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ANALELE STIINTIFICE ALE UNIVERSITATII AL.I.CUZA IASITomul XLVI, s.I a, Matematica, 2000, f.1.

A THEORY OF NONSIMPLE MICROSTRETCH FLUIDS

BY

D. IESAN

and R. QUINTANILLA

Abstract. The paper is concerned with a theory of microstretch fluids in which

the secondorder velocity gradient is added to the classic set of independent constitutivevariables. First, the field equations and a uniqueness result are established. Then a

linearized theory is derived. A representation of Galerkin type for the solutions to the

field equations is given. The effect of concentrated body loads is also studied.

1. Introduction. The theory of microfluids was introduced by ERIN-GEN [1] in order to study fluids whose microelements can deform indepen-dently from their centroidal motions. The theory has been the subject ofan enormous number of investigations. Various reviews of the subject havebeen presented by ERINGEN [2], [3], ERINGEN and KAFADAR [4], ARIMAN,SILVESTER and TURK [5] and BRULIN [6]. In [7], ERINGEN has introducedthe theory of microstretch fluids where the micromotions consist of the in-trinsic rotations and stretch. Recently,ERINGEN[8] has extended this theoryto include heat conduction and dependence of constitutive equations on themicroinertia tensor. The intended applications of the theory are to suspen-sions in viscous fluids, bubbly fluids, blood and liquid crystals.

In various papers [6], [9], [10], the authors have advocated the proposalthat a consistent grade level for a micropolar theory, containing the velocityvector v and the angular velocity vector as independent constitutive va-riables, should contain field derivatives of one order higher in vk than ink.The arguments for this are drawn from general dimensional considerationsas well as from the analysis of special micro-model cases.

In the first part of this paper we present a theory of microstretch fluids

which includes the velocity gradients of second order as independent consti-tutive variables. The theory is a generalization of the theory presented by

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172 D. IESAN and R. QUINTANILLA 2

ERINGENin [8]. In the context of elastic continua,TOUPIN[11],MINDLIN[12]and GREEN and RIVLIN [13] have established a theory of nonsimple mediawhich is characterized by the inclusion of higher gradients of displacement inthe basic postulates. The theory of nonsimple fluids in which velocity gra-

dients of the first and second order are present in the constitutive equationshas been developed by BLEUSTEINand GREEN [14].In the second part of the paper we present a uniqueness theorem within

the framework of the theory of isothermal incompressible fluids which occupybounded domains. Then we derive a linear theory of nonheatconductingcompressible fluids appropriate to small departures from an equilibriumstate. We establish a representation of Galerkin type for the solutions tothe field equations. Finally, we use the Galerkin representation to study theproblem of concentrated body loads in the case of steady vibrations.

2. Basic Equations. Throughout this paper we consider a contin-

uum that in the present configuration at time t occupies a regular regionB(t) of Euclidean threedimensional space. We denote by B the boundaryofB and designate by n the outward unit normal ofB . We assume thatB is a smooth surface. Letters in boldface stand for tensors of an order

p 1, and ifu has the orderp, we writeuij...s (psubscripts) for the rectan-gular Cartesian components ofu. We refer the motion of the body to a fixedsystem of rectangular Cartesian axesOxi (i= 1, 2, 3). We shall employ theEuler representation and usual summation and differentiation conventions:Latin subscripts are confined to the range (1,2,3); summation over repeatedsubscripts is implied and subscripts preceded by a comma denote partial dif-ferentiation with respect to the corresponding spatial coordinate. In all thatfollows, we use a superposed dot or d/dt to denote the material derivative.

The place occupied by the material point X in the current configuration isx.We present a theory of nonsimple microstretch fluids. The local form

of the law of conservation of mass is

(2.1)

t+ div (v) = 0 ,

where is the mass density at time t, and v is the velocity vector field.We restrict our attention to microisotropic fluids. In this case the law ofconservation of microinertia is (cf. [8])

(2.2) jt

+j,svs 2j= 0 ,

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3 A THEORY OF NONSIMPLE MICROSTRETCH FLUIDS 173

wherej is the microinertia and is the microstretch velocity.We postulate an energy balance in the form

V ( + vivi+ ii+ + jivi,j) dv==V

(Fivi+ Mii+ L+ Fjivi,j+ S) dv+

+

V

(tivi+ mii+ h+ jivi,j+ q) da ,

for every partV ofB and every time. Here, is the internal energy density,i is the microrotation rate, Fi is the body force density, Mi is the bodycouple density,L is the generalized body load, Fij is the dipolar body force,Sis the heat source density, ti is the stress vector, mi is the couple stressvector, h is the microstress, ji is the dipolar surface force and q is theheat flux. Moreover, i is the microrotation spin inertia per unit mass (seeERINGEN[8])

(2.4) i = j(i+ 2i) ,

is the microstretch spin inertia per unit mass ([8])

(2.5) =3

2j(+ 2

2

3ii) ,

and ij is the dipolar spin inertia per unit mass (see BLEUSTEINand GREEN[14])

(2.6) ij =d2 [(vj),i vj,kvk,i],

whered is a given constant.We consider a second motion which differs from the given motion of

the continuum by a constant rigid body translational velocity. In this case viis replaced byvi+ ai, whereai is an arbitrary constant. Under this transfor-mation the functions ,, i, , i, , ij, vi,j, Fi, Mi, L,Fij, S , ti, mi, h , jiand qare not affected. Subtracting the energy balance (2.3) from that forthe new motion, we obtain

V

vidv V

Fidv V

tida ai = 0 ,

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for all arbitrary constants ai. The quantities in the square brackets areindependent ofai so that we get

(2.7) V vidv= V Fidv+ V tida .Let n be the outward unit normal ofV. Using the well-known method,from (2.7) we obtain

(2/8) ti= tjinj,

and

(2.9) tji,j+ Fi = vi ,

where tij is the stress tensor. Taking into account (2.8) and (2.9), the

equation (2.3) reduces to

(2.10)

V

( + ii+ + jivi,j) dv=

=

V

[tjivi,j+ (Mii+ L+ Fjivi,j+ S)]dv+

+

V

(mii+ h+ jivi,j+ q) da .

We now assume that V is a tetrahedral element bounded by a plane witharbitrary unit normal ni, and by planes through the point x, parallel tothe coordinate planes. With an argument similar to that used in obtaining

(2.3), from (2.10) we find that

(2.11) (mi mjinj)i+ (h hjnj)+ (ji rjinr)vi,j+ q qjnj = 0 ,

wheremji is the couple stress tensor,hj is the microstress vector, rji is thehyperstress tensor and qi is the heat flux vector. With the help of (2.11),the equation (2.10) reduces to

(2.12) [ + (i Mi)i+ ( L)+ (ji Fji)vi,j ] ==tjivi,j+ S+ mji,ji+ hj,j+ rji,rvi,j++qj,j+ mjii,j+ hj,j+ rjivi,jr.

Let us now consider a motion of the continua which differs from the givenmotion only by a superposed uniform rigid body angular velocity, the body

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occupying the same position at time t. Velocity in the new motion is givenby vi = vi +irjbrxj where irs is the alternating symbol and bi are ar-bitrary constants. Moreover, i is replaced by i = i +bi, and vi,j byvi,j = vi,j+irjbr. Under this transformation the functions,, , tij, S,

mij, hj , rji and qi are not affected. But the body loads Mi, L and Fijmust be accommodated by corresponding changes in rotatory accelerations(cf. ERINGEN[15]) so that

i Mi = i Mi , L= L , ji Fji = ji Fji .

Subtracting the energy equation (2.12) from the corresponding energy equa-tion for the new motion, we obtain

[(i Mi) + irs(rs Frs) irstrs mji,j imnrmn,r] bi = 0 ,

for all arbitrary constants bi. We conclude that

(2.13) mji,j+ irsrs+ Mi = i ,

where we have used the notation

(2.14) rs = trs+ krs,k (rs Frs) .

With the help of (2.13) and (2.14), the equation (2.12) becomes

(2.15) = ijdij+ mijij+ ijkijk + hi,i+ g+ qi,i+ S ,

where

(2.16) dij =vj,i+ jirr, ij =j,i , ijk =vk,ij,

and

(2.17) g= hi,i+ L .

The entropy production inequality isV

dv

V

1

S dv

V

1

q da 0 ,

where is the entropy density and is the absolute temperature which isassumed to be always positive. If we apply the entropy production inequality

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to an arbitrary tetrahedron bounded by coordinate planes at x and a planewith unit normal n, then we obtain (cf. GREEN and STEEL[16])

(2.18) q= qini .

The local form of the entropy production inequality is

(2.19) S qi,i+1

qi,i 0 .

By eliminating Sbetween (2.15) and (2.19) we obtain

(2.20) ijdij+ mijij+ ijkijk + hi,i+ g +1

qi,i 0 ,

where the Helmholtz free energy is defined by

= .

Let us introduce the notation

A= (,j,dij , ij , ijk , , ,i, , ,i) .

A heatconducting fluid is a medium having the constitutive equations

(2.21)

=(A) , ij =ij(A) , mij =mij(A) , ijk =ijk(A) ,hi =

hi(A) , g=

g(A) , =

(A) , qi =

qi(A) ,

mi =mi(mrs, nj) , h=h(hk, nj) , ji =ji(pqr, ns) ,subject to the axioms of admissibility and objectivity. The axiom of objec-tivity requires that the constitutive functionals must be hemitropic. In whatfollows we restrict our attention to isotropic fluids. It follows from (2.20)and (2.21) that

(2.22) =(,j,) , =

,

and

(2.23) ijdij+mijij+ijkijk +hi,i+g ddt j djdt + 1 qi,i 0 .

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In view of (2.1),d

dt = vi,i ,

dj

dt= 2j ,

so that (2.23) becomes

(2.24) (ij+ ij)dij+ mijij+ ijkijk + hi,i+ (g+p)+1

qi,i 0 ,

where

(2.25) ij =2

ij, p= 2j

j .

We introduce the functions sij andG by

(2.26) ij = ij+ sij, g= p + G .

The inequality (2.24) reduces to

(2.27) sijdij+ mijij+ ijkijk + hi,i+ G+1

qi,i 0 .

If we carry (2.25) and (2.26) into (2.15) then we obtain

(2.28) = sijdij+ mijij+ ijkijk + hi,i+ G+ qi,i+ S .

Following [8], [14] we assume that sij, mij , ijk , hi, G and qi arenonlinear functions of, j and , and linear functions in the variables dij ,ij

, ijk

, ,i

, and,i

. In view of (2.28) we obtain

(2.29)

sij = (drr + 0)ij+ ( + )dij+ dji ,mij =ijk(0,k+ 0,k) + rrij+ ji + ij,

ijk =1

21(rrijk + 2krrij+ rrjik)+

+2(irrjk + jrrik) + 23rrkij+ 24ijk++5(kji + kij) + 1ij,k+ 2(ik,j+ jk,i)++1ij,k+ 2(ik,j+ jk,i) ,

hi =a0,i+ a1irsrs+ a2,i+ a3rri + a4irr,G =b0+ b1drr,qi =k,i+ k1,i+ k2irsrs+ k3rri + k4irr,

where the constitutive coefficients depend on , j and .

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The inequality (2.27) is a restriction on the constitutive coefficients.Using standard techniques of the theory of quadratic forms we can derivethe necessary and sufficient conditions for (2.29) to be satisfied for all inde-pendent constitutive variables (see [8], [14]).

In view of (2.18), the equation (2.11) becomes

(2.30) (mi mjinj)i+ (h hjnj)+ (ji rjinr)vi,j = 0 .

For a given motion,, andvi,j in (2.30) may be chosen arbitrarily so that,on the basis of the constitutive equations (2.21), (2.24), (2.25), (2.26) and(2.29) we find that

(2.29) mi = mjinj, h=hjnj, ji = rjinr.

The total rate of the work over the surface B is given by

(2.30) W = B(tivi+ mii+ jivi,j+ h)da==

B

(tjivi+ mjii+ jrsvs,r+ hj)njda .

Following TOUPIN[11] and MINDLIN[12], we can write

(2.33) W =

B

(Pivi+ RiDvi+ mii+ h)da ,

where

(2.34)Pi = [ji kji,k+ (ji Fji)]nj 2njDrjri

nrnsDrsi+ (brs bmmnrns)rsi ,Ri = rsinrns .

Here Di is the surface gradient, bij is the second fundamental form of thesurface B andDf=f,ini.

3. Field Equations. In view of (2.14) the equations of motion (2.9)become

(3.1) ji,j kji,kj+ Fi (Fji),j =vi (ji),j.

By using (2.4) the equations (2.13) can be written in the form

(3.2) mji,j+ irsrs+ Mi = j(i+ 2i) .

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From (2.5) and (2.17) we obtain the equation

(3.3) hi,i g+ L=3

2j(+ 2

2

3ii) .

The basic equations of the theory consist of the equations (2.1), (2.2), (3.1)

(3.3), (2.28), (2.22), (2.25), (2.26), (2.29) and (2.16) for the determinationof the functions, vi, j ,i, and .

In what follows we assume that the constitutive coefficients from (2.29)are constants. By substituting (2.16), (2.29), (2.26) into (3.1)(3.3) we ob-tain the field equations in the form

(3.4)

t+ (vi),i= 0 ,

j

t+j,ivi 2j = 0 ,

2(1 l2)vk+ [1 2 (1l1 2l2)]vj,jk++kjss,j (1+ 22),k+ [0 (1+ 22)],k+pk++Fk (Fjk),j =vk (jk),j,

i+ ( + )s,si+ irsvs,r 2i+ Mi= j(i+ 2i) ,a0+ a2 [b1 (a3+ a4)]vs,s

b0+p + L=3

2j(+ 2

2

3ii) ,

k+ k1+ (k3+ k4)vj,j+ sij(vj,i+ jikk)++mijj,i+ ijkvk,ij+ hi,i+ G = S ,

where is the Laplacian and

(3.5)

1= + 2 + , 2= + , l2= 2(3+ 4)12 ,

l1 = 211

5i=1

i , pk = jk,j.

For an incompressible fluid we have(3.6) vi,i= 0 ,

and the constitutive equations for non-heat-conducting fluids become

(3.7)

ij = ij+ sij,sij =0ij+ ( + )dij+ dji ,

mij =ijk0,k+ rrij+ ji + ij,ijk = ijk jik+pijk,

pijk =1

21(rrijk + rrjik) + 23rrkij+ 24ijk+

+5(kji + kij) + 1ij,k+ 2(ik,j+ jk,i) ,hi =a0,i+ a1irsrs+ a3rri ,

g =

+ G ,G =b0 , qi = 0 ,

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where

(3.9) = (j, ) , = 20j

j ,

0 is the constant density of incompressible fluid, and and i are arbi-trary functions to be determined in the course of solution of each particularproblem.

4. A Uniqueness Result. In what follows we assume that themotion takes place under isothermal conditions. We consider incompressiblefluids with the constant density 0onB(0, t1) and assume that = cj/20,wherec is a constant. The constitutive equations (3.7) lead to

(4.1)ij = ij+ sij,

ijk = ijk jik+pijk,

g = cj+ G ,

wheresij , pijk and G are given by (3.7). The ClausiusDuhem inequalityreduces to

(4.2) sijdij+ mijij+ pijkijk + hi,i+ G 0 .

In what follows we use the approximation (cf. ERINGEN [7])

(4.3) i = j i , =3

2j .

The basic equations are given by

(4.4)

vi,i= 0 ,j

t+j,ivi 2j= 0 ,

2(1 l2)vi+ ijss,j+ [0 (1+ 22)],i P,i+

+fi = 0(1 d2)

vit

+ vi,svs

+0d

2(vi,svs+ vi,smvs,m) ,i+ ( + )s,si+ irsvs,r 2i+ gi

=0j

it

+ i,svs

,

a0 b0 cj+ l= 32 0jt + ,svs ,

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whereP = 2i,i , fi = 0Fi 0Fji,j,gi =0Mi , l= 0L .

We suppose thatfi, giandlare prescribed continuous functions. We assume

that the fluid occupies the bounded domain B(t),t I.To the field equations we add the initial conditions

(4.5) vi(x, 0) = v

0i (x) , j(x, 0) = j

0(x) ,i(x, 0) =

0i (x) , (x, 0) =

0(x) , x B(0) ,

and the boundary conditions (cf. [8], [14])

(4.6) vi = 0 , Dvi = 0 , i = 0 , = 0 on B I ,

wherev0i ,0i ,j

0 and0 are prescribed continuous functions. The conditions

on the boundary are related with the strict adherence (see [4], p. 62). Theboundary conditions for nonsimple continuous media have been discussed byRAJAGOPAL, MASSOUDIand EKMANN[17] and RAJAGOPALand TAO[18].

By an admissible process on B Iwe mean an ordered array of func-tions (vi, i,,j ,P) with the properties: (i)vi are of class C

4,1 on B I;(ii)vi are of class C

1,0 on B I; (iii)i andare of class C2,1 onB I;(iv) i andare of class C

0 onB I; (v) j is of class C1 onB I; (vi)j is of class C0 on B I; (vii) P is of class C1,0 on B I. By a solutionof the boundary-initial-value problem we mean an admissible process thatsatisfies the equations (4.4), the initial conditions (4.5) and the boundaryconditions (4.6).

We note that by (4.1), (4.4)1, (2.32) and (2.33) we obtain

(4.6)

B

[vi(ji,j kji,kj) + i(mji,j+ irsrs) + (hi,i g)]dv =

=

B

[vi(ji kji,k) + imji + hj] njda

B

[(ji kji,k)vi,j+

+irsirs+ mjii,j+ hi,i+ g]dv=

(4.7)

=

B

{vi[(ji kji,k)nj njDrjri+

+(brs bmmnrns)rsi] + rsinrnsDvi+ mjinji+ hjnj} da

B

(sijdij+ mijij+pijkijk + hi,i+ G) dv+

B

cjdv .

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Theorem 4.1. Assume that

(i) 0 is strictly positive;(ii) the entropy inequality(4.2) holds.

Then for any two solutions

(v(1)i ,

(1)i ,

(1), j(1), P(1)) and(v(2)i ,

(2)i ,

(2), j(2), P(2))

of the boundaryinitialvalue problem(4.4)(4.6) we have

v(1)i =v

(2)i ,

(1)i =

(2)i ,

(1) =(2) ,j(1) =j (2) , P(1) =P(2) + F ,

whereF is an arbitrary function of time.

Proof. Suppose that the functions v()i ,

()i ,

(), j(), P(), d()ij ,

()ij ,

()ijk ,

()ij , m

()ij ,

()ijk , h

()i ,G

() satisfy the equations

(4.8) v()i,i = 0 ,

j ()

t +j

(),i v

()i = 2

()j() ,

(4.9)

()ji,j

()kji,kj+ fi = 0(v

()i

()ji,j) ,

m()ji,j+ irs

()rs + gi = 0j

() ()i ,

h()i,i g

() + l=3

20j

()() ,

and

(4.10)

()

ij = ()

ij+ s

()

ij ,

()

ijk =

()

i jk j

()

ik +p

()

ijk ,g() = cj() + G() ,

s()ij =0

()ij+ ( + )d()ij + d

()ji ,

m()ij =ijk0

(),k +

()rr ij+

()ji +

()ij ,

p()ijk =

1

21(

()rri jk +

()rrjik) + 23

()rrkij+ 24

()ijk+

+5(()kji +

()kij ) + 1ij

(),k + 2(ik

(),j + jk

(),i ) ,

h()i =a0

(),i + a1irs

()rs + a3

()rri , G

() =b0() ,

d()ij =v

()j,i + jir

()r ,

()ij =

()j,i ,

()ijk =v

()k,ij,

where

()ij =d2 (v()j ),i v()j,kv()k,i .

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We introduce the notations

(4.11)

ui = v(2)i v

(1)i , wi =

(2)i

(1)i , =

(2) (1) ,

s=j (2) j(1) , Tij =(2)ij

(1)ij ,

Sij =s(2)ij s(1)ij , Mij =m(2)ij m(1)ij ,

Nijk = (2)ijk

(1)ijk, Pijk = p

(2)ijk p

(1)ijk,

Qi = h(2)i h

(1)i , =g

(2) g(1) , S= G(2) G(1) ,

= (2) (1) , i = (2)i

(1)i .

The functionsui,wi, and s satisfy the initial conditions

(4.12) ui(x, 0) = 0, wi(x, 0) = 0, (x, 0) = 0, s(x, 0) = 0, x B(0),

and the boundary conditions

(4.13) ui = 0 , Dui = 0 , wi = 0 , = 0 on B I .

In view of (4.8)1 we have

(4.14) ui,i= 0 on B I .

By (4.8)2 and (4.11) we obtain

(4.15) s

t+ s,iui+ s,iv

(1)i +j

(1),i ui= 2s

(2) + 2j(1) ,

onB I. It follows from (4.10) and (4.11) that

(4.16)

Tij = ij+ Sij, Nijk = ijk jik+ Pijk, = cs + S ,

Sij =0ij+ ( + )eij+ eji ,Mij =0ijk,k+ rrij+ ji + ij,

Pijk =1

21(rrijk + rrjik) + 23rrkij+ 24ijk+

+5(kji + kij) + 1ij,k+ 2(ik,j+ jk,i) ,Qi =a0,i+ a1irsrs+ a3rri , S= b0 ,

where

(4.17) eij =uj,i+ jirwr, ij =wj,i , ijk =uk,ij.

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The equations (4.9) imply that

(4.18)

Tji,j Nkji,kj =0

uit

+ (v(1)i + ui),sus+ ui,kv

(1)k

0d2Qji,j,

Mji,j+ irsTrs = 0j(2) wit

+ wi,sv(2)s + (1)i,s us+ 0sVi ,

Qi,i =3

20j

(2)

t + ,sv

(2)s +

(1),s us

+ 0sW ,

where

(4.19)

Qji = 2uitxj

+ ui,sj(us+ v(1)s ) + v

(1)i,sjus ,

Vi =

(1)i

t +

(1)i,s v

(1)s , W =

3

2

(1)

t + (1),s v

(1)s

.

It follows from (4.14) and (4.18) that

(4.20)

(Tji,j Nkji,kj)ui = 0

1

2

t(u2) +

1

2(u2uk),k+

+v(1)i,k uiuk+

1

2(u2v

(1)k ),k

0d2(Qjiui),j+

+0d2

t(ui,j) + ui,sjus+ v

(1)i,sjus+ ui,sjv

(1)s

ui,j,

(Mji,j+ irsTrs)wi=1

20j

(2)

t(w2) +

1

20(j

(2)w2v(2)s ),s+

+j(2)(1)

i,sus

wi+

0Viwis

1

2

0j

(2)

,k w2v

(2)

k ,

(Qi,i )=3

40j

(2)

t(2) +

3

40(j

(2)2v(2)s ),s+

+3

20j

(2)(1),s us + 0sW3

40j

(2),k

2v(2)k .

Clearly,

(4.21)

ui,sjusui,j =1

2(ui,jui,jus),s ,

ui,sjv(1)s ui,j =

1

2(ui,jui,jv

(1)s ),s ,

v(1)i,jsusui,j = (v

(1)i,jsusui),j ujuiv

(1)i,j (v

(1)i,jus,jui),s+

+v(1)i,jus,jui,s .

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In view of (4.7), (4.20), (4.21) and the boundary conditions (4.13) we obtain

(4.22)

B

(Sijeij+ Mijij+ Pijkijk + Qi,i+ S) dv=

=12B

0 t

(u2 + d2ui,jui,j) +j(2)

t(w2) +3

2j(2)

t(2) dv+

+

B

0

v(1)i,j(uiuj+ d

2us,jui,s) d2uiujv

(1)i,j+

+j(2)(1)i,s wius+

3

2j(2)(1),s us + Viwis + sW

+ cs

dv

1

4

B

0(2w2 + 32)j(2),s v

(2)s dv .

By (4.8), (4.14) and (4.15),

(4.23) 12

t

(s2) = 2(2)s2 + 2j(1)s ( 12

s2),jv(2)j j(1),k uks .

In view of (4.12), (4.22), (4.23) and the entropy inequality we find that

(4.24)

1

2

B

0

t(u2+d2ui,jui,j)+j

(2)

t(w2)+

3

2j(2)

t(2) +

t(s2)

B

0

v(1)i,j(uiuj+ d

2us,jui,s) d2uiujv

(1)i,j+

+j(2)(1)i,s wius+

3

2j(2)(1),s us +j

(1),k uks + Viwis+

+ (W+c1

0 2j(1)

)s2(2)

s2 dv+ 14B0(2w2+32)j(2),k v(2)k dv.

By the arithmetic-geometric mean inequality

(4.25)

(1)i,s wius

1

2(w2 +

(1)i,s

(1)i,jusuj) ,

(1),k uk

1

2(u2 +

(1),k

(1),k

2) ,

Viwis 1

2(w2 + ViVis

2) ,

j(1),k uks

1

2(u2 +j(1),p j

(1),p s

2) ,

(W+ c10 2j(1))s 122 + (W+ c10 2j(1))2s2 .

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Since (v()i ,

()i ,

(), j()), ( = 1, 2) are admissible processes on B I,

the functionsv()i,s, v

(1)i,j,

()i,s,

(),s ,j

(),s ,Vi,W,j

() and() are boundedonB I. If we define the function EonIby

E= 12B

0(uiui+ d2ui,jui,j+ wiwi+ 2 + s2) dv

then from (4.24) and (4.25) we conclude that there exists a positive constantmsuch that

dE

dt mE .

By integration on (0, ) and recalling that E(0) = 0, we findE()exp(m) 0. Thus we conclude that E= 0 on B I. This fact im-plies that u= 0, w= 0, = 0 ands = 0 onB I. From (4.16) and (4.17)we findSij = 0, Nijk = 0 so that from (4.18)1 we obtain (P

(1) P(2)),i = 0.This completes the proof.

The weighted energy method of GALDI and RIONERO[19] can be usedto establish a uniqueness result for unbounded domains.

5. Concentrated Body Loads. In this section we consider non-heat-conducting compressible fluids and assume that the free energy de-pends on the density and the microinertia j. First we derive the lineartheory appropiate to small departures from an equilibrium state and esta-blish a representation of Galerkin type for the solutions to the field equa-tions. Representations of this type for micropolar fluids have been presentedin [20]. Then we use the Galerkin representation to study the problem of

concentrated body loads in the case of steady vibrations.We assume that there exists an equilibrium state of the fluid in whichthe density and the microinertia have the uniform values 0 andj0, repec-tively. We introduce the notations

(5.1) = 0 , = j j0 .

We assume that , ,vi, i andare small, i.e. =,= , vi = vi,i =

i,=

where is a constant small enough for squares and higherpowers to be neglected, and , , v i,

i,

are independent of.To the second order, the free energy is taken in the form

(5.2) = 0+ 0 +p0+12

a2 +12

b2 + c ,

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where 0, 0, p0, a, b and c are prescribed constants. Without loss of

generality we assume that 0 = 0, p0 = 0. It follows from (2.25), (3.5) and(5.2) that

(5.3) pi = (a + b),i , p= c1 + d1 ,

where

a=20a , b= 20c

, c1= 2j00c , d1= 2j00b

.

From (3.4) we obtain the following equations for the functions vi, i, , and

(5.4)

2(1 l2)vi+ [1 2 (1l1 2l2)]vj,ji+ ijss,j+

+[0 (1+ 22)],i a,i b,i+ fi = 0(1 d2)vit

,

i+ ( + )s,si+ irsvs,r 2i+ gi = Iit

,

a0 [b1 (a3+ a4)]vs,s b0+ c1 + d1+ l= J

t ,

t 0vi,i= 0 ,

t 2j0= 0 ,

whereI=0j0, J= 3I/2. The system (5.4) can be written in the form

(5.5)

D1v+T1grad div v+curl +T2grad agrad bgrad = f,D2+ ( + )grad div + curl v= g,D3 T3div v + c1 + d1= l,

t 0div v= 0,

t 2j0= 0,

where we have used the notations

(5.6)

D1 =2(1 l2) 0(1 d2)

t,

D2 = I

t 2 , D3= a0 J

t b0 ,

T1 =1 2 (1l1 2l2) , T2 = 0 (1+ 22) ,T3 =b1 (a3+ a4) .

In the linearized theory a superposed dot will be used to denote the partialderivative with respect to the time.

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In what follows we establish a representation of Galerkin type for thesolutions of the field equations (5.5). We introduce the notations

(5.7)

P = (D1+ T1)

t a0 ,

M1 =c10 T3 t

, M2= D3 t

+ 2j0d1 ,

Z =T2

t 2j0b , N = PM2 ZM1 ,

L = (T1D2 2)M2

t D2(a0M2+ ZM1) ,

=D2+ ( + ) , Q= c1D1 (aT3 c1T1) ,H =T3Z + (D1+ T1)M2 .

Theorem 5.1. Let

(5.8)

v= (ND2+ Lgraddiv)U + curl W + Zgrad u(aM2+ c1Z)grad v (bM2+ d1Z)grad w ,

= Ncurl U D1 [( + )D1 2]grad divW ,= (D1D2+

2)M1div U Pu + Qv+ (d1P+ bM1)w ,= 0(D1D2+2)M2div U+0ZuHv0(bD3+d1T2)w,= 2j0(D1D2+

2)M1div U2j0Pu+2j0Qv+(T2M1D3P)w,

whereU, W, u, v andw satisfy the equations

(5.9)(D1D2+

2)NU= f,(D1D2+ 2)W= g ,Nu= l , Nv= 0 , Nw= 0 .

Thenvi, i, , andsatisfy the equations(5.5).

Proof. It follows from (5.5) and (5.8) that

(5.10)

D1v + T1grad div v + curl + T2grad agrad bgrad = (D1D2+ 2)NU + {D1L + T1(L ND2)+

+2N+ T2(D1D2+ 2)M1

t+ a0(D1D2+

2)M2

2j0b(D1D2+ 2)M1}grad div U + (2j0bP a0Z

T2P

t+ T1Z

t+ D1Z

t)grad u + [aH 2j0bQ+

+T2Q

t (T1 + D1)(aM2+ c1Z)]grad v+

+[(bM2+ d1Z)(D1+ T1) + T2(d1P+ bM1)+

+a0(bD3+ d1T2) b(T2M1 D3P)]grad w== (D1D2+ 2)NU .

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Similarly we obtain

(5.11)

D2+ ( + )grad div + curl v= (D1D2+ 2)W ,D3 T3div v + c1 + d1= Nu ,

t 0div v= Nu ,

t 2j0= Nw .

In view of (5.9), from (5.10) and (5.11) we obtain the desired result.

In the case of steady motions of micropolar fluids the Galerkin repre-sentations were presented in [21].

We now consider a fluid which occupies the entire three-dimensionalEuclidean space. We use the representation (5.8) to study the problem ofconcentrated body loads in the case of steady vibrations. We assume that

f= Re [f(x)exp(it)] , g= Re [g(x)exp(it)] ,

l= Re [l(x)exp(it)] ,

where is the frequency of vibration and i = (1)1/2. If we take

(5.12)v = Re [v(x; )exp(it)] , = Re [(x; )exp(it)] , = Re [(x; ) exp(it)] , = Re [(x; ) exp(it)] , = Re [(x; )exp(it)] ,

then the field equations reduce to a differential system for the amplitudesv,,, and. We denote

(5.13)

U = Re [U(x; )exp(it)] , W= Re [W(x; )exp(it)] ,

u = Re [u

(x; )exp(it)] , v= Re [v

(x; ) exp(it)] ,w = Re [w(x; )exp(it)] .

Then, from (5.8) we obtain

(5.14)

v = (N1A2+ L1graddiv )U + 1curl W iZ1grad u(aM2+ c1Z1)grad v (bM2+ d1Z1)grad w ,

=N1curl U {A11 [( + )A1 2]grad div }W , = (A1A2+ 2)M1idiv U + iP1u iQ1v+

+(d1P+ bM1)w , = 0(A1A2+ 2)M2div U + 0Z1u H1v

0(bA3+ d1T2)w ,

= 2j0(A1A2+ 2

)M1div U

2j0P1u

+ 2j0Q1v

++(T2M1 A3P1)w ,

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(5.15)

A1 =2(1 l2) + i0(1 d2) ,A2 = + iI 2 , A3= a0 + iJ b0 ,

P1 = i(A1+ T1) a0 ,M1 =c10+ iT3 , M2= 2j0d1 iA3 ,Z1 = iT2 2j0b , N1 = M2P1 Z1M1 ,L1 = i(T1A2 2)M2 (a0M2+ Z1M1)A2 ,1 =A2+ ( + ) , Q1= c1A1 (aT3 c1T1) ,H1 =T3Z1 + (A1+ T1)M2 .

The functions U, W, u,v andw satisfy the equations

(5.16)(A1A2+

2)N1U =f ,(A1A2+ 2)1W =g ,N1u =l , N1v = 0 , N1w = 0 .

Let us assume that f =0, g =0,l =(x y) whereis the Diracdelta and y is a fixed point. In view of (5.16) we take Ui = 0, W

i = 0,

u =(x; y),v = 0,w = 0, where satisfies the equation

(5.17) N1= .

From (5.14) we obtain the amplitudes

(5.18) v = iZ1grad , =0 ,

=iP1 , =0Z1 , = 2j0P1 .

In view of (5.6) and (5.15) we obtain

N1= 2 + i(s2 1e3 3e4) + (s3 13) + 02

2 ,


e1 =1 2 , e2= 1l1 2l2 , e3= 1+ 22 , e4 = a3+ a4 , =1l1a1 e3e4 , 1 = c10+ ib1 , 2 = J 2 + 2j0d1+ ib0 ,3 = 2j0b + i0 , s1 = 02d2 i1 , s2= 21l1 a0(s1 a0) ,s3 = (s1 a0)2 i30a0 .

Clearly, we can write

(5.19) N1 = 2( + 21 )( +

22)( +

23) ,

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where 21,22 and

23 are the roots of the equation

(5.20) 2y3 i(s2 1e3 3e4)y2 + (s3 13)y 02

2 = 0 .

In what follows we denote by 1, 2 and3 the roots with nonnegative realparts. We assume that 1 =2 =3 =1, and = 0. Let us consider theequation

(5.21) ( + 21)( + 22)( +

23) = F .

If the functions Qj satisfy the equations

(5.22) ( + 2j )Qj =F , (no sum;j= 1, 2, 3) ,

then the function can be expressed as

(5.23) =

3

j=1

BjQj,

where

(5.24) B11 = (

21

22 )(

21

23) , B

12 = (

22

21)(

22

23 ) ,

B13 = (23

21)(

23

22) .

If we take F=/(2), then from (5.22) we get

(5.25) Qj = 1

4 2rexp(ijr) ,

wherer = |x y|. Thus, from (5.17), (5.19), (5.23) and (5.25) we concludethat

(5.26) = 1

4 2r

3j=1

Bjexp(ijr) .

If we substitute the function from (5.26) into (5.18) then we obtain theamplitudes corresponding to the considered concentrated load.

We now assume that gi = mij , (j fixed), fi = 0 and l

= 0. If wetakeUi = 0, W

i = ij, u

= 0, v = 0, w = 0 then the equations (5.16)are satisfied if is a solution of the equation

(5.27) (A1A2+ 2)1 = m .

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We introduce the notations

1 = 2 + iI , 2 = ( 1l2)2 i0d2 ,3 =

2 + 12+ i0( 1d2) .

In what follows we denote by2j the roots of the equation

2l2z3 + 2z

2 3z+ i01= 0 .

Let24 =1/( + + ) .

The equation (5.27) can be written in the form

(5.28) ( + 21 )( + 22 )( +

23 )( +

24 ) =

1

2l2m .

We denote bys(s= 1, 2, 3, 4) the roots with positive real parts and assume

that 1=2 =3=4 =1, 2=4,1=3.If the functions Yr (r= 1, 2, 3, 4) satisfy the equations

( + 2s )Ys = 1

2l2m , (no sum;s= 1, 2, 3, 4) ,

then the function can be expressed as

=

4s=1

qsYs ,

where

q1k =4

j=1(j=k)

(2j 2k ) , (k= 1, 2, 3, 4) .

Ifm = (x y) then we find that =E, where

(5.29) E= 1

42l2r

4s=1

qsexp(isr) .

It follows from (5.14) that

vk =1ksjE,s ,

k = [( + )A1 2

]E,kj kjA11E , = 0 , = 0 , = 0 .

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In a similar way we can establish the solutions corresponding to a concen-trated body force.

In the context of the nonpolar theory of fluids the fundamental so-lutions have been established in various papers (see e.g. [22], [23]). The

problem of concentrated loads in the theory of micropolar fluids has beenstudied in [20], [24].

Acknowledgement. The first author was supported by the SpanishMinistry of Education and Culture as a visiting professor to the Centrede Recerca Matematica. The second author was supported by the projectPB960497 of the DGES of the Spanish Ministry of Education and Culture.

REFERENCES

1. ERINGEN, A.C. Simple microfluids, Int. J. Engng. Sci., 2(1964), 205217.

2. ERINGEN, A.C. Mechanics of micromorphic continua. In Mechanics of Genera-

lized Continua (E. Kroner, ed.), SpringerVerlag, Berlin, 1967.

3. ERINGEN, A.C. An assessment of director and micropolar theories of liquid crys-

tals. Int. J. Engng. Sci. 31(1993), 605616.

4. ERINGEN, A.C. and KAFADAR, C.B. Polar Field Theories. In Continuum Phy-

sics (A.C. Eringen, ed.), vol. IV, Part. I, Academic Press, New York, 1976.

5. ARIMAN, T, SYLVESTER, N.D. and TURK, M.A. Microcontinuum fluid mecha-

nics. A review. Int. J. Engng. Sci. 11(1973), 905925.

6. BRULIN, O Linear Micropolar Media. In Mechanics of Micropolar Media (O.

Brulin and R. K. T. Hsieh, eds.), World Scientific, Singapore, 1982.7. ERINGEN, A.C. Micropolar fluids with stretch. Int. J. Engng. Sci. 7(1969),

115127.

8. ERINGEN, A.C. Theory of thermomicrostretch fluids and bubbly liquids. Int. J.

Engng. Sci. 28(1990), 133143.

9. BRULIN, O. and HJALMARS, S. Linear gradeconsistent micropolar theory, Int.

J. Engng. Sci., 19(1981), 17311743.

10. RYMARZ, CZ. On the model of a nonsimple medium with rotational degrees of

freedom. Bull. Acad. Polon. Sci. Ser. Sci. Techn. 16(1968), 271277.

11. TOUPIN, R.A. Theories of elasticity with couplestress. Arch. Rational Mech.

Anal. 17(1964), 85112.

12. MINDLIN, R.D. Microstructure in linear elasticity, Arch. Rational Mech. Anal.

16(1964), 5178.

7/27/2019 Iesan

24/24


13. GREEN, A.E. and RIVLIN, R.S. Simple forces and stress multipoles. Arch. Ra-

tional Mech. Anal. 16(1964), 325353.

14. BLEUSTEIN, J.L. and GREEN, A.E. Dipolar fluids. Int. J. Engng. Sci. 5(1967),

323340.

15. ERINGEN, A.C. Balance laws of micromorphic continua revisited. Int. J. Engng.Sci. 6(1992), 805810.

16. GREEN, A.E. and STEEL, T.R. Constitutive equations for interacting continua.

Int. J. Engng. Sci. 4(1966), 483500.

17. RAJAGOPAL, K.R., MASSOUDI, M. and EKMANN, J.M. Mathematical mode-

ling of fluid-solid mixtures. In Recent developments in structured continua, vol.

II (D. De Kee and P. N. Kaloni, eds.) pp. 236248. Pitman Research Notes in

Mathematics Series, vol. 229, Longman Scientific and Technical, Essex, 1990.

18. RAJAGOPAL, K.R. and TAO Mechanics of Mixtures, World Scientific, Singapore

(1995).

19. GALDI, G. and RIONERO, S. Weighted Energy Methods in Fluid Dynamics and

Elasticity. Lecture Notes in Mathematics, Springer, Berlin, 1985.

20. RAMKISSON, H. and MAJUMDAR, S.R. Representations and fundamental sin-gular solutions in micropolar fluid mechanics. ZAMM 56(1976), 197203.

21. AYDEMIR, N.U. and VENART, J.E.S. Flow of a thermomicropolar fluid with

stretch. Int. J. Engng. Sci. 28(1990), 12111222.

22. DRAGOS, L. Fundamental matrix for the equations of ideal fluids. Acta Mechanica

33(1979), 163168.

23. DRAGOS, L. and HOMENTCOVSCHI, D. Stationary fundamental solution for an

ideal fluid in uniform motion. ZAMM 60(1980), 343345.

24. EASWARN, C.V. and MAJUMDAR, S.R. Causal fundamental solutions for the

slow flow of a micropolar fluid. Int. J. Engng. Sci. 28(1990), 843850.

Received: 16.XII.1999 Department of MathematicS

Al.I. Cuza University

6600 Iasi

ROMANIA

Department of Applied Mathematics II

Polytechnical University of Catalonia

Terrassa, Barcelona

SPAIN

Iesan

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