dt
tc(0) = xm,
/z (0) =/~m,
0 ~ t < 1.
(3.1)
The numerical solution of system of Eq. (3.1) is straightforward. As in the two-component case, when K m = (4/3)/tm and x~ = (4/3)/t~, 7 = 1,..., n, then the system is decoupled, giving a composite material of Poisson's ratio of 1/5 at all volume fractions. 3.2. T w o - p h a s e materials with m a x i m u m
v< 1
In two-phase materials with limited inclusions' sizes, the volume fraction of the included phase cannot approach one arbitrarily, since there are interstitial spaces that cannot be filled with inclusions. In fact, the maximum volume fraction of randomly packed monodispersed spheres is about 0.6 [30]. For a multimodal packing of spheres, the maximum volume fraction of the included phase can be higher, since more spheres can be packed in a given volume. This maximum allowable volume fraction can be incorporated into a differential model by considering a special three-component material, where one inclusion phase is actually made up from the matrix material (which does not contribute to the increase of the effective moduli). More specifically, we consider a composite material consisting of a matrix phase, of volume fraction Vm (shear and bulk moduli Jim and Km, respectively), and two inclusion phases. One inclusion phase is a fictitious one having a volume fraction of Vo and the matrix moduli ~tm, Xm. The second inclusion phase is rigid, of volume fraction vl and moduli ~q, ~---, ~ - - w e will be interested in the limit of infinite inclusion moduli. Thus, since v0 + Vl + vm = 1, the maximum allowable volume fraction for the rigid inclusion phase is 1 - v0 = Vm,x. In this manner, the effects of a limited size range, which limits the maximum allowable volume fraction of the inclusions can be simulated. We expect that Vmax- 0.6 for a random distribution of rigid particles. Now, under the conditions stipulated here, Eq. (3.1) become dx dt
to+K, { --
_
_
1 - vlt
UI +
V0
Km_ZX__.~ K m 71- K , )
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
313
d/t _ # + #* {v, + v0 ~__~_-~-.1 dt
1
- - VI t
¢tm + I t * )
~c(0) = Xm,
/~(0) =/zm,
'
0 _< t _ 1.
(3.2)
These two equations become uncoupled in the special case where x~ = (4/3)ktm in which the solution is given implicitly by 1 + ~
-1
=(l_v0_v0
-2,
x=4/t.
(3.3)
As /)l approaches the maximum allowable volume fraction /)max, /~//~m>> 1, and we find ~ (Vn~ax-- V0- 2(2,'r,ax- 1)
(3.4)
/tm The case of Xm, x--' or, which corresponds to an incompressible matrix, is also of great interest, since it would be physically identical to the case of a suspension of rigid particles in a viscous solvent. In this case # can also be regarded as the effective viscosity of a suspension of rigid spheres in a viscous solvent of viscosity /Zm. The differential equation for ~t becomes
_d/~ dt
5
/~
2 1 - vtt
{ /)~ +v0 /tin + (3/2)/t
/~/(0) = ]Am,
0 ~ t ~ l,
(3.5)
which yields the implicit solution ¢z
]/m
[
1 +
3vl - 2Vo
-
1
5~o~(3,,,- 2~,0)
= (1 - v 0 -
Vl)-S/2
(3.6)
51)1
At high volume fractions, vl ~ Vmax, /~/Ftr~>> 1, and we find I/ ~ (/)max -- /)1)- 5(5Vmax 2)/6
(3.7)
/Zm With /)max~ 0.6, this singular behaviour scales as (/)max- Vl) -°83, which is less singular than the empirical Krieger's formula (/)max- V0-~.8 [31] for monodispersed spheres. In Fig. 3, we show the prediction of the differential model for the case where 1 - v0 = Vmax= 0.58, together with the experimental data of Ilic and Phan-Thien [32] for spheres, Gardala-Maria and Acrivos [33] for spheres and coal, and the experimental bounds of Thomas [34]. The bounds represent the range of the scatter in the data obtained from different sources. The prediction of the differential model (solid line) is closer to Thomas' lower bound, presumably due to the diverse size range of the inclusion phase. Near the maximum volume fraction, the differential model displays the correct asymptotic behaviour. It is interesting to plot there also Roscoe's formula /t//z m = (1 - 1.35v0 - 5/2
(3.8)
Eq. (3.6) and Eq. (3.8) represent two competing processes in suspensions: while a certain amount of fluid is frozen between the inclusions and behaves like the rigid phase, some amount of fluid tends to occupy some space equal in geometric sense to that of inclusions (the fictitious
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997)305-318
314
inclusions) giving rise to a locally-inhomogeneous distribution of inclusions. One feels that Eq. (3.8) overestimates the amount of frozen fluid, while Eq. (3.6) treating the solvent only as the lubricant plus fictitious inclusions underestimates this effect. Perhaps a combined formula might better fit experimental data. It is interesting to note that Eq. (3.6) is weakly dependent on Vo, at low volume fractions of the inclusion phase (Vl < 0.3, v0 < 0.5). The main effect of the fictitious phase is to limit the space available for the inclusion phase. The case of a suspension of droplets of limited size range can be treated in the same manner. Here if V/s is the viscosity of the solvent, r/1 is the viscosity of the droplets (of volume fraction vl), and Vo is the volume fraction of the fictitious solvent phase, to account for the maximum allowable volume fraction Vmax= 1 - V0, then the effective viscosity is given by solving the differential equation J" (5/2)t/(r/s- I/) r/+ (5/2)q] "~ Vo + v~ t/ , 1 - vlt ~ ~s + (3/2) v/ v/+ rh
dr/
1
dt
v/(0) = r/s,
0 < t < 1.
(3.9)
The limit v/] = 0 corresponds to a suspension of bubbles. In this limit
dq dt
~ {5Vo v&- q + v,} , 1 - vzt 2Vls + 3~ 100
. . . .
~
. . . .
~
'
Differential ......
'
'
'
i
,
,
,
.
,
. . . .
,
. . . .
o o
. . . .
model
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bounds
.' o°
• spheres, Ilic & Phan-Thien O coal,Gardala-Maria& Aerivos • spheres,Gardala-Maria&Aerivos
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,
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.....----" .
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'
'
'
'
!
0.1
'
'
'
'
!
0.2
'
•
'
'
!
'
0.3
'
'
i
!
0.4
i
.
i
.
!
0.5
.
•
.
.
0.6
Volume fraction Fig. 3. Reduced viscosity for a suspension of rigid spheres of diverse sizes. The experimental data of Gardala-Maria and Acrivos, Ilic and Phan-Thien and the experimental bounds of Thoas are also plotted for comparison.
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
I/(0) = q s,
315
(3.1o)
0 < t < 1,
which has the implicit solution q-- 1 +
571
- 1
= (1 - Vo - / ) 1 )
-{5vO+2t'l)/2(v0+vl)
(3.11)
At high volume fractions where v~ ~ Vm,x, we find L
, ~ O ) m a x __ i ) l ) --{8 . . . . . --
5)/3,
(3.12)
which is m u c h weaker than a suspension of rigid particles.
3.3. Three-phasematerials It is clear that a suspension of combined rigid spheres and bubbles of the same size range can be treated in the same manner. In particular, let v~, r/l be the volume fraction and viscosity of rigid phase, v2, r/2--the volume fraction and viscosity of the bubble phase. We have the differential equation:
vzt
d5r / -vq ~1° - a t {
2rlsqS-*l+ 3q ~-~vl+v2},
r/(0) = r/s,
0 1< , _t _<
(3.13)
which has the implicit solution
=
(/')max - - Vl - - U2) -- (Sv0 + 5vl + 2v2)/2(vo + vl + v2)
(3.14)
In case vo = 0, (3.14) reduces to q-- = (1 - v, - v2) (s,,,+2~2)/20~+,~2)
(3.15)
It is interesting to note that a small a m o u n t of bubbles would significantly increase the effective viscosity of a rigid sphere suspension. In Fig. 4, the reduced viscosity of a suspension of rigid spheres, with 10% bubbles, is c o m p a r e d to those of a rigid sphere suspension, a bubble suspension, and a bubble suspension with 10% rigid spheres. The increase in the effective viscosity of a rigid sphere suspension with a small a m o u n t of bubble can be significant, especially at high volume fractions of the rigid phase. It is of interest to compare Eq. (3.15) with the respective formulae for the cases where the rigid and bubble phases are considerably different in sizes. Let us assume that the rigid phase is c o m p o s e d of spherical particles of a scale smaller than that of those of the bubble phase. Then the suspension of rigid particles in the solvent should be treated first as an effective m e d i u m with the viscosity (with the actual inclusion volume fraction of Vl/(1-V2) ) --= r/~
1
vl 1 ----v2
(3.16)
N. Phan-Thien, D.C. Pham / J . Non-Newtonian Fluid Mech. 72 (1997) 305-318
316
1000
rigid spheres . . . . . . rigid spheres & 10% of bubbles ,P
- - bubbles & 10% of rigid spheres
lOO
,t
---.
0
bubbles
._~
..-"
10
n"
s ~'" °
/
//
.
°°°w
..-S> 1 t f _ . j / / / 0 0
0.0
J
0.2
0.4
0.6
0.8
1.0
V o l u m e fraction
Fig. 4. Reduced viscosity of a suspension of rigid spheres with 10% volume fraction of bubbles, and of a suspension of bubbles, with 10% of rigid spheres of the same sizes.
In the next step we consider the suspension of bubbles in the solvent with a viscosity given by Eq. (3.16) and obtain the effective viscosity of the overall suspension r/
(1
-- =
t/s
UI
1 --
~ -5/2 (1 -
v2)-'
/)2//
(3.17)
"
In the reverse case when the rigid particles are of a scale larger than that of the bubbles we get -I1~
1
/)2 1 -
(1
-- I)1) -5/2
(3.18)
v]/
A comparison between Eq. (3.15) and Eqs. (3.17) and (3.18) is given in Fig. 5, where the volume fraction of the bubbles is kept fixed at 0.2, and the reduced viscosity is plotted as a function of the volume fraction of the rigid phase. The increase in the effective viscosity is largest when the rigid spheres are of a smaller size compared with the bubbles, and smallest when the bubbles are of a smaller size.
N. Phan-Thien, D.C. Pham /J. Non-Newtonian Fluid Mech. 72 (1997) 305-318
1000
,
.
*
,
I
,
,
i
,
I
,
*
*
.
I
,
*
.
*
I
*
.
J
,
I
,
,
i
,
I
,
J
,
.
I
Bubbles & rigid spheres of same sizes
.
.
*
'
'
/
...... Small rigid spheres, large bubbles
"'/
i
,..'/
- - - - Small bubbles, large rigid spheres
1O0
,
317
//
8 > "o ° m
~
10
•
0.0
'
'
'
I
0.1
,
'
'
'
I
0.2
'
'
'
"
!
'
0.3
'
'
•
I
0.4
'
'
'
•
I
0.5
'
'
'
•
I
0.6
'
"
'
'
I
0.7
'
'
0.8
Volume fraction Fig. 5. Reduced viscosity of suspensions of rigid spheres and bubbles, of the same and different sizes.
Acknowledgements We thank the University of Sydney for providing support to PDC through an International Development Link Program grant.
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