Jourml o/ Non - Newroman Flurd Zfechartrtr, I3 ( 1983) 63-78 Elsek~er Science Pubh5her.s B V . Amsterdam - Prmted In The Netherlands

63

SUSPENSION OF SOLID PARTICLES IN A NEWTONIAN FLUID

T LEVY

Deparfmenr de Marhemarlques. L’trrversrre de Rouen. “6130 Mont Sam Argnun (L A 2.Y C VRS) (France)

and

E SANCHEZ-PALENCIA

Laboratorre de Mecanrque Theorrque, LSwersrre Pans ti 1. 4, place Jussreu 7.5230 Parrs (France)

(Recewed December I4? 1982 In rewed form Februaq 16. 1983)

Summaq

We consider the motion of a Newtoman fluld contamIng sohd particles rn the case of high concentration (the partial volume of sohd IS comparable to that of fluld) by using a homogemzatlon technique associated with the small parameter E (the ratio of the particle length to the characterlstlc macroscopl- cal length) The hmlt behavlour as E -+ 0 IS that of fluld wrth amsotropy properties associated with a mlcrostructure The evolution equations for the hmlt flow and for the nucrostructure are glren

1. Introduction

It IS known that the homogemzatlon method [ I.21 IS an asymptotic two-scale method for studying physlcal processes In composite media Under the hypothesis of a perlodlc structure of the medium, It furnishes a rigorous deductive procedure for obtammg the macroscopic equations and constltu- tire laws of the hmrt phenomenon as the ratio E of the nucrostructure to the macrostructure tends to zero Generally speakmg the constltutl\e lags depend on the geometrlcal and physlcal properties of the n-ucrostructure (see for Instance [2]. chapters 6 to 8 for some examples of such a dependence) and euhblt quahtatlve properties bery different from those of the mlcro- SCOPIC structure (hnear and non-linear Darcy’s lau for porous media. vlscoe- lastlc bulk properties m the theory of mixtures and so on)

0377-0257/83/$03 00 < 1983 Else\lcr Science Pubhshers B V

64

The method IS based on a two-scale procedure with the standard macro- scop~c variable x and the rrucroscoplc one y, any solution IS supposed to have an expansion m functions of x and J, perlodlc with respect to J’ (of the same penod as the structure, m order to express the Influence of the perlodlc structure) taken at the pomt J = X/E The result 1s a “locally penodlc” function, takmg almost the same values m nelghbounng periods but Lery dlffcrent values at distant periods The asymptottc expansion IS cart-led out with the classlcal formula

d 3 Ia -=- dx, ~Y,+;F

and consequently the gradlent of the expansion IS not obtamed m a stralght- forward manner Moreover. a rigorous proof of the asymptotic behavlour obtamed m this way has been gl\en by functlonal analysis methods. at least for some model problems It IS to be notlced that shght modlflcatlons of the method [I, sect I 61 allow us to deal with geometric structures H hlch are only locally penodlc (not necessarily perlodlc)

The problems m fluld mechamcs worked out to date with these methods concern geometric patterns which are given Independently of the solution of the problem either flow m porous. rlgld media. or small vlbratlon problems. where the perturbed and unperturbed posmons are supposed to be comcl- dent (see for Instance [2]. chapters 7 and 8) Very different sltuatlons appear If we consider large deformations. the geometric pattern. and consequently the local period of the structure. depend on the actual solution consldered (at least when workmg with Euler’s variables) In fact the local period evohes m time and depends on both Its conflguratlon at time I = 0 and on the flow for subsequent time

In this paper we study a problem m this context We consider a viscous Newtoman fluid contammg sohd particles m suspension. the particles at the lmtlal time are consldered to be perlodlcally (or locally perlodlcally) dlstnb- uted It IS not hard to see (sect 6) that the locally perlodlc structure e\ol\es m tmie by keeping the locally perlodlc character. and consequentI> the evolution of the system 1s consistent with a two-scale method If the imposed perrod m J IS the same as that of the structure ur the wme tt~.strnt~~ It IS to be emphasized that the process E JO describes the smallness of the particles and their mutual distances with respect to a macroscopic length. but the (not small) concentration IS Independent of E As for perlodlclty, ue consider It as a mathematical hypothesis (or a very particular physlcal sltuatlon) allowmg us to fmd In a rigorous, deductIke way the equations descrrbmg the flow

As a result the hrnlt homogemzed flow (I e the first term of the asymp- totlc ercpanslon) satlsfles a vlscoslty Idw which IS m general anIsotropic and depends on the local mIcrostructure The evolution equations for the micro-

65

structure are gtben These equations may be consldered wlthm the frame-

work of the constltutlce equations proposed by Hmch and Lea1 [6] In the

case of slow motion. the e\olutlon of the mlcrostructure IS Irrelevant for

fmrte time and we recover a homogrmzed \lscoslty law analogous to that

obtamed In Stokes flow (for a heterogeneous fluid Instead of a suspension) In [I. sect I IO] Moreover, the non-hnear terms In the expression of the

acceleration of the hmlt flow also mvolie coefflclents depending on the

microstructure It IS to be noticed that the results obtamed bq homogemzatlon techniques

are highly dependent on the presence of other small parameters apart from

the P one (associated alth the small geometric scale) In thrs paper. we consider the other parameters (In particular rhe Reynolds number of the

macroscopic flow and the concentration) to be of order unity In thrs

connection we point out that for a model problem, the andlogy of Emstem’s

Llscoslty formula for dilute suspensions may be obtained as clsymptotlc

behavlour of the homogenrzed law for concentration tending to zero [7. sect

4

2. Formulation of the problem

We consider the three-dlmenslonal flow of an mcompresslble. VISCOUS

fluid contammg t-lgld particles S We assume that. at each Instant. the

geometry of the suspension IS locally penodlc. the period of the structure contains one ngd particle S and IS small compared to the size L of the

region m which the system IS to be studied (see Fig I for a two dlmenslondl pattern)

Note that at this stage. each period IS not completely defined. It ma) be

0

Fig I Tmo-drmrnslonal pattern

XI

66

shlfted by a translation, but for the sake of slmphclty. we shall consider each particle S entirely contamed m Its period

Moreover. the small penods are associated with a small parameter E .vhlch characterizes the size of the penod as well as the size of the particles

At a point of the fluid part of the medium. the velocity V’(r. x) and the pressure P’( r, X) satisfy the equations

dlv V’= 0. (1)

with

a f = -P’6,,+2@,,(V’). D,,(V)=; z+% , I/ i I / ax,

(2)

(3)

where p0 IS the constant fluid density and F the volumlc density of forces. ever-where defined

In each particle S. V’( I. X) IS a rlgld solid beloclty field. so

D,,(V)=O. (4)

and on the boundary i3S of S the velocity IS contmuous because of the no-slip condmon

For each particle S. submitted to the density of forces F and to the fluld

action. the dynamic laws Imply

(5)

J &(“-%)A Tdu=/S( x-x,)r\Fdt - J as(X-Xo) A a,E/tl,e, da. (6)

s

where pS denotes the solid density, xc IS the coordinate of the mass centre G of the particle considered. and n IS the unit normal (outer to the fluId)

According to the general features of the homogemzatlon method. we

Introduce a new mlcroscoplc variable

J’= (X-.X,)/E

In order to describe vanatlons within a period. we search for an expansion of

the form

V(r, X) = V”(r. x._V)+EV’(f,X.~)+ .

P’(r.x)=PO(t.x.~)+EP’(I.X.S’)+ .

with the functions V’. P'. I = 0. I. , periodic with respect to the J’ variable

67

We can consider the $ varlahle defmed m a basic period Y. homothetlc to an actual penod of the medium with the ratlo E-‘. then a function perlodlc with respect to the E vanable takes equal values on opposite faces of the basic penod Y

Fig 2 Permd pattern

We wnte expansions of the equations of motion and of boundary condl- tlons using the asymptotic expansions of V’ and P’ and keeping In mmd that. ahen applied to a function +(x. J) with y = (x - Q)/E. the operator a/ax, becomes a/ax, + (l/~) a/a,,, thus we obtain, at the different orders with respect to E. problems to be solved consecutl\ely

3. Asymptotic expansions of the solution

3 I Fmt term of the expansron

From (1) at order EC’ and (2) at order pm2 we can wnte the followlng equations for the fluld

dl\, ~ ,I” = 0.

g+,,,J~“)l =O, /

where the Index (_y) (or (x)) denotes the variable, the other one. x (or L*). being constant. namely

D ,,(,) (V”)=~(ar”/a~,+av,O,‘a~,)

Thus we obtam a dlfferentlal problem with respect to the y vanable, x being a parameter, III the fluid part YF. of Y. and we look for a solution V”( I. X. y). Y-perlodlc in J’ From (4) at order E- ‘. V” must satisfy In Ys

Moreover. V” IS contmuous on the boundary r of Y, because of the no-shp

68

condltlon and satlsfles, from (5) and (6) at orders E and 6’ respectively

J r D,,,,,(V")n,du,=O.

/ & Dvw nrkr I-

(VO)n,y,da, =0

We easily obtam that, m the sense of dlstrlbutlons. on 1

D,,, , ,( v”) = 0.

and, on account of the Y-penodlclty.

VO(r. x,y)= V”(r. X). (7)

whxh 1s a classIcal result of most problems m homogemzatlon the first order does not depend on the nucroscoplc variable It IS clear that this does

not mean that the first term of the expansion IS the same as the one correspondmg to the fluld wlthout particles The equations satlsfled depend on the presence of the particles (see at the end of sect 6)

3 Z Second lerm of rhe evpansron

The relation

d~v,,,V”+d~vJ” =0 (8)

IS satlsfled m the fluid (from ( I) at order E’ ) and In the sohds ( from (4) at

order EO). so we can write

J ,PY V) I’” + dl\; \ ,V’ ) d?s = 0

on Y Thus, takmg Into account the perlodlclty of V’ the last term vamshes. and we obtam the macroscopic (I e independent of ~9) relation

dlv I”( t. X) = 0. (9)

which expresses the fact that the hmlt flow Y” IS mcompresslble Thus (8) becomes

dlv, , ,V’ = 0 (IO)

In Y We also have m );, from (2) with (3) at order E-’

30,; -=O, a V,

(11)

u ‘= 11 -P”6,,+2~[D,,(F’o)+D ,,,,, (I”)] (12)

69

On the other hand. m Fs, from (4) at order 8

&w”)+D ,,,,, (V’)=O (13)

Moreover. on r. V’ IS continuous and. from (5) and (6) at orders E’ and E’ respectwely

J a,: II, e, da, = 0.

r (14

J y A a,“, 11, e, da, = 0

r

At the moment. we are searchmg for a Y-penodlc solution V’. P” of eqns (IO)-( 15) We have to solve a dlfferentral problem alth respect to the J’

variable m the basic period 1. x bemg a parameter It IS clear that the solution for V’( ~9) ~111 be deternuned up to an addltwe rlgld sohd. I-pen- odlc velocity field A(x). and that ( 10) lmphes that the stress tensor a,‘, In 1’: ~‘111 be determined up to an addltwe pressure tensor [-T”(X) S,,] The solution depends on the D,,( Y”)

In order to stud) this problem. we gl\e a \anatlonal formulation of (lo)-( 15) We first define the Hllbert space

w,,, ( 1 ) = ( WV. w E [ H’( 1 )] ‘. I’-perrodlc. dn w = 0 m 1 F)

and the closed convey set

K=(c.rE It,,,(I ).0,,(u)= -D,,(V”).r.j= 1.2.3.m Ys)

Then Y’ E K. by multrplyng (I I) by a test function DE K. Integrating III

YF, taking Into account the perlodlclty of the functions of the-r Larlable and the equations (12). (14) and (15). we arrive at the relation

~/JD ,,,, @‘)(D,,(t’)--D ,,,,, (v’)}d~=O ‘dt~k’ (16)

Conversely If V’( I. x. J’) satlsfles (16). V’ IS a Y-perlodlc vector which satisfies problem (IO)-( 15) m the dIstrIbutIonal sense, m particular there exists a Y-penodlc function PO( I. X. y) defined III Y, up to an additive constant 7~“( I. X) [2. chap 71, satlsfyng rn the dlstrlbutlonal sense

2p+JD,,,,,W)]=~. I

(17)

le (Il)and(l2) In order to express the dependence of the solutron on D,,( V”). we define

70

the vectors P’J with components

P” = b L J 6 IA ’

then

We consider the set of adnusslhle solutmns (analogous to k’)

&(P”) = ($. #E U;,,( 1’). DA,($) = D,,( P”). X. I. = I. 2. 3. m Ys)

and the correspondmg solutlons x’/ of

\xfJ E f&( P’J)

\a($‘.+-x’J)=O v+E U,,(P’J). (18)

where

We note that x” = xJ’ We shall prove m the followmg the existence and umqueness of the x”.

solution of (18). up to an addltlve constant (with respect toy) vector. hence the exclstence and uniqueness. up to an addltl\e constant vector of V’.

solution of ( 16)

V’ = -Q1( V”)x’J( y)+A(x)

The quotient space ‘I= Wd,,( Y)/C. where C IS the set of constant vectors m a Hllbert space when equipped with the scalar product

In order to sohe (18). we use the theory of varIatIonal InequalItIes (In particular theorem 8 2 of [3. sect 2 8 21) Nom ‘X,, = Lid( P”)/C IS a closed convex unbounded set of q Let ‘1’ be the dual of 5 and A E L’( ‘X,,. ? ‘) the operator associated with a A 1s monotonous and coercive. for

in cP” E A,,.

(A(@).@-@O) II@11 ‘,

-+ cc lfpqI;-) m.@di,,

71

Then the solution ~‘1 of (18) exists. and as

(A(~‘)-A(~*),~‘-~*)>OIf~‘#0*.Q)‘.~*E’3L ‘J ’

x’J 1s umque In :X.,, Thus Ne have found, for I”( t, x). the asymptotic Y-penodlc expansion

P(r,x)= v”(f..Y)+E[--,J(VO)X’J(4’)+A(I.x)]+o(E) (19)

4. Extension of the oEl and averaged stresses

As we have seen. P”( t. X. y) satlsfymg (17) IS well defined up to an

additive function ~‘(1. x). we shall consider rn the sequel

I

lyl 1, / P’(x.J’) dy=O

Thus, In the fluid. the first term of the expansion of the stre%s tensor u” IS

known

0 - OhI - -?rO(X) s,,- P~~x.P~s,,+2~[D,,~~“~-D,,~~‘~~D,,~x”~] (20)

In order to define the averaged stresses

we see from (20) that rt IS only defined In the fluld part ( J* E IF) The stress

tensor IS not defined on the sohd part Y,, nevertheless. taking mto account

that

aa,‘),/a 19, = 0 in YF

and

J a,Yn,e,da, =0 I

/ I J’ A up/ II/~, do, = 0.

(cf ( I I). ( 14) and (15)). then we can prove [4] the existence of a symmetrrc

extension 6” of u” defined all over Y satlsfyng

i36,y/i3); = 0 In YS,

S,:n,=u,:n,on r

Thus we define

72

In I’. the 6:, are defmed. as a&. up to [ -n’(r, x) S,,] Furthermore rn YS, extstence of 6’ IS known but 6’ IS not urnquely determmed Moreoser, it IS to be noticed that the averaged value 6:, Just defmed IS Independent of the extenston (see later expresston (25))

FIN 3 Control \&me D

We obtam the relation between the macroscoplcal vauables by the flux method [5] We apply the momentum conservatton law at the mixture contamed m an arbitrary volume D. wth dunenslons 0( L). consrstmg of a great discrete number of perrods

/

dV, p-du+

D ar s p V, D

zdu =/,Fk do +i,+,do

At the ftrst order PO, u hen E + 0. we deduce contrrbutlon of each term. usmg the asymptotrc expanstons of V’. up and the pre\~ous results

J W; -dd, = np a!

where p denotes the mean densrty

As for the non-lmear term, on account of (19). we have

av; P ‘/’ ax

--dd= ,jV;$ J dL + c E’ v/o I D / permds c D

J avp = P V’OF do -

D J D &i/l D,,(V”) V,‘dc+o(l).

I

with the coefflclents

73

(21)

Also

J F,dc= &du+o(l), D J D

As for the vlscoslty term.

J u;, n, da = /

af,n,da+o(l) an C3D

= ~,b$qdo+o(l)=/ a’,O,n,da+o(l). dD

where the last equahty arises. rn a classlcal manner [2.5]. from the diver- gence-free property of the stress tensor 6’ m the $ variable

Thus we obtain. smce D IS arbitrary. at the first order when e + 0. the following macroscopic law

i3Vp

p ar ‘---+p v,O ax =&,, k;OD,,(V”)=$fj+pk.

/ / (22)

which IS the homogemzed form of the momentum equation In addition to

the homogenized stresses df,. It Involves the new term wrth the coefflclents

P k,,, which IS a correction of the non-hnear term The pk,,, depends on the

mlcrostructure of the medium. and from (21) It can also be written

P,,,,=j$J (PCP,)g$?. \

ph,,, = Ph ,rl Since x’/ = xJ’

and

P hlJ/ + &,,A = (‘A, 6, +b%J$+,/ (Ps-pddr

‘5

Equatlotls (9) and (22) are t\*‘o macroscoprc IGUT rm?olLlng rhe macroscoprcal quanrrrres V”( 1. x) and a’( I. x) We have ro find the relalrotl herw*een the hulh stress tensor 6” and the bulb deformatron of rhe medrum ( ree later relation

(25))

74

5. Homogenized constitutive equations

From Its defuutlon

and @, IS. as CT:, and c?:,. defmed up to the addltwe function [ - a’( t. x) a,,,]

Up to the pressure ,rr’( t. x). m YF, ut, IS gown by (20) with /,,P’ d v = 0, so

J u” d),= - hI . j- P”6,,d~+2pD,,(Vo)/ D,,(P”-~“)dr,

‘F )F ‘F

= 2~ D,,( Y”)I,FDP,( P'* - x") D,,(P") d I

Up to the pressure T’( t. x),

smce m Ys. DP,( xh’) = DP,( PA’) = i( +,A a,, + $,,6,, ).

J 6;, d>s = - c&xi’tlydu,

1, J $‘qxyn,do, = -

r J r

(23)

The last equahty arlses from the stress contmulty on r of the extension 6” (cf sect 4) Takmg mto account the I’-perlodlclty ot the functions m\olved. we may H nte r u al Instead of r m the last Integral

= J )F P” SPY D,,( x”) d \ - 2/1 D,,( I"')

X J D,,( P” - x” ) D,,( xh’ ) d 1. (24) )F

and the fu-st term vamshes because of X”E Md,,( 1 ) Collecting (23) and

(24), we fmd. up to [ - n”( r. X) a,,]

so the homogemzed constltutwe relation IS

-0 - ukl- -p’(t.-+~,+~,,,,~,,(~‘). (25)

75

with

a P” - x”) D,,,( PA’ - x”) d 1’

It generalizes the constltutlve law obtamed m [l. sect I IO 21. for the homogemzatlon of Stokes equations, to Include the problem of a suspension of ngd partlcles m a VISCOUS fluld The coefflcrents a,,,, depend on the nucrostructure. and It IS easy to check that

a l/hi = aJ,,, = a l//l, = ah,,J’

“,JhI D,J D,,2m, D,J D,, “,J= O,,? a,‘o

6. Deformation of the structure

The macroscopic relations (9). (22), (2 1) and (25) for the unknowns V”( I, x), r”( t. X) contam the homogemzed coefflclents PA,,1 and CI,/~, de- pendmg on the mlcrostructure of the medium by the Integration domam and bq the local solutions x’/ of problem (18) At any tnne these coefflcrents are calculated usmg the current basic period at that time

At the lmtlal time ro. the structure of the medium IS perlodlc Hlth period ~1 (or locally penodlc. I e Y dependmg on X) and It IS driven by the flop ulth the velocity V’ gILen at any time by (19)

In order to descnbe the deformation of the structure we must Identify appropriate variables s. these carlables depend on x and r We are Interested here m the three-dImensIonal case, so He can choose for s three vectors d’. d’. d3 characterlzmg a penod (for example d’ = AB, d2 = AD. ). one vector A P, Jommg a summit of the period to d determmed pomt of the particle S. and three angles for the rotation of the particle S

The deformation rate of the nucrostructure can be evaluated. takmg mto account the expansion (19) of the veloclt! field

dd’ -=y= V(B)- V’(A).

dr tdem ford’. d3.

F = V’(P) - Vf( A).

S(S)APM= V(M)- V(P).

where P, M E S. O(S) IS the rotation rate of S and d/dr IS the “particulate denvatlve” I e dnven by the flow. we shall see later that. at the first order. vanables 5 are m fact driven by V”( f. x) We obtain

76

FIN 4 Deformation of the penod

dd’ -= d/

P(B)- VO(A)+E~[--,,(YO)XI~+A]B

- [-D,,(YO)$‘+A] 4; +0(E)

tlVO =Td! +0(e).

I

because xB - x4 = O(E) and x”( yA) = x”( ys) by I’-perrodlcltl To the first

order m E, we have

ddl‘ i3V” -=a,.d;= ‘iI” dh. k= I. 2. 3 dr I

(26)

In an analogous way. we ohtam at O(p)

(27)

O(S)APM= TV0 PM-FD,,(v”)[Xf~(~.,r)-Xf’(J.P)]

We know that x” E Uud( P”). so on the sohd particle S the dlfference x’J( y) - P”( J) IS. wth respect to the J’ variable. a rtgld solld velocity

0~” + p” A J’. then we obtain

C?(S)=! rot V”-D,,(V’)p” (28)

Relattons (26)-(28) are evolution equations \% htch relate the deformatton of the mxrostructure to the bulk flow (the celoclty of whrch IS V’-‘(t. x)) They are of the form

dS -=n(S) TV’). dr

(29)

and consequently the deformation rate of the mlcrostructure depends Iln- early on Y V” (and on the mlcrostructure Itself)

77

The expresslon (29) that we deduced from (19). that IS to sal from the study of the motion at a mlcroscoplc level. IS of the form predlcted bq Hmch and Lea1 [6] using a phenomenologlcal approach of the problem The mam result of ths study IS that the unknowns ,rr’( 1. A-). V”( I. X) satlsf? (9) (22). wluch Involve the coefflclents PA,,, and a,,,, g :ven by (21). (25) as functions of the nucrostructure and the parameters defining this one evolve m time accordmg to (29)

7. Concluding remarks

When V”( t. X) IS small, because the boundary velocltles for the suspen- sion are small or because the fluid wscoslty IS large. eqn (2) becomes the stationary Stokes equation The macroscopic eqn (22) becomes

(30)

and the evolution of the mlcrostructure given by (29) IS neghglble at the first order Equations of the bulk motion are thus (9). (30). (25) Bulk motion IS

steady and the coefflclents of the constltutlve law (25) do not depend at the fu-st order on the medium deformation Thus result IS slmllar to that obtained m [I] for the homogemzatlon of the Stokes equations. m the case of slog motion of a nuxture of viscous flulds Here. for a suspenston of sohd particles m a viscous fluid. we have the macroscopic beha\ lor of a New to-

man non-Isotropic fluid When V”( t. X) IS not small. the deformation of the medium IS not

neghglble. the equations of the bulk motion are (9). (22). (25). (29) In particular the constltutwe law IS

-0 ohI= -71.0(t. x) Sk,+ a,,,,(l) D,,( VO)

and the mlcrostructure variables 5 depend on the bulk motion b> (29) The homogemzed medium IS an mcompresslble simple fluid

We emphasize that. from this study. It IS clear that the non-Newtonian behavlour of the bulk medium arlses from the actlon of the bulk flop on the mlcrostructure

References

I A Bensoussan J L Lions and G Papamcolaou Asymptotic Analws for Perrodlc Struc-

tures North-Holland Pubhshmg Company Amsterdam 1978

2 E Sanchez-Palencra Non-Homogeneous Media and Vlbratlon Theon Spnnger i’erlag

Berlin 1980

78

3 J L Lions Quelques Methodes de Resolution des Problemes aux Lmutes Non Lmeawes.

Dunod Paw 1969 4 F Lene. In S Nemat-Nasser (Ed ). Vanatlonal Methods In the Mechanics of Sohds.

Pergamon Prebs Oxford 1980 p 45 5 T LeL> Int J Eng Scr . 17 (1979) 1005 6 E J Hmch and L G Lea1 J FluId Mech 71 (1975) 481 7 E Sanchez-Palencla In Verhulst (Ed ) Asymptotic Analqbis II (Lecture Notes In Maths.

To appear 1983)