- Email: [email protected]
Application of the homogenization method to a suspension of fibres
Pergamon
wo-7225(93)E0014-V
1~. 1. Engng Sci. Vol. 32, No. 8, pp, 1253-1269, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0020-7225/94 $7.00+ 0.M)
APPLICATION OF THE HOMOGENIZATION TO A SUSPENSION OF FIBRES F&DgRIC
PfiRIN
and THeRl?SE
METHOD
L&Y?
Laboratoire de ModClisation en Mdcanique, UniversitC Pierre et Marie Curie, Tour 66, 4 place Jussieu, 75252 Paris Cedex 05, France (Communicated
by G. A. MAUGIN)
Abstract-The homogen~tion method is apptied to the determination of a ~ntinuum model for an assembly of cylindrical parallel fibres suspended in the longitudinal flow of a viscous incompressibie fluid. Whatever the solids concentration is, the homogenization gives the variations of the velocity and the pressure at the level of the microstructure and furnishes a rigorous deductive procedure for obtaining the governing equations of the homogenized bulk medium. The effective constitutive relation is that of a Newtonian incompressible anisotropic fluid. The viscosity is expressed by a second order tensor completely determined by the microstructure. Lower bounds of the viscosity coefficients are found when the micr~~u~ture exhibits symmetries. Numerical results are obtained for two cases of macroscopic isotropy. They emphasize the good appro~mation arising from the lower bounds and the importance of the microstructure geometry for nondilute suspensions. As a matter of fact the solids concentration is not able to describe correctly the influence of the microstructure on the bulk behaviour of concentrated suspensions.
1. INTRODUCTION
The investigation of the flow of rigid particles in a Newtonian fluid has been the focus of a number of studies. Many of the results are summarized in [l], or in a recent paper [2]. The ultimate aim is the determination of the effective properties of the bulk material considered as homogeneous on a macroscopic scale. Theoretical works have been successful in the case of dilute suspensions, where the relative motion of the fluid near one particle is unaffected by the presence of the others, or in the case of small volume fraction of the particles [3,4]. Less numerous attempts exist for concentrated suspensions, among them [5], which gives rise to many further studies for the determination of the effective coefficients, [6] in the frame of a micropolar continuum theory, and [7] which uses the homogenization process for periodic media. Without any phenomenolo~ic~ assumption at the level of the microstructure this latter proves that generally the bulk medium is an incompressible anisotropic non Newtonian fluid. The effective viscosity of the mixture is expressed by a fourth-order tensor, the instantaneous coefficients of which are completely determined by the microstructure at the same instant. Conjointly the microstructure evolution is determined by the bulk velocity gradient. Numerical calculations of the effective viscosity coefficients for a three-dimensional evolving suspension are not available at the present time. Some special instantaneous dist~butions of particles can be examined. The suspension of an array of spheres centred on the points of a periodic cubic lattice has been investigated by Nunan and Keller [8]. Then the effective instantaneous viscosity tensor has a particular form in agreement with the general situation described in [7] and numerical results can be performed. Likewise [Z] has considered a periodic cubic array of particle clusters in a bulk simple shearing flow. In this paper we are interested in the effective constitutive relations for an oriented fibres assembly suspended in a viscous incompressible fluid. The study of a nondilute suspension of fibres is almost so much complicated than the general case of a three-dimensional suspension. In order to obtain more simple results, especially numerical values of the viscosity coefficients, TTo whom all correspondence
should be addressed. 1253
F. PeRIN
1254
and T. L&Y
we consider the particular case of a bidimensional suspension. As a matter of fact an assembly of parallel, infinite in length fibres is studied. In the physical situation when this suspension is carried along a straight channel by a viscous fluid flow, one can presume that the bulk flow is a Poiseuille-like one with a modified viscosity. This paper is concerned with this problem. The asymptotic homogenization process for periodic structures is used. Section 2 is devoted to the main features of this method. The problem is formulated in Section 3. It presents a bidimensional microstructure. The determination of the local variations of the flow (at the level of the microstructure) is performed in Section 4. The macroscopic equations for the bulk flow are obtained in Section 5. A second-order viscosity tensor, completely determined by the microstructure geometry at the considered instant, appears. In this particular study the microstructure does not evolve in time (Section 6). So the viscosity tensor remains independent of time. The last section is devoted to the study of this tensor, especially the influence of microstructure symmetries, the display of lower bounds for the viscosity coefficients, and numerical results.
2. HOMOGENIZATION
The method concerns media with a microstructure on a scale much smaller than the macroscopic scale of interest. The description of finite volumes of so complicated media with a high density of strong heterogeneities is practically possible at the macroscopic level only when an equivalent continuous medium is defined. Homogenization [9-111 is a double scale asymptotic process related to a small parameter E = l/L, where 1 and L are the characteristic lengths of the microstructure and the macroscopic medium respectively. Two space variables x and y =X/E are introduced and all the quantities entering the problem are postulated of the form f(x, y), which emphasizes the fact that the medium varies rapidly on the small scale I and may also vary slowly on the larger scale L. We look for the unknown quantities in the form of double scale asymptotic expansions: u’(x)
=
u”(x, y) + EU’(X,y) + e2u2(x,y) + . * .
Introducing these expansions in the equations of the physical problem, and identifying the like powers of E gives differential problems in the x and y variables. Generally speaking, equations in y are solvable if the microstructure is in some sense periodic, and terms u’(x, y) in the postulated asymptotic expansions have to be periodic in the y variable with the same period as that of the structure. The macroscopic equations, with respect to the macroscopic variable x, are obtained by taking mean values of the local solutions in a period or by writing the asymptotics of conservation laws in a macroscopic arbitrary domain [ll]. This leads to a rigorous deductive procedure for achieving the equations for the bulk behaviour of the medium. Thus homogenization gives the passage from a microscopic description to a macroscopic discription of the problem, without the need of phenomenological hypotheses. As a matter of fact, by applying the classical equations of fluids and solids mechanics to the components of a mixture, we can deduce its bulk behaviour without any presumed description of the components interactions at the level of the microstructure. In the study of media with large deformations, as suspensions, the geometry pattern evolves in time and depends on both its configuration at the initial time and the flow for the subsequent time. In the case of a suspension of three-dimensional rigid particles it has been proved [7,11], that if the particles at the initial time to are locally periodically distributed in the fluid [that means with a period .eY(to, x) which may vary slowly on the larger scale L], the microstructure evolves in time by keeping the locally periodic character. Consequently at any instant in the
Homogenization
method applied to a suspension of fibres
1255
evolution of the system the required periodicity conditions of the homogenization techniques are satisfied with a period which is that of the structure at the same instant. Finally let us remark that according to the required periodicity condition of the microstructure, the small characteristic length I = EL characterizes both the smallness of the particles and their mutual distance. So the investigated suspensions are nondilute.
3. FORMULATION
OF THE PROBLEM
The two-dimensional flow of a nondilute fibres suspension in a large straight channel is investigated. The thin fibres are assumed to be rigid cylindrical infinitely long bodies which are parallel to the Poiseuille-like macroscopic flow. Denoting by L the characteristic width of the channel and by 1 the characteristic thickness of the fibres, the small parameter of the homogenization process is E = l/L. At the considered instant t the distribution of the fibres in the incompressible viscous fluid is supposed locally periodic. The e3 axis of a rectangular Cartesian system of coordinates X(X,, x2, x3) is chosen to
Fig. 1. Locally periodic distribution of the fibres and basic period Y(t, x, , x2),
F. PBRIN and T. LhY
1256
coincide with the direction of fibres. Henceforth we assume that the section of the periodicity cell by a plane e,, e2 is a parallelogram cY(t, x,, x2). It can be considered homothetic with the ratio E, E << 1, of Y(t, x,, x2) which is called the basic period in the stretched coordinates. For the sake of simplicity we consider that the solid part Ys of Y, corresponding to the fibre section, is completely surrounded by the fluid filling the complementary part Y,, and that Y is centred in the mass centre G of the fibre section. So the stretched coordinates may be defined as: Xl
Yl =
-xcd
x2
&
’
-
Y2 =
xc2
&
The motion equations are written using the Einstein summation convention. the medium the velocity V” and the pressure P’ satisfy the equations
In the fluid part of
(1)
div V” = 0 po
=?s
8Vg r+v;$
(
I
>
(2)
axj
with U; = -P&6, + 2/~Dij(V”),
Dij(VE) = ; ($
+ $) I
. I
(3)
In each fibre, V’ is a rigid velocity field, so Dij(VE) = 0.
(4)
Because of the non-slip condition, on the solids surfaces V” is continuous.
(5)
For each arbitrary portion S’ of a fibre defined by xX E [a, b], the dynamic law implies
(6)
I
-I -
p;(x-x,)n$dv=
(X
s
r).s
~0)
A
ri+jei da.
(7)
In the two latter relations, n denotes the unit inward to the solid normal. The Sk denote the components of the stress tensor in the whole medium defined as 88 = a; given by (3) in the fluid and extended symmetrically in the solids with the stress continuity condition on the fibres surfaces and the momentum equation SFfl,
= c&n,,
.dV; PST=
3 ax; .
Let us remark that 6” is not uniquely defined in the solids. It is submitted only to the influence of the fluid flow by (8) and to the bidimensional geometry of the fibres. So we can assume that the form of the asymptotic expansion (when E is small) of the stress tensor in the solids is analogous to the fluid one. In the following we write x for (x,, x2, xg) and y for (yr, y2). Correlatively latin indices i, j - - * can take values 1,2,3, whereas greek indices cy, p 1 . . take values 1,2. According to the general features of the homogenization method, the solid density pg is supposed, at instant t, of the
Homogenization
method applied to a
suspensionof fibres
form p&,, x2, y). We search asymptotic expansions of the equations (l)-(7), zero, of the form: V” = Vg(t, X, y)e, + EV:(f, X, y)ei + * * * P” = P”(t, x, y) + EP*(t, x, y) + * * *
1257
when 6 tends to (9) (10)
with V’:, Vf, . . . , PO, Pi,. . . , Y-periodic with respect to the local stretched variable y. In (9) we have taken into account that, in first approximation, the macroscopic flow will be parallel to the fibres and the channel walls.
4. LOCAL
STUDY
We insert expansions (9) and (10) in equations (l)-(7) taking care that, when applied to a a 1 a function f(x, y), $ becomes g + --. In the following, index x (or y) specifies partial a n E ay, a2 derivative, for example A,, = By identifying the like powers of E, we obtain successive a~, ay,’
differential problems with respect to the local variable y, at this stage x is considered as a parameter. Let us remark that according to the required condition of Y-periodicity, we may consider that y varies in the basic period Y, then the periodi~ity of a function means that it takes equal values on the opposite faces of dY. 4.1 Local variation of V,”
It is determined by the first approximation of equations (Z), with (3), (4) and (5). From the O(F-‘) approximation of (2), the O(E-I) approximation of (4), and the 0(&O)approximation of (5), we obtain AyV;=O
av:
for y E Yr
-0
for y E Ys
ayuV!j is continuous
for y E au,.
With the conditions of Y-periodicity, these equations determine the local variation of V! up to an additive function independent of y. As a matter of fact, with the appropriate space of functions W = w, w E H”(Y), w Y-periodic, g = 0 in Ys, f w dy = O] i Y CX they are equivalent to the following variational problem:
vl: E w, The Lax-Milgram is obviously
VW E
theorem proves existence and uniqueness of the solution in W. The solution V% -K y) = V$, X).
Furthermore,
from the O(E’) approximation
2 av’ g-+&o 3 ES S&S-S
w.
ah
of (1) and (4) for y E Y.
(11)
F. PeRIN
12.58
and T. L&Y
Integrating this relation in Y, taking into account (11) and the Y-periodicity
v: = v@,XI,
of V’, we obtain
x2).
(12)
in Y.
(13)
Consequently div,, V’ = 0
From (3) the stress tensor in the fluid admits the asymptotic expansion a;=(T;+q!i+.”
with according to (9) and (12) a:0 = -P%,,
+ 2/&,,(V’), (14)
The next approximation independently of Vi.
of equations
(l)-(7)
gives the local variations of VL, (Y = 1, 2, and
4.2 Local variations of V:, Vi They are connected by (13) the 0(&-l) approximation of (2) and the 0(&O)approximation (4) the O(c) app roximation of (5), and the O(E’) approximation of (7).
of
for y E Yr
Q&V’) = 0
for y E Ys
V’ continuous for y E aY,
Let us remark that the first approximation of (6) is identically satisfied taking into account the Y-periodicity and the divergence free property of the tensor 0’. The vector of components Vi, Vq is determined up to a Y-periodic vector U = (V,, U,) satisfying D++,(U) = 0 in Y, that is to say, a vector independent of y. In this frame, like in the former section, the variational formulation of the problem is obtained by multiplying the second equation by a test function w’ and integrating in Yr using the previous equations. The appropriate space of functions is W’ =
w’,
w’
E
[H'(Y)]*,
w’Y-periodic, div,, w’ = 0 in Yr, D&w’)
= 0 in Ys, /.w’dy
=0}
and the variational problem is
w:,Vi) E W’,
J‘&pyW’Pa,(w’) dy = 0
VW’
E
W’.
YF
It follows that K&P X, y) = VXr, x).
(15)
1259
Homogenization method applied to a suspension of fibres
Furthermore,
according to the expression (14) of a$,
we deduce that grad, P0 = 0, thus
P(f, -%y) = PO@,x). 4.3 Local variation
(16)
of V.{
The O(E-‘) approximation of (5) are
of (2) the 0(&O)approximation
AJ:
=0
av: dy,=
of (4) and the O(E) approximation
for y E Y,
avg
for y E Ys
-z
Vi continuous for y E aus. These equations determined convex subset of H’(Y)
Vi up to a function independent
K = u, u E H’(Y), u Y-periodic, 2
of y. Introducing
in &, I,~dy
= -2 CE
the closed
=0}
the variational problem for Vi is
From a theorem by Stampacchia [12], Vi is uniquely determined
v:(tYx7 Y)
= -
3~~+A(t, ax
in K, and we can write
x)
(17)
n
with X.&, x1, x2, y) the unique Y-periodic solution, continuous on aYs, of: A,,x~ =0 for y E YF Xa=ya for YE Ys.
(18)
According to the previous results and (14), the stress tensor in the fluid is of order so, with $2 = 0,
ff:, = a& = & = -P(t,
x),
(19) We also remark that the first approximations calculations are identically satisfied.
of (6) and (7) which are not used in the previous
5. MACROSCOPIC
EQUATIONS
5.1 Macroscopic conservation law First of all, it should be noted that V” = (0,0,Vs) is a macroscopic quantity (depending only on the macroscopic variable x) and (12) establishes that the macroscopic flow is incompressible. As classically in homogen~ation process [ll], we obtain the macroscopic momentum equation by applying the momentum conservation law to the mixture contained in an arbitrary
F. PeRIN
1260
and T. L&Y
Fig. 2. Section T of the macroscopic cylinder D.
macroscopic cylinder D parallel to e3, with x3 E [a, b] and the section consisting of a great number of periods EY. This law is
T by a plane e,, e2
(20) where density p’ is p,, in the fluid and p$ in the fibres. We look for the first order approximation of each term when E tends to zero, using the asymptotic expansions of V” and 6’. The lateral boundary dT X [a, b] of D lies in the fluid, so the first approximation of the stress tensor CV is (+Ogiven by (19). In the solid cylinders the first approximation of B” is of order co as in the fluid. Only the 5ii appear in (20). According to (8)
a@,
-=
aYC2
furthermore
0
10
from the (E-‘) approximation
a& -=
0
= u3sa
u3sa
and
for y E Ys
for y E ar,
(21)
of (2), already used in Section 4.3: 0
(22)
for y E YF,
dY,
so the c?!, defined in the whole Y are divergence free. With regard to (ii3 according to the remark connected with (8) we can assume that this undetermined component in the cylinders depends on x3 like ai3, that is to say A0 u 33 =
-p”(t,
x>
+
G(t,
Xl,
x2,
y).
(23)
With calculations analogous to [ll] we can write with (12), (15) and (17)
PET&,=
I
$$e3du
+ o(l).
D
The mean density defined by
is, when the cylinders are homogeneous
(ps independent 6 = PO+ c(Ps - PO)
where c(x, t) = lYs{/lYl is the solids concentration.
of y)
(24)
Homogenization
applied to a suspension
method
of fibres
1261
The projections of the right-hand side of (20) on e, and e2 are, according to (19) h I i)D
6&dda=
-
P’n, ds, dx3 +
If(1
ilr
I Ti
where T, and Th are the sections of D corresponding Let us examine Jr&t3 da. As in [ll] we obtain:
S”,, da -
I r,
a”,, da + o(l)
to x3 = a and x3 = 6.
fT
The last term can be expressed using relations (21)
with ~0,~given by (19). This proves that _fT&i3 dy does not depend on x3. So we have:
f JD
The projection
- 1” l,P’n_
Qridu=
dsx dx3 + o(l) = -I,
a
g
dx + o(1). LI
(25)
of the right-hand side of (20) on e3 is:
If b
f l3D
&pi da =
(1
The stress component solids,
fT
&g3du=
c &YCT
u~,n,drdr,+lbg,du-Ibpdo+o(l).
.3T
Si, on a section T is expressed by (19) in the fluid and by (23) in the
x) fv, If-I’“@,
c2
dy +
[-P”(f, x) + u’$(~ ~1, x2,
~11 dy + 41)
YF
and then
We have to calculate the stress on dT. The boundary of T is made of sides cay, (with normal denoted II“, k = 1,2,3,4) of the neighbouring periods of T, so the following approximation occurs:
f aT
u$Jl,ds=
c
EJY&dT
E
f aY,
s;Jznl;d.sy+ o(1).
As we have seen before, (21) and (22), c?:,, (u = 1,2), in the whole period Y is divergence free. Thus we know [ll] that in the integral on a side aY,, B$zk may be replaced by &n”,, with a:, the mean macroscopic stress tensor component defined by
(26) So we can write
F. PeRIN
1262
and T. LhVY
And we have (27)
Taking into account (24), (25) and (27). The first approximation macroscopic momentum equations.
aP" -= 0
(Y= 1,2
axa
_av:’ PT=-~ 5.2 Homogenized
of (20) leads to the following
aP
f-
aa;, ax, ’
(29)
constitutive equation
It appears in (29) a bulk plane stress tensor defined by (26). In this section we find the macroscopic constitutive equation and prove that the average value (26) is independent of the extensions [not uniquely determined by (21)] of the & in Y,. As a matter of fact, as we have found in the previous section, and using (18)
=-
I
~~,x~n, hy = -
3%
=-
I a(&d YF
aYY
dy
axa
o
-dy. u3Y ay,
In the latter equalities we have taken into account the Y-periodicity (22). Then we obtain for (26):
We recall that a& is given by (19), so the homogenized -0
u3,
= aLlo
aV’:
constitutive
of the xa and a!?, and
equation is
with ao,p
(30)
8x0 Taking into account the Y-periodicity of the &, and (18), it can be written: (31) The macroscopic constitutive equation involves a second order viscosity tensor, the coefficients of which are completely defined by the microstructure at the considered instant t. Let us note that, like the basic period Y the asp may depend on the macroscopic variables x,, x2. It is easy to prove that the viscosity tensor is symmetric and positive definite. The bulk medium behaviour is that of an anisotropic generally inhomogeneous fluid. 6. DEFORMATION
OF THE
STRUCTURE
AND
CONSEQUENCES
The macroscopic relations (30) contain homogenized coefficients aap depending on the microstructure by the averaging domain and the local solutions X~ of the problems (18). These coefficients are calculated using the basic period Y(t, x1, x2). The microstructure of the medium is locally periodic at instant t and driven by the flow with the velocity V’ V” = V;(t, x,, xz)e3 + eV’(t, x, y) + O(E)
Homogenization
1263
method applied to a suspension of tibres
d.
Fig. 3. Geometrical characteristics of the microstructure.
with V’ Y-periodic and given by (15) and (17). Like in [ll], in order to describe the deformation of the structure we introduce the appropriate variables: two vectors d’ and d* of order E characterizing the form of a period in the plane e r, e2, and the rotation of a fibre. dd” dt for (Y= 1,2. We have to calculate e, - -g- and e2 sdd” dd’
dAB - = V(B) dtdt
- V(A) = V@(B) - v(A)
+ s[V’(B) - V’(A)] + O(E).
The terms containing V’ give contribution of order E [because AB = O(E)] and the terms containing V’ give contribution of order E* because the y dependence is the same in A and B by periodicity. We have at the first order of approximation: dd’ dt=
dV:(t, x1, x2)
ax,
dbe, + 0(c)
and generally for (Y= 1,2 dd” *-=o(E) e’ dt
and
e2 * 5
= O(E),
(32)
the form of the period Y in the e,, e2 plane does not vary. The rotation rate of a fibre is
the first approximation
is, with y E Ys
nc =f(rotXV@+rot,V1)+o(l)=~[~e,-~~e2-~~e,+~~e2]+o(l)
a
(I
and taking into account xU = y, in Ys [see (US)], nE = O(l), then in first approximation,
the rotation
of the fibres is zero. The microstructure
(33) remains
1264
F. PeRIN
and T. Ll?VY
bidimensional and according to (32) and (33) the periodic distribution of the fibres does not evolve in time. In this case we have proved that the microstructure does not evolve in time, contrary to the three-dimensional case [7,11]. Consequently the homogenization process (Sections 3-5) can be performed at any instant with the given basic period Y(x,, x2). Furthermore, since the microstructure is independent of the macroscopic velocity V!e,, so it is for the viscosity tensor and the bulk fluid is then Newtonian. According to (28), (29) and (30), the steady flow is Poiseuille-like, and the longitudinal bulk velocity is governed by
a avl: = constant. -ax, [aap -1 ax, In the next section we specifically study the viscosity tensor asp using its expression (31).
7. THE
VISCOSITY
TENSOR
In this section the basic period is supposed to be rectangular. 7.1 Influence of period symmetries First we suppose that Y admits Oyl as a symmetry axis [Fig. 4(a)]. The local problem (18) reduces to finding x, and x2 2A periodic with respect to y, satisfying: Ax, = 0 in Y,/2; Ax2 = 0 in YJ2;
x1 = y1 Y E a&/2; x2 = y2
Y E a&./2;
ax1 ---g=o
yES,USg
x*=0
y ESlUSX.
(34)
In this case, the viscosity tensor (31) is given by a,2 = azl = 0
(35)
y/2
Y/4
Y
;3
O3
B
aYs/4
Y$2
=4
I-LIPe,
) S.I 2A
(4
t
2
B
-=2
<
YF/4
a4
aYs/2
Yl
I&
Yl
a1 <
A
)
(b)
Fig. 4. (a) Half of the basic period Y admitting Oy, as a symmetry axis; (b) quarter of the basic period Y admitting Oy, and Oy, as symmetry axis.
Homogenization
method applied to a suspension of fibres
1265
If we assume that Oyl and Oy, are symmetry axis in the period Y [Fig. 4(b)], x, and x2 are then the solutions of Ax, = 0 in YJ4;
xl =Y, on aYJ4;
X,=Oona,Ua,;
~=Oon~,Uo,
Ax2 = 0 in Y,/4; x2 =
x2 = y2 on aYJ4;
ax2
z=O
0 on (T, U a3;
on g2Uf14
and the viscosity components are given by (35) in which Y/2 may be replaced by Y/4. If in addition, the period Y is left invariant by a II/2 rotation we obtain a macroscopic isotropy in the el, e2 Plane: aI1 = a22 = ph, because Xl(Yl, Y2) = X2(Y2, Yt). So we obtain by (31)
This proves that if the microstructure is such that it exists a macroscopic isotropy in the plane perpendicular to the fibres’ direction, the macroscopic viscosity of a suspension of parallel fibres in a bulk longitudinal flow is greater than ~(1 + c), with p the fluid carrier’s viscosity and c the fibres concentration. A better lower bound will be found in the next section. 7.2 Lower bounds for the viscosity coefficients We define: Yt = .;f;
{Y,>
y; = ym$ {y,}. E
and
We call “upper function” of aYs with respect to y,: fi:]Y;,Y:l+R yl+fdyJ
=
max
(Ylm)EaYS
{y2)
and in an analogous way f2 the “upper function” of aYs with respect to y2. Let us remark that generally the fa are not continuous. Now we consider the case when the basic period Y admits Oy, as a symmetry axis. Then we can find a lower bound for a22lcL.
Fig. 5. Graph of the “upper function” fi in the case when the basic period admits Oy, as a symmetry axis.
F. Pl?RIN and T. L&Y
1266
Looking at the expression (35) of uZ2/~, we have:
I
6x2 ax2 5
yl2 ay,
ay,
sc$($2dy2dy,.
We have [let us recall that x2 E H’(Y)]: h(h)
I
ax2
ayz dy2 = XZ(Y,,
0
fi(yl))
- XZ(Y,,
0).
From (18) x2(y) = y2 for y E Ys and using (34) we obtain: XZ(Yl,
h(Yl))
and
=h(yJ
x2(y,,
0) = 0
then fi(Yd
I
ax2
~dy’=fi(yd
0
By the Cauchy-Schwarz
inequality:
’dy2>-fh). We have also, using again (18) and (34)
B
ax2 - dyz= -fi(~J i fi(Yl)aY2 and by the Cauchy-Schwarz
inequality:
6,,, (3
2
f?(Yl) dyz%
-fi(y,)’
so:
and
Let us remark that this lower bound depends only on fi and is not determined by aYs/2, as a matter of fact it is the same for all the graphs of the fibre section having the same “upper function” f, with respect to y,. In some sense the lower bound of a22 is not influenced by the concave parts of the fibre section. When Oyi and Oy, are symmetry axis of Y, we obtain lower bounds for a,,/~ and a221p: &1+CL
&I+-w
1 B
ys Io A
1 A0I
h(Y2)
dy2
-h(y2)
yT fi(Y1) B -I
dy
(37) ‘*
Homogenization
method applied to a suspension of fibres
If in addition the period Y is left invariant by a z/2 rotation, medium is isotropic, and we have 1
&I+I-L
yi
fi(Y*)
1267
as we have proved the bulk
dy
A I o A -.L(Y,)
(38)
”
The results can be applied to particular simple cases (see Fig. 6). In all the envisaged geometries Y admits Oy, and Oy, as symmetry axis, so aI2 = 0 and (37) is: Case 6a:
ab
&I+ P
B(A-a)’
ab
%>I+ p
A(B -b)
1
Case 6b:
Case 6c:
f&1+ P
Case 6d:
ab B(A-a)’
arctg
A+a
-
J A-a
J
B+b B-b
---nA
1
xB 2b
’
2a
1 1 1’
A
(4
)
A
(b)
YF/4
(cl
(d)
Fig. 6. Case of a fibre with (a) rectangular section; (b) cross shaped section; (c) section involving concave parts; (d) elliptic section.
F. PBRIN and T.
1268
Fig. 7. (a) Square shaped period and square shaped fibre section; (b) square shaped period and circular fibre section.
7.3 Numerical results
We study two sorts of geometry for the period Y which leads to macroscopic isotropy [Figs 7(a) and (b)]. The values of p,,/p [see problems (36) and (35)] and the values of its lower bound m [see (391, determined in Sections 7.2 are calculated using the MODULEF code. The results are given in the following tables. Table 1. Values of the concentration c, of the relative viscosity coefficient of the suspension p,,/p and of its lower bound m. Case of a square shaped fibre section of side a. a
c =a2
PhlP
m
0.10 0.20 0.30 0.40 0.50 0.60 0.65 0.70 0.75 0.80 0.85 0.90
0.01 0.04 0.09 0.16 0.25 0.36 0.42 0.49 0.56 0.64 0.72 0.81
1.03 1.09 1.22 1.42 1.73 2.23 2.58 3.06 3.73 4.73 6.40 9.77
1.01 1.05 1.12 1.26 1.50 1.90 2.20 2.63 3.25 4.20 5.81 9.10
Table 2. Values of the concentration c, of the relative viscosity coehicient of the suspension p,,/p and of its lower bound m. Case of a circular section with diameter a. a
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.98 0.99
c = na2/4
0.0078 0.0314 0.0706 0.1256 0.1963 0.2827 0.3848 0.5026 0.6361 0.7088 0.7542 0.7697
PJP
m
1.02 1.06 1.15 1.28 1.48 1.78 2.26 3.10 5.08 7.97 13.73 20.19
1.008 1.037 1.095 1.19 1.34 1.59 2.01 2.79 4.70 7.52 13.22 19.70
Homogenization
method applied to a suspension of fibres
1269
From these numerical results we can conclude that the lower bound found in Section 7.2 is a rather good approx~ation of the homogenized viscosity whatever the solids concentration c is. The numerical values of j.&halso emphasize the dependence of the homogenized viscosity on the geometry of the microstructure. When the solid concentration is smaller than 0.5 the influence of the form of fibre section is not determinative, but for c greater than 0.5 the geometry is a basic parameter.
8. CONCLUSIONS
We have applied the homogenization process to the determination of a continuum model and its governing equations for an oriented fibres assembly suspended in a viscous fluid in a Foiseuille-like situation. The study emphasizes the incompressible anisotropic Newtonian fluid behaviour of the bulk medium. In this special two-dimensional microstructure the viscosity of the homogenized medium is determined by a second order tensor depending only on the microstructure. As a matter of fact a suspension of fibres in a viscous fluid is relevant to the general case of a three-dimensional suspension of rigid particles, It displays a macroscopic viscosity described by a fourth-order tensor ([7,11]), such that the homogenized constitutive equation for the macroscopic viscous stress is
(with the notations of the Section 5). In the present paper, the assumption of a macroscopic flow parallel to an assembly of infinitely long fibres leads to a two-dimensional problem which allows a more simple study. The macroscopic viscous stress is given by the constitutive equation (30). In this relation the viscosity coefficients aap (a!,@ = 1,2) are approximations of the -4aP3 when the fibres are infinite in length in the e3 direction. Lower bounds of these coefficients are found when the microstructure admits some symmetry properties. The effective numerical results in two simple cases suggest that these lower bounds may be good approximations of the viscosity c~fficients. They also emphasize the influence of the microstructure geometry on the values of the bulk viscosity for nondilute suspensions. We find, as [2] in another particular problem, that up to moderate concentration of particles, the viscosity of the suspension displays only a weak dependance on the particles geometry. For solids concentration greater than 0.5 the dependance becomes drastic, this is related to the packing of the fibres.
REFERENCES [i] [Z] [3] [4] [5] [6] [7] [8] [9]
H. BRENNER, A. Rev. Fluid Mech. 2,137 (1970). H. PHAN-THIEN, T. TRAN-CONG and A. L. GRAHAM, J. Fluid Mech. 228,275 (1991). G. K. BATCHELOR, f. F&id Mech. 46,813 (1971). A. ACRIVOS and E. S. G. SHAQFEH, Phys. Fluids 31, 1841 (1988). G. K. BATCHELOR, 1. Fluid Mech. 41,545 (1970). A. C: ERINGEN, Rheol. Acru 30,23 (1991). T. LEVY and E. SANCHEZ-PALENCIA, J. Non-Newr. Fluid Mech. 13,63 (1983). K. C. NUNAN and J. B. KELLER, J. Fluid Me& 142,269 (1984). A. BENSOUSSAN, J. L. LIONS and G. PAPANICO~OU, Asymptotic Analyst for Periodic Srructures. North-HoIIand, Amsterdam (1978). [lo] E. SANCHEZ-PALENCIA, Non-Homogeneous Media and Vibrations Theory. Lectures Notes in Physics, Vol. 127. Springer, Berlin (1980). [ll] T. LEVY, in Homogenization Techniques for Composite Media, Lecture Notes in Physics (Edited by SANCHEZ-PALENCIA and A. ZAOUI), Vol. 272, pp. 63-119. Springer, Berlin (1987). [12] H. BREZIS, Anaiyse FonctionneNe. Masson (1992). (Received and accepted 12 October 1993)
wo-7225(93)E0014-V
1~. 1. Engng Sci. Vol. 32, No. 8, pp, 1253-1269, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0020-7225/94 $7.00+ 0.M)
APPLICATION OF THE HOMOGENIZATION TO A SUSPENSION OF FIBRES F&DgRIC
PfiRIN
and THeRl?SE
METHOD
L&Y?
Laboratoire de ModClisation en Mdcanique, UniversitC Pierre et Marie Curie, Tour 66, 4 place Jussieu, 75252 Paris Cedex 05, France (Communicated
by G. A. MAUGIN)
Abstract-The homogen~tion method is apptied to the determination of a ~ntinuum model for an assembly of cylindrical parallel fibres suspended in the longitudinal flow of a viscous incompressibie fluid. Whatever the solids concentration is, the homogenization gives the variations of the velocity and the pressure at the level of the microstructure and furnishes a rigorous deductive procedure for obtaining the governing equations of the homogenized bulk medium. The effective constitutive relation is that of a Newtonian incompressible anisotropic fluid. The viscosity is expressed by a second order tensor completely determined by the microstructure. Lower bounds of the viscosity coefficients are found when the micr~~u~ture exhibits symmetries. Numerical results are obtained for two cases of macroscopic isotropy. They emphasize the good appro~mation arising from the lower bounds and the importance of the microstructure geometry for nondilute suspensions. As a matter of fact the solids concentration is not able to describe correctly the influence of the microstructure on the bulk behaviour of concentrated suspensions.
1. INTRODUCTION
The investigation of the flow of rigid particles in a Newtonian fluid has been the focus of a number of studies. Many of the results are summarized in [l], or in a recent paper [2]. The ultimate aim is the determination of the effective properties of the bulk material considered as homogeneous on a macroscopic scale. Theoretical works have been successful in the case of dilute suspensions, where the relative motion of the fluid near one particle is unaffected by the presence of the others, or in the case of small volume fraction of the particles [3,4]. Less numerous attempts exist for concentrated suspensions, among them [5], which gives rise to many further studies for the determination of the effective coefficients, [6] in the frame of a micropolar continuum theory, and [7] which uses the homogenization process for periodic media. Without any phenomenolo~ic~ assumption at the level of the microstructure this latter proves that generally the bulk medium is an incompressible anisotropic non Newtonian fluid. The effective viscosity of the mixture is expressed by a fourth-order tensor, the instantaneous coefficients of which are completely determined by the microstructure at the same instant. Conjointly the microstructure evolution is determined by the bulk velocity gradient. Numerical calculations of the effective viscosity coefficients for a three-dimensional evolving suspension are not available at the present time. Some special instantaneous dist~butions of particles can be examined. The suspension of an array of spheres centred on the points of a periodic cubic lattice has been investigated by Nunan and Keller [8]. Then the effective instantaneous viscosity tensor has a particular form in agreement with the general situation described in [7] and numerical results can be performed. Likewise [Z] has considered a periodic cubic array of particle clusters in a bulk simple shearing flow. In this paper we are interested in the effective constitutive relations for an oriented fibres assembly suspended in a viscous incompressible fluid. The study of a nondilute suspension of fibres is almost so much complicated than the general case of a three-dimensional suspension. In order to obtain more simple results, especially numerical values of the viscosity coefficients, TTo whom all correspondence
should be addressed. 1253
F. PeRIN
1254
and T. L&Y
we consider the particular case of a bidimensional suspension. As a matter of fact an assembly of parallel, infinite in length fibres is studied. In the physical situation when this suspension is carried along a straight channel by a viscous fluid flow, one can presume that the bulk flow is a Poiseuille-like one with a modified viscosity. This paper is concerned with this problem. The asymptotic homogenization process for periodic structures is used. Section 2 is devoted to the main features of this method. The problem is formulated in Section 3. It presents a bidimensional microstructure. The determination of the local variations of the flow (at the level of the microstructure) is performed in Section 4. The macroscopic equations for the bulk flow are obtained in Section 5. A second-order viscosity tensor, completely determined by the microstructure geometry at the considered instant, appears. In this particular study the microstructure does not evolve in time (Section 6). So the viscosity tensor remains independent of time. The last section is devoted to the study of this tensor, especially the influence of microstructure symmetries, the display of lower bounds for the viscosity coefficients, and numerical results.
2. HOMOGENIZATION
The method concerns media with a microstructure on a scale much smaller than the macroscopic scale of interest. The description of finite volumes of so complicated media with a high density of strong heterogeneities is practically possible at the macroscopic level only when an equivalent continuous medium is defined. Homogenization [9-111 is a double scale asymptotic process related to a small parameter E = l/L, where 1 and L are the characteristic lengths of the microstructure and the macroscopic medium respectively. Two space variables x and y =X/E are introduced and all the quantities entering the problem are postulated of the form f(x, y), which emphasizes the fact that the medium varies rapidly on the small scale I and may also vary slowly on the larger scale L. We look for the unknown quantities in the form of double scale asymptotic expansions: u’(x)
=
u”(x, y) + EU’(X,y) + e2u2(x,y) + . * .
Introducing these expansions in the equations of the physical problem, and identifying the like powers of E gives differential problems in the x and y variables. Generally speaking, equations in y are solvable if the microstructure is in some sense periodic, and terms u’(x, y) in the postulated asymptotic expansions have to be periodic in the y variable with the same period as that of the structure. The macroscopic equations, with respect to the macroscopic variable x, are obtained by taking mean values of the local solutions in a period or by writing the asymptotics of conservation laws in a macroscopic arbitrary domain [ll]. This leads to a rigorous deductive procedure for achieving the equations for the bulk behaviour of the medium. Thus homogenization gives the passage from a microscopic description to a macroscopic discription of the problem, without the need of phenomenological hypotheses. As a matter of fact, by applying the classical equations of fluids and solids mechanics to the components of a mixture, we can deduce its bulk behaviour without any presumed description of the components interactions at the level of the microstructure. In the study of media with large deformations, as suspensions, the geometry pattern evolves in time and depends on both its configuration at the initial time and the flow for the subsequent time. In the case of a suspension of three-dimensional rigid particles it has been proved [7,11], that if the particles at the initial time to are locally periodically distributed in the fluid [that means with a period .eY(to, x) which may vary slowly on the larger scale L], the microstructure evolves in time by keeping the locally periodic character. Consequently at any instant in the
Homogenization
method applied to a suspension of fibres
1255
evolution of the system the required periodicity conditions of the homogenization techniques are satisfied with a period which is that of the structure at the same instant. Finally let us remark that according to the required periodicity condition of the microstructure, the small characteristic length I = EL characterizes both the smallness of the particles and their mutual distance. So the investigated suspensions are nondilute.
3. FORMULATION
OF THE PROBLEM
The two-dimensional flow of a nondilute fibres suspension in a large straight channel is investigated. The thin fibres are assumed to be rigid cylindrical infinitely long bodies which are parallel to the Poiseuille-like macroscopic flow. Denoting by L the characteristic width of the channel and by 1 the characteristic thickness of the fibres, the small parameter of the homogenization process is E = l/L. At the considered instant t the distribution of the fibres in the incompressible viscous fluid is supposed locally periodic. The e3 axis of a rectangular Cartesian system of coordinates X(X,, x2, x3) is chosen to
Fig. 1. Locally periodic distribution of the fibres and basic period Y(t, x, , x2),
F. PBRIN and T. LhY
1256
coincide with the direction of fibres. Henceforth we assume that the section of the periodicity cell by a plane e,, e2 is a parallelogram cY(t, x,, x2). It can be considered homothetic with the ratio E, E << 1, of Y(t, x,, x2) which is called the basic period in the stretched coordinates. For the sake of simplicity we consider that the solid part Ys of Y, corresponding to the fibre section, is completely surrounded by the fluid filling the complementary part Y,, and that Y is centred in the mass centre G of the fibre section. So the stretched coordinates may be defined as: Xl
Yl =
-xcd
x2
&
’
-
Y2 =
xc2
&
The motion equations are written using the Einstein summation convention. the medium the velocity V” and the pressure P’ satisfy the equations
In the fluid part of
(1)
div V” = 0 po
=?s
8Vg r+v;$
(
I
>
(2)
axj
with U; = -P&6, + 2/~Dij(V”),
Dij(VE) = ; ($
+ $) I
. I
(3)
In each fibre, V’ is a rigid velocity field, so Dij(VE) = 0.
(4)
Because of the non-slip condition, on the solids surfaces V” is continuous.
(5)
For each arbitrary portion S’ of a fibre defined by xX E [a, b], the dynamic law implies
(6)
I
-I -
p;(x-x,)n$dv=
(X
s
r).s
~0)
A
ri+jei da.
(7)
In the two latter relations, n denotes the unit inward to the solid normal. The Sk denote the components of the stress tensor in the whole medium defined as 88 = a; given by (3) in the fluid and extended symmetrically in the solids with the stress continuity condition on the fibres surfaces and the momentum equation SFfl,
= c&n,,
.dV; PST=
3 ax; .
Let us remark that 6” is not uniquely defined in the solids. It is submitted only to the influence of the fluid flow by (8) and to the bidimensional geometry of the fibres. So we can assume that the form of the asymptotic expansion (when E is small) of the stress tensor in the solids is analogous to the fluid one. In the following we write x for (x,, x2, xg) and y for (yr, y2). Correlatively latin indices i, j - - * can take values 1,2,3, whereas greek indices cy, p 1 . . take values 1,2. According to the general features of the homogenization method, the solid density pg is supposed, at instant t, of the
Homogenization
method applied to a
suspensionof fibres
form p&,, x2, y). We search asymptotic expansions of the equations (l)-(7), zero, of the form: V” = Vg(t, X, y)e, + EV:(f, X, y)ei + * * * P” = P”(t, x, y) + EP*(t, x, y) + * * *
1257
when 6 tends to (9) (10)
with V’:, Vf, . . . , PO, Pi,. . . , Y-periodic with respect to the local stretched variable y. In (9) we have taken into account that, in first approximation, the macroscopic flow will be parallel to the fibres and the channel walls.
4. LOCAL
STUDY
We insert expansions (9) and (10) in equations (l)-(7) taking care that, when applied to a a 1 a function f(x, y), $ becomes g + --. In the following, index x (or y) specifies partial a n E ay, a2 derivative, for example A,, = By identifying the like powers of E, we obtain successive a~, ay,’
differential problems with respect to the local variable y, at this stage x is considered as a parameter. Let us remark that according to the required condition of Y-periodicity, we may consider that y varies in the basic period Y, then the periodi~ity of a function means that it takes equal values on the opposite faces of dY. 4.1 Local variation of V,”
It is determined by the first approximation of equations (Z), with (3), (4) and (5). From the O(F-‘) approximation of (2), the O(E-I) approximation of (4), and the 0(&O)approximation of (5), we obtain AyV;=O
av:
for y E Yr
-0
for y E Ys
ayuV!j is continuous
for y E au,.
With the conditions of Y-periodicity, these equations determine the local variation of V! up to an additive function independent of y. As a matter of fact, with the appropriate space of functions W = w, w E H”(Y), w Y-periodic, g = 0 in Ys, f w dy = O] i Y CX they are equivalent to the following variational problem:
vl: E w, The Lax-Milgram is obviously
VW E
theorem proves existence and uniqueness of the solution in W. The solution V% -K y) = V$, X).
Furthermore,
from the O(E’) approximation
2 av’ g-+&o 3 ES S&S-S
w.
ah
of (1) and (4) for y E Y.
(11)
F. PeRIN
12.58
and T. L&Y
Integrating this relation in Y, taking into account (11) and the Y-periodicity
v: = v@,XI,
of V’, we obtain
x2).
(12)
in Y.
(13)
Consequently div,, V’ = 0
From (3) the stress tensor in the fluid admits the asymptotic expansion a;=(T;+q!i+.”
with according to (9) and (12) a:0 = -P%,,
+ 2/&,,(V’), (14)
The next approximation independently of Vi.
of equations
(l)-(7)
gives the local variations of VL, (Y = 1, 2, and
4.2 Local variations of V:, Vi They are connected by (13) the 0(&-l) approximation of (2) and the 0(&O)approximation (4) the O(c) app roximation of (5), and the O(E’) approximation of (7).
of
for y E Yr
Q&V’) = 0
for y E Ys
V’ continuous for y E aY,
Let us remark that the first approximation of (6) is identically satisfied taking into account the Y-periodicity and the divergence free property of the tensor 0’. The vector of components Vi, Vq is determined up to a Y-periodic vector U = (V,, U,) satisfying D++,(U) = 0 in Y, that is to say, a vector independent of y. In this frame, like in the former section, the variational formulation of the problem is obtained by multiplying the second equation by a test function w’ and integrating in Yr using the previous equations. The appropriate space of functions is W’ =
w’,
w’
E
[H'(Y)]*,
w’Y-periodic, div,, w’ = 0 in Yr, D&w’)
= 0 in Ys, /.w’dy
=0}
and the variational problem is
w:,Vi) E W’,
J‘&pyW’Pa,(w’) dy = 0
VW’
E
W’.
YF
It follows that K&P X, y) = VXr, x).
(15)
1259
Homogenization method applied to a suspension of fibres
Furthermore,
according to the expression (14) of a$,
we deduce that grad, P0 = 0, thus
P(f, -%y) = PO@,x). 4.3 Local variation
(16)
of V.{
The O(E-‘) approximation of (5) are
of (2) the 0(&O)approximation
AJ:
=0
av: dy,=
of (4) and the O(E) approximation
for y E Y,
avg
for y E Ys
-z
Vi continuous for y E aus. These equations determined convex subset of H’(Y)
Vi up to a function independent
K = u, u E H’(Y), u Y-periodic, 2
of y. Introducing
in &, I,~dy
= -2 CE
the closed
=0}
the variational problem for Vi is
From a theorem by Stampacchia [12], Vi is uniquely determined
v:(tYx7 Y)
= -
3~~+A(t, ax
in K, and we can write
x)
(17)
n
with X.&, x1, x2, y) the unique Y-periodic solution, continuous on aYs, of: A,,x~ =0 for y E YF Xa=ya for YE Ys.
(18)
According to the previous results and (14), the stress tensor in the fluid is of order so, with $2 = 0,
ff:, = a& = & = -P(t,
x),
(19) We also remark that the first approximations calculations are identically satisfied.
of (6) and (7) which are not used in the previous
5. MACROSCOPIC
EQUATIONS
5.1 Macroscopic conservation law First of all, it should be noted that V” = (0,0,Vs) is a macroscopic quantity (depending only on the macroscopic variable x) and (12) establishes that the macroscopic flow is incompressible. As classically in homogen~ation process [ll], we obtain the macroscopic momentum equation by applying the momentum conservation law to the mixture contained in an arbitrary
F. PeRIN
1260
and T. L&Y
Fig. 2. Section T of the macroscopic cylinder D.
macroscopic cylinder D parallel to e3, with x3 E [a, b] and the section consisting of a great number of periods EY. This law is
T by a plane e,, e2
(20) where density p’ is p,, in the fluid and p$ in the fibres. We look for the first order approximation of each term when E tends to zero, using the asymptotic expansions of V” and 6’. The lateral boundary dT X [a, b] of D lies in the fluid, so the first approximation of the stress tensor CV is (+Ogiven by (19). In the solid cylinders the first approximation of B” is of order co as in the fluid. Only the 5ii appear in (20). According to (8)
a@,
-=
aYC2
furthermore
0
10
from the (E-‘) approximation
a& -=
0
= u3sa
u3sa
and
for y E Ys
for y E ar,
(21)
of (2), already used in Section 4.3: 0
(22)
for y E YF,
dY,
so the c?!, defined in the whole Y are divergence free. With regard to (ii3 according to the remark connected with (8) we can assume that this undetermined component in the cylinders depends on x3 like ai3, that is to say A0 u 33 =
-p”(t,
x>
+
G(t,
Xl,
x2,
y).
(23)
With calculations analogous to [ll] we can write with (12), (15) and (17)
PET&,=
I
$$e3du
+ o(l).
D
The mean density defined by
is, when the cylinders are homogeneous
(ps independent 6 = PO+ c(Ps - PO)
where c(x, t) = lYs{/lYl is the solids concentration.
of y)
(24)
Homogenization
applied to a suspension
method
of fibres
1261
The projections of the right-hand side of (20) on e, and e2 are, according to (19) h I i)D
6&dda=
-
P’n, ds, dx3 +
If(1
ilr
I Ti
where T, and Th are the sections of D corresponding Let us examine Jr&t3 da. As in [ll] we obtain:
S”,, da -
I r,
a”,, da + o(l)
to x3 = a and x3 = 6.
fT
The last term can be expressed using relations (21)
with ~0,~given by (19). This proves that _fT&i3 dy does not depend on x3. So we have:
f JD
The projection
- 1” l,P’n_
Qridu=
dsx dx3 + o(l) = -I,
a
g
dx + o(1). LI
(25)
of the right-hand side of (20) on e3 is:
If b
f l3D
&pi da =
(1
The stress component solids,
fT
&g3du=
c &YCT
u~,n,drdr,+lbg,du-Ibpdo+o(l).
.3T
Si, on a section T is expressed by (19) in the fluid and by (23) in the
x) fv, If-I’“@,
c2
dy +
[-P”(f, x) + u’$(~ ~1, x2,
~11 dy + 41)
YF
and then
We have to calculate the stress on dT. The boundary of T is made of sides cay, (with normal denoted II“, k = 1,2,3,4) of the neighbouring periods of T, so the following approximation occurs:
f aT
u$Jl,ds=
c
EJY&dT
E
f aY,
s;Jznl;d.sy+ o(1).
As we have seen before, (21) and (22), c?:,, (u = 1,2), in the whole period Y is divergence free. Thus we know [ll] that in the integral on a side aY,, B$zk may be replaced by &n”,, with a:, the mean macroscopic stress tensor component defined by
(26) So we can write
F. PeRIN
1262
and T. LhVY
And we have (27)
Taking into account (24), (25) and (27). The first approximation macroscopic momentum equations.
aP" -= 0
(Y= 1,2
axa
_av:’ PT=-~ 5.2 Homogenized
of (20) leads to the following
aP
f-
aa;, ax, ’
(29)
constitutive equation
It appears in (29) a bulk plane stress tensor defined by (26). In this section we find the macroscopic constitutive equation and prove that the average value (26) is independent of the extensions [not uniquely determined by (21)] of the & in Y,. As a matter of fact, as we have found in the previous section, and using (18)
=-
I
~~,x~n, hy = -
3%
=-
I a(&d YF
aYY
dy
axa
o
-dy. u3Y ay,
In the latter equalities we have taken into account the Y-periodicity (22). Then we obtain for (26):
We recall that a& is given by (19), so the homogenized -0
u3,
= aLlo
aV’:
constitutive
of the xa and a!?, and
equation is
with ao,p
(30)
8x0 Taking into account the Y-periodicity of the &, and (18), it can be written: (31) The macroscopic constitutive equation involves a second order viscosity tensor, the coefficients of which are completely defined by the microstructure at the considered instant t. Let us note that, like the basic period Y the asp may depend on the macroscopic variables x,, x2. It is easy to prove that the viscosity tensor is symmetric and positive definite. The bulk medium behaviour is that of an anisotropic generally inhomogeneous fluid. 6. DEFORMATION
OF THE
STRUCTURE
AND
CONSEQUENCES
The macroscopic relations (30) contain homogenized coefficients aap depending on the microstructure by the averaging domain and the local solutions X~ of the problems (18). These coefficients are calculated using the basic period Y(t, x1, x2). The microstructure of the medium is locally periodic at instant t and driven by the flow with the velocity V’ V” = V;(t, x,, xz)e3 + eV’(t, x, y) + O(E)
Homogenization
1263
method applied to a suspension of tibres
d.
Fig. 3. Geometrical characteristics of the microstructure.
with V’ Y-periodic and given by (15) and (17). Like in [ll], in order to describe the deformation of the structure we introduce the appropriate variables: two vectors d’ and d* of order E characterizing the form of a period in the plane e r, e2, and the rotation of a fibre. dd” dt for (Y= 1,2. We have to calculate e, - -g- and e2 sdd” dd’
dAB - = V(B) dtdt
- V(A) = V@(B) - v(A)
+ s[V’(B) - V’(A)] + O(E).
The terms containing V’ give contribution of order E [because AB = O(E)] and the terms containing V’ give contribution of order E* because the y dependence is the same in A and B by periodicity. We have at the first order of approximation: dd’ dt=
dV:(t, x1, x2)
ax,
dbe, + 0(c)
and generally for (Y= 1,2 dd” *-=o(E) e’ dt
and
e2 * 5
= O(E),
(32)
the form of the period Y in the e,, e2 plane does not vary. The rotation rate of a fibre is
the first approximation
is, with y E Ys
nc =f(rotXV@+rot,V1)+o(l)=~[~e,-~~e2-~~e,+~~e2]+o(l)
a
(I
and taking into account xU = y, in Ys [see (US)], nE = O(l), then in first approximation,
the rotation
of the fibres is zero. The microstructure
(33) remains
1264
F. PeRIN
and T. Ll?VY
bidimensional and according to (32) and (33) the periodic distribution of the fibres does not evolve in time. In this case we have proved that the microstructure does not evolve in time, contrary to the three-dimensional case [7,11]. Consequently the homogenization process (Sections 3-5) can be performed at any instant with the given basic period Y(x,, x2). Furthermore, since the microstructure is independent of the macroscopic velocity V!e,, so it is for the viscosity tensor and the bulk fluid is then Newtonian. According to (28), (29) and (30), the steady flow is Poiseuille-like, and the longitudinal bulk velocity is governed by
a avl: = constant. -ax, [aap -1 ax, In the next section we specifically study the viscosity tensor asp using its expression (31).
7. THE
VISCOSITY
TENSOR
In this section the basic period is supposed to be rectangular. 7.1 Influence of period symmetries First we suppose that Y admits Oyl as a symmetry axis [Fig. 4(a)]. The local problem (18) reduces to finding x, and x2 2A periodic with respect to y, satisfying: Ax, = 0 in Y,/2; Ax2 = 0 in YJ2;
x1 = y1 Y E a&/2; x2 = y2
Y E a&./2;
ax1 ---g=o
yES,USg
x*=0
y ESlUSX.
(34)
In this case, the viscosity tensor (31) is given by a,2 = azl = 0
(35)
y/2
Y/4
Y
;3
O3
B
aYs/4
Y$2
=4
I-LIPe,
) S.I 2A
(4
t
2
B
-=2
<
YF/4
a4
aYs/2
Yl
I&
Yl
a1 <
A
)
(b)
Fig. 4. (a) Half of the basic period Y admitting Oy, as a symmetry axis; (b) quarter of the basic period Y admitting Oy, and Oy, as symmetry axis.
Homogenization
method applied to a suspension of fibres
1265
If we assume that Oyl and Oy, are symmetry axis in the period Y [Fig. 4(b)], x, and x2 are then the solutions of Ax, = 0 in YJ4;
xl =Y, on aYJ4;
X,=Oona,Ua,;
~=Oon~,Uo,
Ax2 = 0 in Y,/4; x2 =
x2 = y2 on aYJ4;
ax2
z=O
0 on (T, U a3;
on g2Uf14
and the viscosity components are given by (35) in which Y/2 may be replaced by Y/4. If in addition, the period Y is left invariant by a II/2 rotation we obtain a macroscopic isotropy in the el, e2 Plane: aI1 = a22 = ph, because Xl(Yl, Y2) = X2(Y2, Yt). So we obtain by (31)
This proves that if the microstructure is such that it exists a macroscopic isotropy in the plane perpendicular to the fibres’ direction, the macroscopic viscosity of a suspension of parallel fibres in a bulk longitudinal flow is greater than ~(1 + c), with p the fluid carrier’s viscosity and c the fibres concentration. A better lower bound will be found in the next section. 7.2 Lower bounds for the viscosity coefficients We define: Yt = .;f;
{Y,>
y; = ym$ {y,}. E
and
We call “upper function” of aYs with respect to y,: fi:]Y;,Y:l+R yl+fdyJ
=
max
(Ylm)EaYS
{y2)
and in an analogous way f2 the “upper function” of aYs with respect to y2. Let us remark that generally the fa are not continuous. Now we consider the case when the basic period Y admits Oy, as a symmetry axis. Then we can find a lower bound for a22lcL.
Fig. 5. Graph of the “upper function” fi in the case when the basic period admits Oy, as a symmetry axis.
F. Pl?RIN and T. L&Y
1266
Looking at the expression (35) of uZ2/~, we have:
I
6x2 ax2 5
yl2 ay,
ay,
sc$($2dy2dy,.
We have [let us recall that x2 E H’(Y)]: h(h)
I
ax2
ayz dy2 = XZ(Y,,
0
fi(yl))
- XZ(Y,,
0).
From (18) x2(y) = y2 for y E Ys and using (34) we obtain: XZ(Yl,
h(Yl))
and
=h(yJ
x2(y,,
0) = 0
then fi(Yd
I
ax2
~dy’=fi(yd
0
By the Cauchy-Schwarz
inequality:
’dy2>-fh). We have also, using again (18) and (34)
B
ax2 - dyz= -fi(~J i fi(Yl)aY2 and by the Cauchy-Schwarz
inequality:
6,,, (3
2
f?(Yl) dyz%
-fi(y,)’
so:
and
Let us remark that this lower bound depends only on fi and is not determined by aYs/2, as a matter of fact it is the same for all the graphs of the fibre section having the same “upper function” f, with respect to y,. In some sense the lower bound of a22 is not influenced by the concave parts of the fibre section. When Oyi and Oy, are symmetry axis of Y, we obtain lower bounds for a,,/~ and a221p: &1+CL
&I+-w
1 B
ys Io A
1 A0I
h(Y2)
dy2
-h(y2)
yT fi(Y1) B -I
dy
(37) ‘*
Homogenization
method applied to a suspension of fibres
If in addition the period Y is left invariant by a z/2 rotation, medium is isotropic, and we have 1
&I+I-L
yi
fi(Y*)
1267
as we have proved the bulk
dy
A I o A -.L(Y,)
(38)
”
The results can be applied to particular simple cases (see Fig. 6). In all the envisaged geometries Y admits Oy, and Oy, as symmetry axis, so aI2 = 0 and (37) is: Case 6a:
ab
&I+ P
B(A-a)’
ab
%>I+ p
A(B -b)
1
Case 6b:
Case 6c:
f&1+ P
Case 6d:
ab B(A-a)’
arctg
A+a
-
J A-a
J
B+b B-b
---nA
1
xB 2b
’
2a
1 1 1’
A
(4
)
A
(b)
YF/4
(cl
(d)
Fig. 6. Case of a fibre with (a) rectangular section; (b) cross shaped section; (c) section involving concave parts; (d) elliptic section.
F. PBRIN and T.
1268
Fig. 7. (a) Square shaped period and square shaped fibre section; (b) square shaped period and circular fibre section.
7.3 Numerical results
We study two sorts of geometry for the period Y which leads to macroscopic isotropy [Figs 7(a) and (b)]. The values of p,,/p [see problems (36) and (35)] and the values of its lower bound m [see (391, determined in Sections 7.2 are calculated using the MODULEF code. The results are given in the following tables. Table 1. Values of the concentration c, of the relative viscosity coefficient of the suspension p,,/p and of its lower bound m. Case of a square shaped fibre section of side a. a
c =a2
PhlP
m
0.10 0.20 0.30 0.40 0.50 0.60 0.65 0.70 0.75 0.80 0.85 0.90
0.01 0.04 0.09 0.16 0.25 0.36 0.42 0.49 0.56 0.64 0.72 0.81
1.03 1.09 1.22 1.42 1.73 2.23 2.58 3.06 3.73 4.73 6.40 9.77
1.01 1.05 1.12 1.26 1.50 1.90 2.20 2.63 3.25 4.20 5.81 9.10
Table 2. Values of the concentration c, of the relative viscosity coehicient of the suspension p,,/p and of its lower bound m. Case of a circular section with diameter a. a
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.98 0.99
c = na2/4
0.0078 0.0314 0.0706 0.1256 0.1963 0.2827 0.3848 0.5026 0.6361 0.7088 0.7542 0.7697
PJP
m
1.02 1.06 1.15 1.28 1.48 1.78 2.26 3.10 5.08 7.97 13.73 20.19
1.008 1.037 1.095 1.19 1.34 1.59 2.01 2.79 4.70 7.52 13.22 19.70
Homogenization
method applied to a suspension of fibres
1269
From these numerical results we can conclude that the lower bound found in Section 7.2 is a rather good approx~ation of the homogenized viscosity whatever the solids concentration c is. The numerical values of j.&halso emphasize the dependence of the homogenized viscosity on the geometry of the microstructure. When the solid concentration is smaller than 0.5 the influence of the form of fibre section is not determinative, but for c greater than 0.5 the geometry is a basic parameter.
8. CONCLUSIONS
We have applied the homogenization process to the determination of a continuum model and its governing equations for an oriented fibres assembly suspended in a viscous fluid in a Foiseuille-like situation. The study emphasizes the incompressible anisotropic Newtonian fluid behaviour of the bulk medium. In this special two-dimensional microstructure the viscosity of the homogenized medium is determined by a second order tensor depending only on the microstructure. As a matter of fact a suspension of fibres in a viscous fluid is relevant to the general case of a three-dimensional suspension of rigid particles, It displays a macroscopic viscosity described by a fourth-order tensor ([7,11]), such that the homogenized constitutive equation for the macroscopic viscous stress is
(with the notations of the Section 5). In the present paper, the assumption of a macroscopic flow parallel to an assembly of infinitely long fibres leads to a two-dimensional problem which allows a more simple study. The macroscopic viscous stress is given by the constitutive equation (30). In this relation the viscosity coefficients aap (a!,@ = 1,2) are approximations of the -4aP3 when the fibres are infinite in length in the e3 direction. Lower bounds of these coefficients are found when the microstructure admits some symmetry properties. The effective numerical results in two simple cases suggest that these lower bounds may be good approximations of the viscosity c~fficients. They also emphasize the influence of the microstructure geometry on the values of the bulk viscosity for nondilute suspensions. We find, as [2] in another particular problem, that up to moderate concentration of particles, the viscosity of the suspension displays only a weak dependance on the particles geometry. For solids concentration greater than 0.5 the dependance becomes drastic, this is related to the packing of the fibres.
REFERENCES [i] [Z] [3] [4] [5] [6] [7] [8] [9]
H. BRENNER, A. Rev. Fluid Mech. 2,137 (1970). H. PHAN-THIEN, T. TRAN-CONG and A. L. GRAHAM, J. Fluid Mech. 228,275 (1991). G. K. BATCHELOR, f. F&id Mech. 46,813 (1971). A. ACRIVOS and E. S. G. SHAQFEH, Phys. Fluids 31, 1841 (1988). G. K. BATCHELOR, 1. Fluid Mech. 41,545 (1970). A. C: ERINGEN, Rheol. Acru 30,23 (1991). T. LEVY and E. SANCHEZ-PALENCIA, J. Non-Newr. Fluid Mech. 13,63 (1983). K. C. NUNAN and J. B. KELLER, J. Fluid Me& 142,269 (1984). A. BENSOUSSAN, J. L. LIONS and G. PAPANICO~OU, Asymptotic Analyst for Periodic Srructures. North-HoIIand, Amsterdam (1978). [lo] E. SANCHEZ-PALENCIA, Non-Homogeneous Media and Vibrations Theory. Lectures Notes in Physics, Vol. 127. Springer, Berlin (1980). [ll] T. LEVY, in Homogenization Techniques for Composite Media, Lecture Notes in Physics (Edited by SANCHEZ-PALENCIA and A. ZAOUI), Vol. 272, pp. 63-119. Springer, Berlin (1987). [12] H. BREZIS, Anaiyse FonctionneNe. Masson (1992). (Received and accepted 12 October 1993)