A Continuum Theory of Dense Suspensions Eringen2005

8/12/2019 A Continuum Theory of Dense Suspensions Eringen2005

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Z. angew. Math. Phys. 56 (2005) 5295470044-2275/05/030529-19DOI 10.1007/s00033-005-3119-2c 2005 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

A continuum theory of dense suspensions

A. Cemal Eringen

Abstract. A continuum theory is introduced for viscous fluids carrying dense suspensions (suchas blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertiathat make the mixture an anisotropic fluid whose viscosity changes with motion and orientation ofsuspensions. The microinertia balance law coupled with the equations of motion of an anisotropicfluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutiveequations are obtained in terms of the frame-independent tensors. In a special case, a theoryof bar-like suspensions is obtained. The field equations, boundary and initial conditions aregiven for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory isdemonstrated with the solution of the channel flow problem. The mean viscosity of the fluidwith suspensions is determined. The motions of suspensions down flow are demonstrated.

Keywords. Suspensions, emulsion fluids, blood, anisotropic fluids.

1. Introduction

When isotropic fluids are mixed with particulate suspensions, the fluid viscos-ity is increased. In the case of dilute suspensions, the increase in viscosity wasdetermined by Einstein in his Ph.D. dissertation as early as 1906:

= 0(1 +),

where and 0 are, respectively, the viscosities of the fluid with and withoutsuspensions, and is the volume fraction. Since that time, a large number ofresearch papers have appeared in literature, improving Einsteins formula some-what. Nevertheless, the state of dense suspensions remains unresolved. The sub-

ject has been approached from both molecular and continuum viewpoints, Doiand Ohta [1], Almusallam et al [2, 3], Lee and Denn [4], for immiscible blendsof polymers in dispersed emulsions. Some of these quasi-molecular approachesare explored in a recent book by Fuller [5], for dilute and dense suspensions,by considering dumbbell molecules of rigid or flexible bars, and polymer seg-ments in viscous fluids. These approaches lead to some equations for suspen-sions that are not closed. Another modeling employed by Phan-Tien et al [6]uses the expression of the center-to-center vector between two spheres in a vis-cous fluid that is due to Kim and Karrila [7]. These authors employ the concept


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530 A. C. Eringen ZAMP

stresslet introduced by Batchelor [16], as an extra stress arising from the pres-ence of suspensions, to be added to the viscous stress. In these approaches, equa-tions of motion of suspensions remain open, and ad hocclosures are suggested.One may also question the concept of stresslet superposition in a nonlinear the-ory.

The continuum approach presented by Eringen [8, 9, 10] employs the conceptthat suspensions and surrounding fluid together constitute a continuum with mi-crostructures. This continuum is endowed with a microinertia tensor at each point,just like rigid bodies. The most general model of such fluids is the micromorphicfluid, introduced by Eringen [12], see also [11]. Micromorphic fluids are proper

candidates to model flexible viscous suspensions like polymer melts, or blood flowin capillaries. If the suspensions are rigid, the proper model is the micropolar fluid,[11]. The balance laws of these fluids involve two additional equations of motion;one for the microinertia tensor and one for the stress-moment tensor. Here, weare interested in suspensions in Newtonian fluids.

The presence of suspensions dictates that we must retain the microinertia bal-ance law of micromorphic mechanics. In addition, we must obtain constitutiveequations for the stress-moment tensor and the heat vector that are functions ofthe deformation rate tensor and the microinertia tensor, with no superpositioninvolved. Here, suspensions can be of any arbitrary shape whose shape and inertiachanges with the motion. This is particularly relevant to the emulsion of viscousdroplets in fluids and blood flow in arteries.

In Section 2, we derive a balance law for the microinertia tensor, and show that,for bar-like molecular melts, this balance law reduces to that given by Jeffery [13],for rigid suspensions in a shear flow.

Section 3 contains the statements of the balance laws and the second law ofthermodynamics. In Section 4, we develop constitutive equations for both thecompressible and the incompressible thermofluids carrying suspensions, subjectto the restrictions of frame-independence and the second law of thermodynamics.Under these restrictions, we construct properly invariant constitutive equations, inwhich a pressure tensor appears, in addition to a scalar thermodynamic pressurethat depends on the suspension inertia. These new effects appear not to have beennoticed before.

The constitutive equations for rod-like suspensions are obtained in Section 5,both for compressible and incompressible fluids. All constitutive equations display

the anisotropic nature of the mixture due to the microinertia of the suspensions.The field equations, boundary and initial conditions for arbitrarily-shaped suspen-sions, are given in Section 6, and those of bar-like suspensions, in Section 7.

By way of providing an example solution, in Section 8, we give the solution ofthe channel flow problem, for deformable, elliptic-shaped suspensions in viscousfluid. The mean viscosity is determined. The motions of suspensions down flow aredemonstrated. This solution may be used to determine some of the new materialconstants.


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Vol. 56 (2005) A continuum theory of dense suspensions 531

2. Microinertia balance law

We consider a viscous fluid with dense suspensions of arbitrary shape. Phenomeno-logically, this mixture is equivalent to an anisotropic fluid with mass density andmicroinertia tensorikl, defined by

ikl =kl (2.1)where is the mass density of microelements of a particle of the fluid mixture,and is the position vector of a point in the particle extending from its centroid.The angular bracket denotes the space mean.

The microinertia tensor ikl was introduced by Eringen [12] to account for themicrodeformations and microrotations of a material particle of a micromorphicfluid (see also Ref. [10, 11]). It is a subject to a balance law obtained by Eringenand verified by Oevel and Schroer [14], through statistical mechanical means fromthe atomic theory.

Di

Dt iT i g= 0, (2.2)

where is the gyration tensor. A superscript Tdenotes the transpose.It is well-known that, in micromorphic fluid mechanics, for a special case, when

the stress-moment tensor vanishes, and

kl

wkl =1

2(vk,l

vl,k), (2.3)

we obtain the equations of classical fluid (c.f. Ref. 11). Here and throughout,an index after a comma denotes the partial derivative with respect to Cartesiancoordinate xk, e.g., vk,l = vk/xl. With (2.3), (2.2) reduces to

Di

Dt iwT wi g(,, i, d, w) =0, (2.4)

whereDi/Dt denotes the material time rate, is the absolute temperature and, dand w are, respectively, the deformation-rate tensor and vorticity tensor, definedby

dkl = (vk,l +vl,k)/2, wkl = (vk,l vl,k)/2 (2.5)Eq. (2.4) represents a microinertia balance law, which can be arrived at without

reference to the theory of micromorphic fluids. In fact, the microinertia flux,defined by

= Di

Dt iwT wi (2.6)

is a frame-independent tensor, so that

=g(,, i, d, w) (2.7)

represents a tensor balance law, where the tensor g is also frame-independent.The generators of a symmetric tensor, for a polynomial representation, in terms


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ofi, dand w, may be read from Eringern [15], App. [B]. Here, we give the generalexpression ofg that is of the first degree in d and w

g= g01 +g1i +g2i2 +g3d +g4(id + di)

+g5(i2d di2) +g6(iw wi) +g7(i2w wi2).

(2.8)

The second degree term in i (coefficient ofg5) represents the stretch of a poly-mer melt. It is a dominant term which must be included for the discussion ofpolymer melts and blood flow.

Equation (2.8) can be simplified by considering special cases: for i = 0, gvanishes. Hence, g

0 = g

3 = 0. When the fluid is at rest, Di/Dt cannot change.

This implies thatg1= g2= 0. The presence of terms, with coefficients g6and g7in(2.8), destroys the frame-independence of , upon combining, unless g6 =g7 = 0.With these, (2.8) reduces to

g= (id + di) +(i2d + di2), (2.9)

where, we set g4 = and g5 = . These moduli are functions of and . Withthis, the microinertia balance (2.7) reads

Di

Dt iwT wi (id + di) (i2d + di2) =0. (2.10)

Equations similar to (2.10) have been obtained by various authors (c.f., [1, 5]),under the terminology equation of structure factor. However, most of these

equations are not closed, since they involve a fourth order unknown tensor. Notethat Eq. (2.10), jointly with the equations of motion, given in section 6, areclosed. Moreover, the tensor i has a meaningful physical significance, namely themicroinertia of the internal structure. The consideration of the surface tension andsplitting of suspensions are not considered here. Away from the state of failure ofsuspensions, (2.10) may be simplified further by setting = 0, namely

Di

Dt iwT wi (id + di) = 0 (2.11)

For rod-like suspensions, i can be expressed in terms of a unit vector n:

ikl = i0

2(kl nknl), n n= 1. (2.12)

Substituting (2.12) into (2.10), we obtain

1

2

Di0Dt

(kl nknl) i02

(nknl nknl) i02

wki(il ninl)

i02

(ki nkni)wli ai02

(2dkl nknidil ninldki) = 0 (2.13)

where we set

a= i0+i202

(2.14)


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The scalar product of (2.13) by n, leads to

nk+wlknl+(dklnl dilninknl) = 0, (2.15)where, a superposed dot denotes the material time derivative, e.g., nk = Dnk/Dt,except oni and j . Eq. (2.15) was given in another context in Ref. [8]. Expression(2.15) is identical to the one obtained by Jeffery [13], for rigid bars in shear flow, inan entirely different way. For rigid bars,a = (r21)/(r2+1), where,r = L/d is theaspect ratio of a suspended bar (ratio of its length to its diameter). Consequently,a0. Fora = 0, microinertia is conserved. a= 1 corresponds to rigid dumbbellsuspensions.

3. Balance laws

The balance law are:

Conservation of Mass

+ v= 0 inV , (3.1a)[[(v u)]] n= 0 on . (3.1b)

Balance of Microinertia

Dikl

Dt ikrwlr

ilrwkr

gkl(i, d) = 0 in

V , (3.2a)

[[ikl(v u)]] n= 0, on. (3.2b)Balance of Momentum

tkl,k+(fl vl) = 0 inV , (3.3a)[[tkl vl(vk uk)]]nk = 0, on . (3.3b)

Balance of Energy

+tkldkl+qk,k +h= 0, inV , (3.4a)[[tklvl+qk (+1

2v v)(vk uk)]]nk = 0, on, (3.4b)

where,, vk, ikl, tkl, fk, , q k andh are, respectively, the mass density, the velocity

vector, the microinertia tensor, the stress tensor, the body force density, the inter-nal energy density, the heat vector and the energy (heat) source density. The firstset of each group of equations, marked by an a (e.g., (3.1a)), is valid within thevolume V, excluding the points of a discontinuity surface, which may be sweepingthe body with its own velocity u.V denotesV V. The deformation-ratetensor dkl and the vorticity wkl are given by (2.5). nk denotes the positive unitnormal to . In the microinertia balance equation (3.2a), gkl are given by theconstitutive equations (2.9). The second set of equations, marked by a b (e.g.(3.1b)) are the jump conditions at the discontinuity surface . Double brackets


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denote the difference of their enclosures from positive and negative sides of .These jump conditions provide boundary conditions and they are fundamental tothe discussion of shock waves.

Excluding the microinertia balance law (3.2b), these balance laws are well-known in classical continuum mechanics. Since the material points considered inthe classical theory are geometrical points, they do not posses moments of inertia.In the continuum theory of suspensions, a suspension is also considered to be ageometrical points with an inertia tensor.

To the basic laws, we adjoin the second law of thermodynamics:

(qk/),k h/0 inV , (3.5a)[[(vk uk) qk/]]nk0 on, (3.5b)

where, and are, respectively, the entropy density and the absolute temperature.We introduce Helmholtzs free energy by

= , (3.6)and eliminate and h from (3.5a), by means of (3.4a) and (2.6), leading to

(+) +tkldkl + 1

qk,k0. (3.7)

This inequality is fundamental to the development of the constitutive equations.

4. Constitutive equations

The set of constitutive dependent and independent variables are, respectively,denoted by Z and Y:

Z={ , , tkl, qk}, (4.1)Y ={,,ikl, dkl, ,k/}. (4.2)

We decompose the dependent variable set into static and dynamic parts, de-noted, respectively, by left indices R and D :

RZ={, ,Rtkl}, (4.3)

DZ={Dtkl, qk}, (4.4)where

tkl =Rtkl +Dtkl. (4.5)

does not possess a dynamic part, and the dynamic part of is not relevant here.Of course, qk has no static part. is a function of, and the invariants of iklonly. Since

i is a small quantity as compared to the macroscopic length scale,

we express it as

= 0(, ) +1(, )tri. (4.6)


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We calculate the material time derivative of

=

+

+1

DikkDt

. (4.7)

From (2.11), we haveDikk

Dt = 2ikldlk (4.8)

Substituting this and from (3.1a) into (4.7), we obtain

=

v+

+ 21ikldlk. (4.9)

Carrying this and tkl from (4.5) into the inequality (3.7), we get

+

+ [kl+kl+ Rtkl]dkl+ D tkldkl+

1

qk,k0, (4.10)

where, is the thermodynamic pressure of the fluid with suspension, and kl is apressure tensor arising from the presence of suspensions, defined by,

= 0+1, kl =21ikl, (4.11)Here,0 is the thermodynamic pressure of the fluid without suspensions, and

1 is an additional pressure due to the presence of suspensions, defined by

0= 2 0

, 1= 2 1

i0. (4.12)

At this point, the reader may wonder why we have not included higher orderinvariants of i into the expression (4.6) of. The higher invariants of i, throughthe use of (2.10) and (2.11), produce terms in the resulting expression of, thatare not allowed by the theory of invariants.

The inequality (4.10) must not be violated for all independent variations of,dkl and ,k. Hence, we must have

=0

1

tri, (4.13)

Rtkl =kl kl (4.14)Inequality (4.10) is now reduced to

Dtkldkl +1

qk,k0. (4.15)

Consequently, with the presence of suspensions, both the thermodynamic pressureand the stress tensor are changed.

The solution of the inequality (4.15) is obtained in the usual way (c.f. Ref. [11,Section 3.4]).

Dtkl =

dkl, qk =

(,k/), (4.16)


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where, the dissipation function is a function of the invariants of the set Y, givenby (4.2). The joint invariants ofi and d are

tri, tri2, trd, trd2, tr(id), tr(id2), tri2d, tri2d2.

We express in the form

= 1+ 2, (4.17)

where,

21= (0+1tri +3tri2)(trd)2 + 2(2trid + 24tri

2d)trd +5(trid)2

2(0+1tri +3tri2)tr(d2) + 24tritr(id2) + 42tr(id2) + 45tr(i2d2)(4.18)

22= (K0+K1tri +K4(tri)2 +K5tri

2)(/)2+ (K2+K3tri) i /2 +K6(i ) (i )/2, (4.19)

where,0 to5 and 0 to5 are viscosity moduli, and K0 toK6are heat conduc-tion moduli. In general, they depend on the density and temperature. Substituting into (4.16), we obtain

Dt= [(0+1tri +3tri2)trd +2trid + 24tri

2d]l

+ (2trd +5trid)i + 24i2trd + 2(0+1tri +3tri

2)d

+ (22+4tri)(id + di) + 25(i2

d + di2

), (4.20)

q= [K0+K1tri +K4(tri)2 +K5tri

2]

+ (K2+K3tri)i

+K6

i2

. (4.21)

These constitutive equations account for all contributions (to the stress) ofthe nonlinear interactions of arbitrary-shaped suspensions with the surroundingviscous fluid. The constitutive equation (4.21) for the heat vector is new, not givenbefore, except for rod-like suspensions, Eringen [9, 10, 11].

Suspended particles generally possess smaller length scales as compared to themacroscopic scale L, i.e. i1/2 L. Consequently, consistent with (2.11), theterms involving second powers ofi can be dropped in (4.18) to (4.21), and we have

21= (0+1tri)(trd)2 + 22tr(id)trd + 2(0+1tri)trd

2 + 42tr(id2),(4.22)

22= (K0+K1tri) /2 +K2 i /2, (4.23)Dt= [(0+1tri)trd +2trid)l +2itrd + 2(0+1tri)d + 22(id + di),

(4.24)

q= (K0+K1tri)

+K2

i

. (4.25)


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The second law of thermodynamics requires that the dissipation function must be positive semi-definite for all independent variations ofi, dand/, i.e.,

10, 20. (4.26)These requirements impose restrictions on the material moduli. We observe

that ikl is a positive semi-definite tensor, having non-negative eigenvalues i1, i2and i3. Consequently, 2 will be non-negative, if and only if

K0+K1(i1+i2+i3) +K2ii0, i= 1, 2, 3 (4.27)This is fulfilled, if

K0, K1, K20. (4.28)In order to determine the restrictions imposed by 10, we express 1 in termsof eigenvalues (d1, d2, d3) ofdkl, and (i1, i2, i3) ofikl, i.e.,

21= Kijdidj, (4.29)

where,

Kii = 0+ 20+ (1+ 21)(i1+i2+i3) + 22ii,

i not summed, i = 1, 2, 3

2Kij =0+ 20+ (1+ 21)(i1+i2+i3) +2(ii+ij),

i=j, i, j = 1, 2, 3 (4.30)It follows that, the eigenvalues ofKij must be non-negative, i.e., the roots Ki

of the equationdet(Kij Kij) = 0, (4.31)

must be non-negative. Alternatively,

K110, K11K22 K2120, det(Kij0. (4.32)The case, i1= i2= i3= 0 gives the classical result,

30+ 20, 00. (4.33)

5. Rod-like suspensions

The static parts of the constitutive equations are given by (4.13) and (4.14), thedynamic parts by (4.24) and (4.25).

For rod-like suspensions, from (2.12), it follows that tri= i0, and we have

=

0

+ 1

i0

Rtkl =klRkl,

= 2

0

+1i0

a1i0,


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Rkl = a1i0nknl. (5.1)

The dynamic constitutive equations for stress and heat follow from (4.20) and(4.21), by substituting the expression ofikl from (2.12),

Dtkl = (trd +eninjdij)kl + 2dkl+etrdnknl

+e(nknjdlj + njnldjk) +dijninjnknl,

qk = 1

(Kkl+Kenknl),l, (5.2)

where,

= 0+ (1+2)i0+ (3+ 34+5/2)i2

0/2,= 0+ (21+2)i0/2 + (3+4/2 +5)i

20/2,

e=i04

(22+ 24i0+5i0),

e =[2i0+ (4+5)i20/2, =5i0/4,

K=K0+ (K1i0+K2i02

) +i20

2(K3+ 2K4+K5+K6),

Ke =i02

[K2+K3i0+K6i02

]. (5.3)

We observe that the presence of suspensions changes viscosities 0 and 0of the classical theory. In addition, the stress and the heat are affected by the

orientations of the suspended bars. Thus, the fluid has become anisotropic, theanisotropy changing with the motion.For incompressible fluids, we set trd = 0, in (4.20), or, equivalently, 1 to 5

vanish, and we obtain, for the stress tensor

tkl =pkl + 2dkl+ (+dijninj)nknl+e(dkininl+dlinink), (5.4)where, is introduced from Rtkl, given by (5.1), i.e.,

=a1i0, (5.5)with a scalar a1i0 incorporated with the pressure p.

Equation (5.4) is identical to that given by Ericksen [17] in the case of barsuspensions. We remind the reader that the nonlinear constitutive equations (4.20)and (4.21) are much more general. They are valid for arbitrary-shaped suspensions

in compressible fluids. These constitutive equations are valid for polymeric meltsuspensions as well.

6. Field equations

The field equations are obtained by substituting the constitutive equations for gkl,tkl and qk into the balance laws (3.1a) to (3.4a).

+ v= 0, (6.1)


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DiklDt

12

ikj(vl,j vj,l) 12

ilj(vk,j vj,k)

2

[ikj(vj,l+vl,j + ilj(vj,k+vk,j )] = 0, (6.2)

0,l 1,l kl,k+ [(0+1i0)vj,j+ 2iijvj,i],l+2(iklvj,j),k (6.3)+[(0+1i0)(vk,l +vl,k)],k+2[iki(vi,l+vl,i) + (vk,i +vi,k)iil],k+(flvl) = 0,

202

+21

2 i0

2

20

+21

i0

v

+Dtkl+ 2

1

ikl

dkl +qk,k +h= 0. (6.4)

In (6.3) and (6.4), 0, 1, kl and qk are to be replaced by their expressions(4.11), (4.12) and (4.25). For small temperature variations from the room tem-perature T0, we may replace by

= T0+T , T00, |T|< T0. (6.5)Usually, the first two terms, in (6.4), are linearized, so that

c0 T c1 v+ (Dtkl +c2ikl)dkl+qk,k +h= 0, (6.6)where, the diffusivity c0, at = T0, and c1 and c2 are given by

c0=0T0

2

02

+ 2

12

i00

,

c1= 20T0

20

+21

i0

0

, (6.7)

c2= 20T0

1

0

.

The field equations (6.1) to (6.4) constitute eleven partial differential equationsfor the eleven unknown , vk, ikl and . Hence the system is closed.

The field equations are subject to some boundary and initial conditions. Manydifferent boundary conditions can be derived from the set of equations (3.1b) to(3.4b), by letting coincide with the surface Vof the body. Here, we give a setof mixed boundary conditions that are relevant to a large class of problems.Let Vdenote a regular region of Euclidean space, occupied by the body whoseboundary isV. The interior ofVis denoted byV, and the exterior normal toVbyn. LetSi (i= 1, 2, 3, 4) denote subsets ofV, such that

S1 S2= S3 S4= V,S1 S2= S3 S4= 0.

A set of boundary conditions on these surfaces, at time T+ = [0, ) may beexpressed as


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vk = vk on S1 T+, tklnk =tl on S2 T+,= onS3 T+, qknk = qonS4 T+,

(6.8)

where, quantities carrying a carat are prescribed.The initial conditions usually consist of Cauchy data:

(x, 0) =0(x), v(x, 0) =v0(x),

(x, 0) =0(x), i(x, 0) =i0(x), (6.9)

where, quantities with a superscript zero are prescribed throughout V. Clearly,other possibilities exist.

Incompressible fluids

For incompressible fluids, = 0 = const., and v vanishes in all equations(6.1) to (6.4). In this case, is replaced by an unknown pressurep(x, t). The fieldequations are reduced to

v= 0, (6.10)DiklDt

12

ikj(vl,j vj,l) 12

ilj(vk,j vj,k)

2

[ikj(vj,l+vl,j) +ilj(vj,k+vk,j)] = 0 (6.11)

p,l 2a01ikl,k+2(iijvj,i),l+ [(0+1i0)(vk,l +vl,k)],k+2[iki(vi,l+vl,i) +ili(vk,i+vi,k)],k+(flvl) = 0, (6.12)

c0 T+ (Dtkl +c2ikl)dkl+qk,k+h= 0 (6.13)The boundary conditions (6.8) remain the same, with = replaced by T = T.The initial conditions (6.9) exclude(x, 0) =0, since now = 0= const.

7. Field equations of rod-like suspensions

For rod-like suspensions, the microinertia balance equation is given by (2.13) andthe constitutive equations by (5.1) and (5.2). The field equations are:

+ v= 0, (7.1)nk1

2(vj,k vk,j )nj+

2[(vk,l+vl,k)nl (vi,j +vj,i)ninjnk] = 0, (7.2)

,l kl,k+ (+)vk,lk+vl,kk +e(ninjvi,j),l+e(vj,jnknl),k+

e2

[nknj(vl,j + vj,l) +nlnj(vj,k+vk,j )],k

+(vi,jninjnknl),k+(fl vl) = 0 (7.3)


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2

2 2

2

v+ (Dtkl+ 2a 1

ikl)dkl+qk,k+h= 0, (7.4)

For small temperature variations, (7.4) can be expressed as

c0 T c1 v+ (Dtkl+c2)dkl+ 1T0

[(Kkl +Kenknl)T,l],k+h= 0. (7.5)

Since n n= 1, here we have seven partial differential equations to determine theseven functions , nk, vk and T. Hence the system is closed.

Incompressible fluids with rod-like suspensions

In this case, (7.1) to (7.5) reduce to

v= 0. (7.6)

nk+1

2(vk,j vj,k)nj + a

2[(vk,l+vl,k)nl (vi,j+ vj,i)ninjnk] = 0, (7.7)

p,l kl,k+vl,kk+e(ninjvi,j),l+ e2

[nknj(vl,j+ vj,l)

+nlnj(vj,k+vk,j )],k+(vi,jninjnknl),k+(flv) = 0, (7.8)

c0 T+c2ikldlk+ 2trd

2 + 2enknjdljdkl+ (+dijninj)nknldkl

+ 1T0

[(Kkl +Kenknl)T,l],k+h= 0, (7.9)

wherep(x, t) is an unknown pressure.

8. Channel flow

An examination of the field equations (6.11) and (6.12) reveals that ikl and vdepend on x1 and x2. However, with the linearization of (6.12), it is possible toobtain a Poiseuille velocity profile.

Here, we determine the velocity and microinertia profiles in a channel flow ofsuspended small elliptic plates. We consider a two-dimensional flow of incom-

pressible fluid suspensions, flowing within a channel located ath < x2 < h,0x1 . The usual assumption for the velocity is

vk =v(x2)kl, (8.1)

Of course, the microinertia can depend on x1 and x2, i.e.,

ikl = ikl(x1, x2) (8.2)

However, because of the presence of ikl in the equations of motion, the ve-locity will depend on both x1 and x2. Nevertheless, by means of linearization of


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the equations of motion for the velocity, we may obtain an approximate solutioncompatible with (8.1) and (8.2). With (8.1), the equation of continuity is satisfiedidentically. The field equations read

i11x1

v i12[1 +)]v,2= 0,i22x1

v i12[1 )]v,2= 0,i12x1

+1

2[(i11 i22) 1

2(i11+i22)]v,2= 0, (8.3)

andp,1 201(i11,1+i12,2) + {[(0+ (1+2)i0]v,2},2

+ [(2+ 22)i12v,2],1= 0, (8.4)

p,2 201(i12,1+i22,2) + (2+ 22)(i12v,2),2+ [(0+ (1+2)i0)v,2],1= 0. (8.5)

For the boundary conditions, we take,

v(h) = 0, (8.6)i11(0, x2) = 2c, i22(0, x2) =c, i12(0, x2) = 0. (8.7)

The conditions (8.6) are the usual ones, with the understanding that suspensionswill be, at least, one internal characteristic length away from the boundaries. Theboundary conditions (8.7) express that elliptic suspensions, having microinertias(i11 = 2c, i22 = c, i12 = 0, c 0), are distributed uniformly along the height ofthe channel, at the inlet of the channel.

For moderately dense suspensions, the fluid pressure will not change appre-ciably, so that we can set 1 = 0. With this, and substituting for i0/x1 =2i12v,2/v, from (8.3) into (8.5), we obtain

p,1+ (2+ 22)(i12v,2),1+ {[0+ (1+2)i0]v,2},2= 0, (8.8)p,2+ (2+ 22)(i12v,2),2+ 2(1+2)i12(v,2)2/v= 0. (8.9)

Ignoring the nonlinear term in (8.9), we have

p,2+ (2+ 22)(i12v,2),2= 0. (8.10)Eq. (8.10) integrates to

p+ (2+ 22)i12v,2=p0(x1), (8.11)where,p0(x1) is an integration function. Substituting p from (8.11) into (8.8), andintegrating, we obtain

p,1y = [0+ (1+2)i0]v,2= 0. (8.12)Consideration of the mean viscosity requires that we replace the field equations

(8.8) and (8.9), by their means, over an interval in the x-direction. For this, the


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most appropriate interval is one period ofiij in the x-direction. The mean valueof this equation, over one period, in thex-direction reads

p0,1y+ i0v,2= 0, (8.13)where,

= 0+ (1+2)i0 (8.14)is the mean viscosity of the fluid with suspensions. Angular brackets indicate themean, over one period, in the x-direction. Eq. (8.13) is integrated to obtain

v=

p0,1h2

2

(1

y2), y=

x2

h

. (8.15)

We now carry this into (8.3)

i11x

+ 2y

1 y2 (1 +)i12= 0,i22x

2y1 y2 (1 )i12= 0,

i12x

y1 y2 [i11 i22 (i11+i22)] = 0, (8.16)

where,x= x1/h. (8.17)

The solution of (8.16), subject to the boundary conditions (8.7), is given by

i11c

= 1

2( 1) (3 ++ (1 + 3)cos

21 21 y2 xy

,

i22c

= 1

2(+ 1)(3 ++ (1 + 3)cos

2

1 21 y2 xy

,

i12c

= 1 32

1 2sin

2

1 21 y2 xy

. (8.18)

It remains to show that the nonlinear term in (8.9) is negligible in the mean. Infact, the mean value ofi12 calculated over one period q(y) =(1 y2)/

1 2y

is given by

i12= 1

q q0

i12dx (8.19)

Substituting i12 from (8.18), we find thati12 = 0. Hence, the mean of thenonlinear term vanishes.

i0 = i11+ i22 has the period q(y) = (1 y2)/y

1 2 that depends on y,along thex-direction. We calculate the mean ofi0 over one period, in the interval0y1, by

i0= 10

dy

q(y)

q(y)0

i0dx=3 +1 + c (8.20)


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Figure 1. Mean polar inertia i0/c versus

Consequently, the mean viscosity is obtained to be

= 0+ 3 +1 + , (8.21)

where, = (1+2)chas the same dimension of classical viscosity 0, representingthe viscosity of the polar inertia. Figure 1 displaysi0/c, for = 0.2, 0.4, 0.6 and0.8.

The periods of i11(x, y) at levels y = 0.2, 0.4, 0.6 and 0.8 are calculated andplotted againstx for a typical = 0.5 in Figure 2. These figures display the pathsfollowed by the centers of the discs. It is interesting to note that, starting from ahorizontal position, the long axis of a disc (having an inertia i11= 2c) moves along

its path, remaining tangential to the path described by the center of the disc. Thedisc inertiai11 grows, approachingi11= 2.288cat the peaks of the paths curves.On the descending side of the paths curves, i11 decreases reducing to the initialvalue i11 = 2c at the end of the period x = 2/q. The periods for one completerotation of the discs decrease with increasing y. This is as expected, since shearstress increases with y . The periods, at the levels y = 0.0, 0.2, 0.4, 0.6 0.8 and 1.0are given in Table 1.

Experimental observation of the period would lead to the determination of.Figure 3 illustrates a three-dimensional picture ofi11(x, y), for = 0.5.


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Figure 2. i11(x, y) at various levels ofy , versusx, for = 0.5


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Figure 3. Three-dimensional picture of polar inertiai11(x, y) as a function ofx and y , for= 0.5

y = 0.0 0.2 0.4 0.6 0.8 1.0= 0 15.08 6.6 3.35 1.41 0= 0.2 15.39 6.73 3.42 1.44 0= 0.4 16.45 7.2 3.65 1.54 0= 0.6 18.85 8.24 4.19 1.77 0= 0.8 25.13 10.99 5.58 2.36 0= * 0

* = indeterminant

Table 1. Periods ofi11

Conclusion

We have introduced a continuum theory for a viscous fluid carrying suspensions ofarbitrary shapes and inertia. The fluid is anisotropic, with its anisotropy evolvingwith the motion. We have constructed a set of constitutive equations that areframe-independent, complying with the second law of thermodynamics. We havegiven the field equations for several cases, including bar-like suspensions. We havedemonstrated the theory, by obtaining the solution of the channel flow, carryingelliptic-shaped suspensions.


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[2] A. S. Almusallam, R. G. Larson and M. J. Solomon, A constitutive model for the predictionof ellipsoidal droplet shapes and stresses in immiscible blends, J. of Rheology 44(2000),105583.

[3] A. S. Almusallam, R. G. Larson and M. J. Solomon, Anisotropy and breakup of extendeddroplets in immiscible blends, J. Non-Newtonian Fluid Mech. 113(2003), 2948.

[4] H. S. Lee and M. M. Denn, Rheology of viscoelastic emulsions with a liquid crystallinepolymer dispersed phase,J. Rheology43 (1999), 158398.

[5] G. G. Fuller, Optical Rheology of Complex Fluids, Oxford University Press, 1995.

[6] Phan-Thien, Nhan, Xi-Jun Fan and Boo Cheong Khoo, A new constitutive model formonodispersed suspensions of spheres at high concentrations,Reol. Acta38 (1999), 297304.

[7] S. Kim and S. J. Karrila, Microhydrodynamics: Principles and selected applications,Butterworth-Heinemann, Massachusetts 1991.

[8] A. C. Eringen, Continuum theory of dense rigid suspensions, Rheol. Acta 30 (1991), (seealso [7]).

[9] A. C. Eringen, Rigid suspensions in viscous fluid,Lett. Appl. Engng. Sci. 23 (1985), 49195.[10] A. C. Eringen, A continuum theory of rigid suspensions,Lett. Appl. Engng. Sci. 22 (1984),

137388.[11] A. C. Eringen, Microcontinuum Field Theories II, Fluent Media, Sect. 11.7 Springer, New

York 1998.[12] A. C. Eringen, Simple microfluids, Int. J. Engng. Sci. ]bf 2 (1964), 205.[13] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. R. Soc.

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25 (1981), 645.[15] A. C. Eringen, Mechanics of ContinuaSecond edition, Robert E. Krieger Publishing Co.,

Melbourne, Florida 1980.[16] G. K. Batchelor, The stress system in a suspension of force-free particles,J. Fluid Mech. 41

(1970), 54570.[17] J. L. Ericksen, Transversely isotropic fluids, Kolloid Z. 173 (1960), 117122.

A. Cemal EringenEmeritus Professor of Princeton University15 Reed Tail DriveLittleton CO 80126USA

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A Continuum Theory of Dense Suspensions Eringen2005