On the existence and uniqueness of flows multipolar fluids of grade 3 and their stability

PERGAMON

International Journal of Engineering Science 37 (1999) 75±96

On the existence and uniqueness of ¯ows multipolar ¯uids of grade 3 and their stability Hamid Bellout a, *, Jindrich NecÏas a, K.R. Rajagopal b a

b

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, U.S.A. Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, U.S.A. Received 30 April 1997

(Communicated by C.G. SPEZIALE) Abstract Combining the theory of multipolar ¯uids with the theory of ¯uids of di€erential type we ®nd a new model whose solutions exhibits reasonable stability characteristics # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Most of the models that are used for describing the behavior of ¯uids fall into the category of simple ¯uids (cf. Noll [1]) wherein the stress response is determined by the history of the relative deformation gradient. While many non-simple ¯uid models have been proposed (cf. Green and Rivlin [2, 3], Eringen [4]) they have not been used extensively by the practising ¯uid dynamicist. While plausible arguments can be advanced for the need and usefulness of such theories, no experimental evidence has been provided to support such theories. This is in most part due to the fact that no systematic experimental study has been proposed wherein the material moduli that appear in such ¯uid models can be determined. Also, as such theories lead to much more complicated governing equations and fundamental questions about the boundary conditions that go with these equations, such theories have not found great favor amongst ¯uid dynamicists. * Corresponding author. 0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 8 ) 0 0 0 2 3 - 8

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However, there seems to be a body of evidence emerging that might suggest the need for looking beyond the frame-work of simple ¯uids, or special subclasses of them. If we con®ne our attention to ¯uids of the di€erential type of complexity n (cf. Truesdell and Noll [5]), we ®nd that for these models to exhibit acceptable thermodynamic and stability characteristics we require certain signs for the material coecients from a theoretical standpoint. However, experiments that presuppose that the ¯uids belong to such a class do not lead to the signs for the material moduli which are required for stability. This has led to a great deal of unnecessary confusion and controversy in the ®eld and has been discussed in detail in the recent paper by Dunn and Rajagopal [6]. Fluids of the di€erential type of complexity n are ¯uids whose constitutive response is completely determined by the history of the relative deformation radient. However, such models do not seem appropriate for describing the ¯ows of many dilute polymeric solutions in view of experimental evidence. For such ¯uids, we could adopt a more general simple ¯uid model or on the other hand we could consider non-simple ¯uids in order to reconcile the predictions of experiments and those of theory. As there is an unlimited choice with regard to such non-simple ¯uids we in fact turn to the original choice of ¯uids of the di€erential type for some guidance with regard to such a choice. Amongst the ¯uids of the di€erential type, a subclass that has gained some prommence is the ¯uids of grade n (cf. Truesdell and Noll [5]). The classical Navier±Stokes ¯uid is a special ¯uid of grade one. While these models can be thought of as models in their own right, they have also been regarded to be, within the context of retarded ¯ows, the approximation for the stress of a simple ¯uid (cf. Coleman and Noll [7]), to various orders in the retardation parameter. It is important to recognize that not all ¯ows can be obtained as retardations of other ¯ows and thus this process of approximation does not lead to a hierarchy of models (cf. Dunn and Rajagopal [6]). This fact was recognized by Truesdell and Noll [5] who state, while discussing the ¯ow of simple ¯uids in pipes of non-circular cross-section, that ``It is plausible that the ¯ow will be slow when `a' is small. However, there is no reason to believe that the ¯ow for a small speci®c driving force can be obtained from the ¯ow for a larger speci®c driving force by a mere retardation. Therefore, the asymptotic approximation for the slow ¯ow discussed in Section 40 does not apply drectly''. For Truesdell and Noll [5], `a` is the pressure gradient along the axis of the pipe. Having articulated the need for caution in considering the approximations as models, let us ask ourselves the following two questions. Firstly, is it possible that the approximation procedure, based on the notion of retardation, does not lead to reasonable models? Secondly, is it necessary to start with a more general class of ¯uids and then develop an approximation procedure that leads to physically acceptable models? We investigate the answer to the second question that we might need to consider models wherein not only the history of the deformation gradient, but also spatial gradients of the history of the deformation gradient play a role in determining the stresses. Or to put it more simply, we should be concerned with both history and geography (non-locality) when proposing models of non-linear ¯uids. This is where the polar theories of Green and Rivlin [2, 3] become relevant as they take into account non-locality. The idea of approximations based on spatial dependence is not new. Motivated by the work of Coleman and Noll [7], Coleman has carried out such a spatial approximation. Here, we would like to consider a fusion of the models of the di€erential type due to Rivlin and Ericksen [8] and the multipolar models of Green and Rivlin [2, 3]. For instance, it is well

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known that the ¯uid of grade three, whose Cauchy stress T is given by T ˆ ÿp1 ‡ mA1 ‡ a1 A2 ‡ a2 A21 ‡ b1 A3 ‡ b2 ‰A1 A2 ‡ A2 A1 Š ‡ b3 …trA21 †A1

…1†

where the kinematical tensors A1 and A2 are de®ned through (See Rivlin and Ericksen [8]) A1 ˆ…grad v† ‡ …grad v†T d A2 ˆ A1 ‡ A1 …grad v† ‡ …grad v†T A1 dt is compatible with thermodynamics if and only if (Fosdick and Rajagopal [9]) p mr0; a1 r0; ja1 ‡ a2 jR 24mb3 ; b1 ˆ b2 ˆ 0

…3†

Unfortunately, dilute polymeric ¯uids tested do not meet (3), thereby questioning the appropriateness of using such a model for the ¯uid under consideration. If the ¯uid in question is assumed to be of the form (1) and the data reduced, then one usually ®nds that a1 < 0: However, such models are inherently unstable (Fosdick and Rajagopal [9], Patria [10], Galdi [11]) and call into question our initial assumption that the ¯uid in question can be modeled by (1). Here, we would like to investigate the possibility of melding the models of Rivlin and Ericksen [8] with those of Green and Rivlin [2, 3]. Such a theory would allow for the inclusion of higher-order temporal and spatial derivatives; and in the model (1) it is the higher-order term that determines the stability characteristics of the model. Thus, introducing in (1) other higher-order derivaives that appear in the multipolar models might provide the stabilization that is necessary. We should emphasize here that our choice of model is not based purely on the basis of obtaining a stable modi®cation of (1) as such a stabilization can be achieved in a variety of ways. As we observed before, there is no reason for supposing a priori that higher order time derivatives of the velocity gradient are important, while higher-order spatial derivatives of the velocity gradient are neglectable. We ®nd that an amalgam of the models of Rivlin and Ericksen [8] and Green and Rivlin [2, 3] does lead to a model that exhibits reasonable stability characteristics. However, such a model is not without attendant diculties; namely, the specifcation of boundary conditions in addition to those that are used in classical ¯uid mechanics.

2. Processes in tripolar materials The reader is referred to the papers [2, 3] by Green and Rivlin for a general treatment of multipolar materials and to Bleustein and Green [12] and NecÏas and SÏilhavy [13] and Bellout, Bloom and NecÏas [14] for multipolar ¯uids. Here we restrict ourselves from the outset to the basic equations for a generalized tropolar material (tripolar ¯uid of grade 3). An isothermal process of an incompressible tripolar ¯uid is described by six functions (v, c, T, S, B, b) of position x and time t, whose tensorial nature and interpretation is as follows:

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v ˆ…vi † c T ˆ…Tij † S ˆ…Sijk † B ˆ…Bijkm † b ˆbi Each process is assumed to satisfy the equations of balance of linear and angular momenta, given below in this section. As the ¯uid is incompressible, it can undergo only isochoric motions and thus vi;i ˆ 0 (with the Einstein summation convention over repeated indices). We denote by r>0 the constant density of the ¯uid. The local form of the equation of balance of linear momentum reads rv_i ˆ Tij;j ‡ rbi

…4†

and the equation of balance of angular momentum can be reduced to the assertion that the tensor Tij ‡ Sijk;k

…5†

is symmetric. The second law is taken here to mean that the Clausius±Duhem inequality holds in every process satisfying the laws. For isothermal processes, the Clausius±Duhem inequality, combined with the equation of balance of energy, leads to the dissipation inequality   1 r vi vi ‡ c R…Tij vi ‡ Sik;j vi;k ‡ Bikmj vi;km †;j ‡rbi vi : …6† 2 On combining this inequality with the equation of balance of linear momentum, it leads to the reduced dissipation inequality _ rcR…T ij ‡ Sijk;k †vi;j ‡ …Sijk ‡ Bijkm;m †vi;jk ‡ Bijkm vi;jkm :

…7†

3. Constitutive equations Rather than constructing a general form for the response functions via the representation theorems for isotropic tensor functions, here we restrict ourselves to a particular model of a tripolar ¯uid of grade 3. As the constitutive response function depend on various kinematical quantities we shall de®ne them ®rst. The ®rst two Rivlin±Ericksen tensors A1. A2 are de®ned through (2) as

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…A1 †ij ˆ2Dij ˆ vi;j ‡ vj;i d …A2 †ij ˆ …A1 †ij ‡ Lmi …A1 †mj ‡ Lmj …A1 †im ; dt where Dij is the symmetric part of the spatial gradient of velocity Lij ˆ vi;j : We shall also need an objective time-rate (Jauman derivative) of the symmetric part of the gradient of the symmetric part of the velocity gradient, denoted by M = (Mijk) and given by Mijk ˆ

d Dij;k ‡ Wmi Dmj;k ‡ Wmj Dim;k ‡ Wmk Dij;m dt

with W = (Wij) the spin tensor de®ned as 1 Wij ˆ …vi;j ÿ vj;i †: 2 We shall assume constitutive relations of the form Tij ‡ Sijk;k ˆ ÿ pdij ‡ m…A1 †ij ‡ a1 …A2 †ij ‡ a2 …A21 †ij ‡

…8†

b3 …A21 †mm …A1 †ij ;

Sijk ‡ Bijkm;m ˆm1 …A1 †ij;k ‡ gMijk ; Bijkm ˆm2 …A1 †ij;km ; 1 1 rc ˆ a1 …A1 †ij …A1 †ij ‡ g…A1 †ij;k …A1 †ij;k : 4 8

…9† …10† …11†

Here, m, a1, a2, b3, g, m1, m2 are material moduli that are assumed to be constant and the quantities on both sides of the constitutive equations are evaluated at (x, t). The coecient m is the classical viscosity, a1, a2 are the normal stress coecients, while b3 is a higher-order viscosity coecient (cf. Fosdick and Rajagopal [9]). The coecient g is a new additional viscometric coecient and [13]). The second law of m1, m2 are higher-order viscosities (see NecÏas and SÏilhavy thermodynamics requires that the coecients satisfy certain inequalities (see the following section). Additional inequalities will be derived from the stability considerations (positivede®niteness of total energy), and for the existence of solutions, strict versions of some of these inequalities must be satis®ed. Upon calculating the divergences in (8)±(9), one can obtain the following more explicit form of the constitutive relations:

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Tij ˆ ÿ pdij ‡ m…A1 †ij ‡ a1 …A2 †ij ‡ a2 …A21 †ij ‡ b3 …A21 †mm …A1 †ij ÿ m1 D…A1 †ij ‡ m2 D2 …A1 †ij ÿ gMijk;k d ˆ ÿ pdij ‡ 2mDij ‡ 2a1 Dij ‡ 4…a1 ‡ a2 †…D2 †ij dt d ‡ 8b3 …D2 †mm Dij ‡ g DDij dt ‡ Dmk Dij;mk ‡ 2…Wmi Dmj ‡ Wmj Dim † ‡ Wmi DDmj ‡ Wmj DDim ‡ Wmi;k Dmj;k ‡ Wmj;k Dim;k ‡ Wmk;k Dij;m;

…12†

Sijk ˆ2m1 Dij;k   d Dij;k ‡ Wmi Dmj;k ‡ Wmj Dim;k ‡ Wmk Dij;m ‡g dt ÿ 2m2 DDij;k ; Bijkm ˆ2m2 Dij;km ;

…13† …14†

1 …15† rc ˆa1 Dij Dij ‡ gDij;k Dij;k : 2 A ¯uid given by these constitutive equations satis®es the principle of material frame indi€erence and the equation of balance of angular momentum in the strong sense that not only the sum (5), but also each of the terms Tij, Sijk,k, are symmetric.

4. Thermodynamic compatibility It can be shown using standard arguments in continuum mechanics (Silhavy [15]) that the ¯uid given by the constitutive Eqs. (4)±(7) satis®es the dissipation inequality (6) in every process if and only if the following inequalities hold: p …16† mr0; b3 r0; ja1 ‡ a2 jR 24mb3 ; m1 r0; m2 r0:

…17†

The second law, i.e. the dissipation inequality, imposes no restriction on the signs of the coecients a1, a2 and g. The inequalities (16) were derived by Fosdick and Rajagopal [9] in the context of simple ¯uids of grade 3. The inequalities (17) on higher viscosities are completely analogous to (16). The theorem follows from a straightforward application of the Clausius±Duham inequality. At this juncture, it would be appropriate to highlight the distriction between (16) and (17) and the results of Fosdick and Rajagopal [9] for ¯uids of grade three. Eqs. (16) and (17) remain silent about the sign for the coecient a1, while in a ¯uid of grade three it is necessary that a1>0. Thus, in the tripolar model being considered, it is possible that a1R0. If for such an eventuality, we can

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prove existence of solutions, and the asymptotic stability of such solutions, then we might have a model that might be a candidate for describing the behavior of some dilute polymeric solutions. 5. Existence and uniqueness of a weak solution We begin by de®ning weak solutions. De®nition 5.1. Let a1>0, a2$(ÿ1, 1), b3>0, mr 0, m1r0, m2>0, g r0. A function v $ L4[I, 2 3,2 2 2,2 W1,4 0 (O)], v $ L [I, W (O)] vt$L [I, W (O)] is a weak solution of the problem rv_i ˆ Tij;j ‡ rbi divv ˆ 0 in

in QT

QT

…18† …19†

v…x; 0† ˆ v0 …x†:

…20†

If a.e. 8t>0, 3;2 divf ˆ 0 f 2 W1;2 0 …O† \ W …O†; … … … … @…A1 †ij @vi @Dij;k fi dx ‡ a1 fi;j dx ‡ g fi;jk dx ‡ m …A1 †ij fi;j dx @t O @t O O @t O  …  … @…A1 †ij ‡ a1 vl ‡ Lmi …A1 †mj ‡ Lmj …A1 †im fi;j dx ‡ a2 …A21 †ij fi;j dx @x l O …O … ‡ b3 …A21 †mm …A1 †ij fi;j dx ‡ m1 …A1 †ij;k fi;jk dx O … … O vi f dx ‡ m2 …A1 †ij;km fi;jkm dx ‡ vj @xj i O O … … ÿ bi fi dx ‡ g …Wmi Dmj;k ‡ Wmj Dim;k ‡ Wmk Dij;m †fi;jk dx O

…21†

O

ˆ 0: Next we state our ®rst existence theorem. Theorem 5.1. Assume that @O is smooth enough. Then for any bi$L2(QT) and for any v0$W3,2(O)\W1,2 0 (O), divv0=0 there exists a weak solution to problem (18)±(20). Furthermore, the weak solution v is such that @v 2 L2 ‰0; T; W2;2 …O†Š: …23† @t Proof. The existence will be proved using Galerkin method. For this purpose we consider the sequence of functions wl, which are eigenvalues to the Stokes problem in O with homogeneous Dirichlet boundary conditions. It is well known that this sequence constitutes a Schauder basis for the spaces {w, such that w $ W3,2(O)\W1,2 0 (O), divw = 0.}

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We set Em=span{wl; l = 1, m} and let vm=Sl R m cl(t)wl be such that setting v = vm in (22) the equality is satis®ed for all test functions f $ Em. … l …24† c …0† ˆ v0 wl dx 8l: O

The existence and uniqueness of vm follows from Picard theorem. To ®nish the proof of the theorem we would need to establish some a priori estimates. These are provided in the following Lemmas. Lemma 5.1. There exist C independent of m such that jjvm jjL1 ‰0;T;W2;2 …O†Š RC‰jjv0 jj2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š

…25†

jjvm jjL2 ‰0;T;W3;2 …O†Š RC‰jv0 j2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š

…26†

jjvm jjL4 ‰0;T;W2;4 …O†Š RC‰jjv0 jj2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š:

…27†

Proof of the lemma. Since vm$Em it can be used as a test function and after integration in time we ®nd that: (it is clear that all the estimates in this lemma are for the vm and its derivatives, but for convenience of notations we will leave out the superscript m) … … … …T … 1 1 vi vi dx ‡ a1 Dij Dij dx ‡ g Dij;k Dij;k dx ‡ m …A1 †ij …A1 †ij dxdt 2 Ot 2 Ot 0 Ot Ot …T … ‡ m1 …A1 †ij;k …A1 †ij;k dxdt 0

‡ m2

Ot

…T …

0 O …T …

…A1 †ij;km …A1 †ij;km dxdt

…A21 †mm …A1 †ij …A1 †ij …A1 †ij dxdt ‡ b3 … … 0 O 1 R vi vi dx ‡ a1 Dij Dij dx 2 O0 O0 … …T … 1 ‡ g Dij;k Dij;k dx ‡ j bi vi dxdtj 2 O0 0 O …T … ‰Lmi …A1 †mj ‡ Lmj …A1 †im Švi;j dxdtj ‡ ja1 0

‡ ja2

…T 0

O

…A21 †ij vi;j dxdtj:

We have made use of the following easy identities in deriving the above inequality:

H. Bellout et al. / International Journal of Engineering Science 37 (1999) 75±96

a1

…T … 0

…T …

g

0

… O

vj

83

@…A1 †ij vl vi;j dxdt ˆ 0 O @xl

…29†

…Wmi Dmj;k ‡ Wmj Dim;k ‡ Wmk Dij;m †vi;jk dxdt ˆ 0

…30†

O

vi vi dx ˆ 0: @xj

…31†

It then follows from (28) that jjvjj2W2;2 …OT † ‡jjvjj2L2 ‰0;T;W3;2 …O†Š ‡ jjvjj4L4 ‰0;T;W2;4 …O†Š R c‰jjvjj2W2;2 …O0 † ‡jjvi jjL2 …QT † jjbi jjL2 …QT † ‡ jjDvi jj3L3 …QT † Š; from which it easily follows by standard techniques that jjvjj2W2;2 …OT † ‡ jjv0 jj2L2 ‰0;T;W3;2 …O†Š ‡ jjvjj4L4 ‰0;T;W2;4 …O†Š Rc‰jjvjj2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š;

…33†

which ®nishes the proof of the Lemma. Lemma 5.2. There exists a constant C independent of m such that jj

@vm RC‰jjv0 jj2W3;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š: jj 2 2;2 @t L ‰0;T;W …O†Š

…34†

Proof. Since @vm/@t $ Em it can be used as a test function and after integration in time we ®nd that: (as we did in the proof of the previous lemma we will again leave out the superscript m). …T … …T … @…A1 †ij @…A1 †ij @vi @vi dxdt ‡ a1 dxdt @t @t 0 O @t @t 0 O …T … @Dij;k @Dij;k dxdt ‡g @t 0 O @t … 1 …A1 †ij …A1 †ij dx ‡ m 2 OT … 1 …A1 †ij;k …A1 †ij;k dx ‡ m1 2 OT … 1 ‡ m2 …A1 †ij;km …A1 †ij;km dx 2 OT … 1 R m …A1 †ij …A1 †ij dx 2 O0 … 1 ‡ m1 …A1 †ij;k …A1 †ij;k dx 2 O0

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… 1 ‡ m2 …A1 †ij;km …A1 †ij;km dx 2 O0  …T …  @…A1 †ij @vi;j ‡ ja1 vl ‡ Lmi …A1 †mj ‡ Lmj …A1 †im dxdtj @xl @t 0 O …T … @vi;j dxdtj ‡ ja2 …A21 †ij @t 0 O …T … @vi;j ‡ jb3 …A21 †mm …A1 †ij dxdtj @t 0 O …T … …T … @vi @vi @vi ‡j vi bi dxdtj dxdtj ‡ j @t 0 O @xj @t 0 O …T … @vi;jk …Wmi Dmj;k ‡ Wmj Dim;k ‡ Wmk Dij;m † dxdtj ‡ jg @t 0 O  I1 ‡ I2 ‡ I3 ‡ I4 ‡ I5 ‡ I6 ‡ I7 ‡ I8 ‡ I9 :

…35†

I1+I2+I3 can be bounded using the norm of the initial condition in W3,2(O). The term I7 can easily be estimated by using Cauchy±Schwartz inequality in the usual fashion: 1 @vi I7 R jjbi jj2L2 …QT † ‡ ajj jj2L2 …QT † : a @t

…36†

Choosing a>0 small enough we can absorb the term ak@vi/@tk2L2(QT) in the left hand side of the inequality (35). Similarly, using embedding theorems and Holder inequality each of the reamining terms can be bounded as follows: 1 @vi I3 ‡ I4 ‡ I5 ‡ I6 ‡ I8 R jjvi jj2L2 ‰0;T;W3;3 …O†Š ‡ ajj jj2L2 ‰0;T;W3;2 …O†Š : a @t Using again an appropriate choice of a, we can absorb the second term in the estimate above into the left-hand side of estimate (35), the lemma then follows from Lemma (5.1). Proof of theorem (5.1). From Lemma (5.1) we deduce that there exist a subsequence, denoted again by vm, which converges weakly in L2[0, T; W3,2(O)\W1,2 0 (O)] to a function v. Using the estimate of Lemma (5.2) and Aubin's Lemma we can conclude that there exist a subsequence, which we will again denote by vm, of the sequence vm which converges also strongly in L2[0, T; W2,2(O)1,2 0 (O)]. These convergences are enough to allow for a passing to the limit in all the terms involved in the de®nition of a weak solution. Next we state our main existence and uniqueness theorem. In this next theorem we will weaken the assumptions of Theorem 5.1 on the initial data v0 and end up with a weak solution which would not necessarily satisfy 923). Theorem 5.2. Assume that @O is smooth enough then for any bi$L2(QT) and for any v0$W2,2(O)\W1,2 0 (O), divv0=0 there exists a unique weak solution to problem (18)±(20).

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The proof is based on a density argument. Indeed, we are assuming that O is bounded and smooth, therefore, the set {w, such that w $ W3,2(O)\W1,2 0 (O), divw = 0} is dense in the set {w, 1 (O), divw = 0.} such that w $ W2,2(O)\W1,2 0 k Let vk0 be a sequence of divergence free functionsin W3,2(O)\W1,2 0 (O),} and such that v0 2,2 converges strongly to v0 in W (O) as k 4 1. By Theorem 5.1 for every ®xed k the problem (18)±(20) with intial condition vk0 has a weak solution vk. Since the sequence vk0 is bounded in W2,2(O) independently of k, it follows that there exists a C independent of k such that jjvk jjL1 ‰0;T;W2;2 …O†Š RC‰jjv0 jj2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š

…37†

jjvk jjL2 ‰0;T;W3;2 …O†Š RC‰jjv0 jj2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š

…38†

jjvk jjL4 ‰0;T;W2;4 …O†Š RC‰jjv0 jj2W2;2 …O0 † ‡ jjbi jj2L2 …QT † ‡ 1Š

…39†

Next, for f $ L2[0, T; W3,2 0 (O)] and divergence free it follows from the de®nition of weak solutions that for every k … @vki …fi ÿ a1 fi;jj ‡ gfi;jkjk †dx O @t … ˆ ÿm …A1 †ij fi;j dx O # … " … @…Ak1 †ij k k k k k vl ‡ Lmi …A1 †mj ‡ Lmj …A1 †im fi;j dx ÿ a2 …A21 †kij fi;j dx ÿ a1 @xl O O … … ÿ b3 …A21 †kmm …A1 †kij fi;j dx ÿ m1 …A1 †kij;; fi;jl dx O O … … k @v ÿ vkj i fi dx ÿ m2 …A1 †kij;lm fi;jlm dx @xj O …O … ‡ bi fi dx ÿ g …Wkmi Dkmj;l ‡ Wkim;l ‡ Wkml Dkij;m †fi;jk dx: O

O

From which it follows that by using (37)±(39) that … @vki …fi ÿ a1 fi;jj ‡ gfi;jkjk †dxjRcjjfjjW 3;2 …O† ; j 0 O @t

…40†

and after integration by parts, jj

X @…vki ÿ a1 vki;jj ‡ gvki;jkjk † i

@t

jjL2 ‰0;T;W3;2 …O†Š RC;

…41†

where C is independent of k. 1

An easy way to see this is to recall that the eigenfunctions of the Stokes problem in O are smooth and constitute a basis of the space {w, such that w $ W2,2(O)\W1,2 0 (O), divw = 0}.

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Also, by (37)±(39) X jj …vki ÿ a1 vki;jj ‡ gvki;jkjk †jjL2 ‰0;T;Wÿ1;2 …O†Š RC;

…42†

i

where C is again independent of k. It then follows from Aubin's Lemma (see [16] for example) that the sequence Si(vkiÿa1vki,jj +gvki,jkjk) is in a set which is compact for the strong topology of L2[0, T; Wÿs,2(O)] for all s>1. The uniqueness of the solution will be obtained by the usual method. We assume that we have two distinct solutions v1, and v2 (we will use the superscripts 1 and 2 to refer to the tensors involving v1 and v2, respectively). We will prove a Gronwall inequality for … 1 …v1 ÿ v2 †2 ‡ a1 ‰eij …v1 ÿ v2 †eij …v1 ÿ v2 †Šg…D1ij;k ÿ D2ij;k †…D1ij;k ÿ D2ij;k †dx: …43† 2 Ot For this purpose we take the di€erence of the equations satis®ed by v1 and v2 use the function v1ÿv2 as a test function. The linear terms in the partial di€erential equation do not create any unusual problem and we will just indicate below how to handle the nonlinear terms: … …D1ij;k ÿ D2ij;k †…D1ij;k ÿ D2ij;k †jv2 j2 dxRCjjv1 ÿ v2 jj2W2;2 …O† jj2W2;2 …O† …44† Ot

… Ot

… Ot

…D1ij;k ÿ D2ij;k †…D1ij;k ÿ D2ij;k †jDv2 j2 dxRCjjv1 ÿ v2 jj2W2;2 …O† jjv2 jj2W3;2 …O†

…45†

…Dv1 ÿ Dv2 †2 jD2ij;k †j2 dxRCjjv1 ÿ v2 jj2W2;2 …O† jjv2 jj2W3;2 …O† :

…46†

The uniqueness then easily follows from the usual Gronwall inequality.

6. Boundary conditions It is assumed that the ¯uid occupies a ®xed region O R3 and that the ¯uid adheres to the boundary in the sense that the velocity ®ld v = vi(x, t), x $ O, t $ [0, T] satis®es vi …x; t† ˆ 0; x 2 @O; t 2 ‰0; TŠ: While the above boundary condition suces to obtain a well-posed problem within the context of the Navier±Stokes theory, it is insucient in the case of a tripolar ¯uid of grade three. We now proceed to augment the above boundary condition. To formulate further boundary conditions, let us denote by Tij(v), Sijk(v), Bijkm(v) the stress ®elds corresponding to the velocity ®eld vi=vi(x, t) via the constitutive Eqs. (12)±(15). The boundary conditions require that for every vector valued test function fi=fi(x, t) de®ned on O  [0, T] and vanishing on the boundary of O:

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87

… @O

…Sikj …v†fi;k ‡ Bikmj …v†fi;km †nj dA ˆ 0

…47†

for every t $ [0, T], where n-n(x) is the unit outward normal on @O, t 1 and t 2 are linearly independent tangent vectors to @O and dA denotes the surface measure on @O. The de®nition of a weak solution in Section 7 speci®es appropriate function spaces to make the boundary conditions meaningful. Here we shall derive the formal pointwise form of the above boundary conditions. It is claimed that if the ®elds occurring in (47) are suciently smooth, then the ®elds Bijkl=Bijkl(v) and Sijk=Sijk(v) satisfy Bijkm nj nk nm tli ˆ 0:

l ˆ 1; 2

…48†

and ‰Sijk nj nk ‡4wBijkm nm nk nj ÿ …Bijkm nm †;k nj

ÿ…Bijkm nm nk †;j ‡ …Bljkm nl nj nm nk †;i Štpi ˆ 0

p ˆ 1; 2

at every point of the boundary @O. Here the semicolon followed by an index denotes the surface gradient of a function, de®ned on @O, given by f;j ˆ f~;j ÿ nj nm f~;m ;

…50†

where fÄ denotes any extension of the function f from @O into a neighborhood of @O. This de®nition is independent of the choice of the extension fÄ. Observing that the tensor Pij given by Pij=dijÿninj is a projection onto the tangent plane, one sees that the surface gradient is a projection of the gradient onto the tangent plane. To derive the conditions (48), (49) from (47), we shall need a particular case of a surface divergence theorem asserting that for an vector ®eld wi=wi(x) de®ned on @O one has … … wi;i dA ˆ 2wwi ni dA; …51† @O

O

where w is the mean curvature, de®ned by 2w ˆ ÿvi;i : The formula (51) can be derived from the Stokes formula applied to @O and to the vector ®eld Eijkwjnk. Let us ®rst observe that since the test function fi vanishes on the boundary, one has fi;j ˆ

@fi @f np
nj ˆ i nj ; @xp @n

…52†

where @fi/@n denotes the normal derivative of fi. Let us further denote by @2fi/@n2 the second normal derivative of fi, given by @2 fi @2 fi ˆ np nq : @n2 @xp @xq

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We ®rst transform the second term in the integrand in (47) as follows: Bijkm nm fi;jk ˆBijkm nm dkp fi;jp ˆBijkm nm …dkp ÿ nk np †fi;jp ‡ Bijkm nm nk np fi;jp ˆBijkm nm fi;j;k ‡ Bijkm nm nk np djp fi;qp ˆBijkm nm fi;j;k ‡ Bijkm nm nk np …djq ÿ nj nq †fi;qp ‡ Bijkm nm nk np nj nq fi;qp ˆBijkm nm fi;j;k ‡ Bijkm nm nk np fi;p;j @ 2 fi @n2 ˆ…Bijkm nm fi;j †;k ÿ …Bijkm nm †;k fi;j ‡ Bijkm nj nk nm

‡ …Bijkm nm nk np fi;p †;j ÿ …Bijkm nm nk np †;j fi;p ‡ Bijkm nj nk nm

@ 2 fi dA: @n2

Hence, using (51) twice, we ®nd that … Bijkm fi;jk nm dA @O … … ˆ …Bijkm nm fi;j †;k dA ÿ …Bijkm nm †;k fi;j dA @O @O … … ‡ …Bijkm nm nk np fi;p †;j ÿ …Bijkm nm nk np †;j fi;p dA @O @O … @2 f ‡ Bijkm nj nk nm 2i dA @n … @O ˆ 2wBijkm nm nk fi;j dA @O … ÿ …Bijkm nm †;k fi;j dA …@O ‡ 2wBijkm nm nk np fi;p nj dA @O … ÿ …Bijkm nm nk np †;j fi;p dA @O … @2 f Bijkm nj nk nm 2i dA: ‡ @n @O

…53†

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Furthermore, we have npnp=1 and, therefore, np;knp=0. Using this, one can rewrite the term …Bijkm nm nk np †; fi;p on the right-hand side of the above equation as …Bijkm nm nk np †;j fi;p ˆ …Bijkm nm nk np †;j np

@fi @n

ˆ ‰…Bijkm nm nk †;j np ‡ Bijkm nm nk np;j Šnp ˆ …Bijkm nm nk †;j

@fi ; @n

@fi @n …54†

where also (52) has been used. Using (52) once more, and (53), one ®nally obtains … Bijkm fi;jk nm dA @O … ‰2wBijkm nm nk nj ÿ …Bijkm nm †;k nj ˆ @O

@f ‡ 2wBijkm nk nk nj ÿ …Bijkm nm nk †;j Š i dA @n … 2 @f ‡ Bijkm nj nk nm 2i dA: @n

…55†

The boundary condition (47) now can be written in the form, using (52) again, as … ‰Sijk nj nk ‡ 2wBijkm nm nk nj ÿ …Bijkm nm †;k nj @O

@f ‡ 2wBijkm nm nk nj ÿ …Bijkm nm nk †;j Š i dA @n … @ 2 fi ‡ Bijkm nj nm 2 dA ˆ 0: @n @O

…56†

Since fi is a divergence free vector ®eld the functions @fi/@n, @2fi/@n2 are not completely arbitrary and we cannot conclude that each of the integrands above is zero. For a divergence free vector ®eld fi, which vanishes on the boundary of O the most general form of the normal derivatives on the boundary is given by: @fi ˆ gi on@O @n

…57†

@2 fi ˆ hi ÿ ni divs …g†on@O; @n2

…58†

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where h and g are arbitrary vectors tangent to the surface @O and divS is the surface divergence operator de®ned by divs …v† :ˆ vi;i ; (see [17]). The last term in (56) becomes …

@2 f Bijkm nj nk nm 2i dA ˆ @n @O

… @O

… Bijkm nj nk nm hi dA ÿ

@O

Bijkm nj nk nm divs …gi †ni dA:

…59†

Since the vector g is tangential to @O we have by integration by parts: … ÿ

…

@O

Bijkm nj nk nm ni divs …g† dA ˆ

@O

gl …Bijkm nj nk nm ni †;l dA:

…60†

The integral Eq. (56) then yields … ‰Sijk nj nk ‡ 4wBijkm nm nk nj ÿ …Bijkm nm †;k nj @O

ÿ …Bijkm nm nk †;j ‡ …Bljkm nl nj nk †;i Šgi dA … ‡

@O

Bijkm nj nk nm hi dA ˆ 0:

…61†

Since h and g are arbitrary tangent vectors ®elds, it follows that if t 1 and t 2 are linearly independent vectors, tangent to @O then Bijkm nj nk nm tli ˆ 0:

l ˆ 1; 2p

…62†

and ‰Sijk nj nk ‡ 4wBijkm nm nk nj ÿ …Bijkm nm †;k nj ÿ …Bijkm nm nk †;j ‡ …Bljkm nl nj nm nk †;i Štpi ˆ 0

p ˆ 1; 2:

…63†

Using now the arbitrariness of @Oi/@n, @2fi/@n2, one ®nally obtains, after easy rearrangements, (48) and (49).

7. The energy inequality Integrating the energy inequality m1(A1)ij,k(A1)ij,kr0 over O, using the divergence theorem and the boundary conditions (47) with fi0vi, we obtain the following important identity:

H. Bellout et al. / International Journal of Engineering Science 37 (1999) 75±96

d dt

91

  1 r vi vi ‡ c dx 2 O

…

1 ‡ 2

… O

‰m…A21 †mm ‡ …a1 ‡ a2 †…A31 †mm ‡ b3 …A21 †mm …A21 †mm

‡ m1 …A1 †ij;k …A1 †ij;k ‡ m2 …A1 †ij;km …A1 †ij;km Š dx ˆ 0: The quantity    …  1 1 1 rvi vi ‡ a1 Dij Dij ‡ gDij;k Dij;k dx E…t† ˆ r vi vi ‡ c dx ˆ 2 2 O O 2 …

is the total energy of the ¯uid at time t. Since the second integral is non-negative as a consequence of the second law, we have _ E…t†R0

…65†

for every process in a ¯uid. let us now assume that the rest state of the ¯uid, vi00, is stable in the sense that every perturbation of the rest state is eventually damped by the dissipative mechanisms of the ¯uid. Then it is natural to assume that the energy of the ¯uid during this process tends to the energy of the rest state, which is 0: E…t†40

as t41

…66†

As a consequence of (65), (66) one sees that necessarily E…t†r0 and as this must hold for every initial perturbation of the rest state, one sees that a necessary condition for the formal stability in the sense of (65), (66) is that  …  1 1 E‰v…
†Š ˆ rvi vi ‡ a1 Dij Dij ‡ gDij;k Dij;k dxr0 …67† 2 O 2 for every velocity ®eld vi with vi=0 on @O and vi,i=0. Clearly a necessary condition for (67) to hold for every ®eld of velocity is that g r0. The following proposition gives a sucient condition. Proposition 7.1. Let g>0 and suppose that either a1 r0 or a1 R0

…68† 3 and g > a21 : r

…69†

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Then there exists a constant c>0 such that E‰v…
†Šrcjjvjjw2;2 …O† for every v $

2,2 W1,2 0 (O)\W (O)

…70† such that >vi,i=0 on O.

Proof. The proof in the case (68) is immediate. In the case of (69) we have the following lemma. 2,2 Lemma 7.1. For every f $ W1,2 0 (O)\W (O) one has

jjgradfjj2L2 …O† RjjfjjL2 …O† jjDfjjL2 …O† ;

…71†

where Df is the Laplacian of f. Remark 7.1. It is worth noticing that the constant c = 1 in the inequality jjgradfjj2L2 …O† RcjjfjjL2 …O† jjDfjjL2 …O†

…72†

is optimum. Indeed for eigenfunctions of the Laplacian (71) becomes an equality. Proof of the lemma. We have … @f @f 2 jjgradfjjL2 …O† ˆ dx O @xi @xi   …   @ @f ˆ f ÿ fDf dx @xi @xi … …O @f fvi dA ÿ fDf dx ˆ @O @xi O RjjfjjL2 …O† jjDfjjL2 …O† :

2,2 3 Lemma 7.2. For every vector-valued function v $ W1,2 0 (O)\W (O) with vi,i=0, O $ R one has … … p Dij Dij dxR 3jjvjjL2 …O† Dij;k Dij;k dx: …73†

O

O

Proof of the lemma. One has 1 1 Dij;k ˆ …vi;j ‡ vj;i †;k ˆ …vi;jk ‡ vj;ik † 2 2

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and solving this system of equations gives vi;jk ˆ Dij;k ‡ Dik;j ÿ Djk;i

…74†

from which, using Dmm,i 00 Dvi ˆ 2Dim;m ; …Dvi †2 ˆ 4…Dim;m †2 R4…Di1;1 ‡


‡ Di3;3 †2 ˆ 4
3…D2i1;1 ‡;


; ‡D2i3;3 †:

…75†

On the other hand, we have 1 Dij Dij ˆ …vi;j ‡ vi;i †…vi;j ‡ vj;i † 4 1 ˆ …vi;j vi;j ‡ vi;j vj;i † 2 1 1 ˆ jgradvj2 ‡ vi;j vj;i 2 2 and hence … … 1 1 vi;j vj;i dx Dij Dij dx ˆ jjgradvjjL2 …O† ‡ 2 2 O … 1 1 ˆ jjgradjjL2 …O† ‡ vi;j vj vi dA 2 2 @O … 1 ˆ vi;ji vj dx; 2 O and the last two integrals vanish in view of the boundary condition for v and the constraint of incompressibility. Hence … … 1 Dij Dij dx ˆ jgradvj2 dx: …76† 2 O O Combining (73), (75), and (76) gives (75).

Q.E.D.

Proof of proposition 7.1. We have, by a1R0 and (75) p 1 E‰v…
†Šr rjjvjj2L2 ‡ 3a1 jjvjj2L2 jjgradDjjL2 2 1 ‡ gjjgradDjjL2 2 where jgradDj2 ˆ Dij;k Dij;k :

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The right-hand side of (77) is a quadratic form in kvkL2. kgrad DkL2 and its discriminant is 3a1ÿrg and it is negative if and only if (69) holds. Hence, (69) implies that this form is positive de®nite, and hence, for some constant, E‰v…
†ŠrcjjgradDjj2L2 :

…78†

On the other hand, by (74), one has that jjgradDjjrcjjgrad2 vjjL2

…79†

for another constant c>0. Finally, combining (78), (79) with PoincareÂ's inequality, one sees that (70) holds. Q.E.D. We have thus shown that a tripolar ¯uid of grade 3 is stable even when a1<0, provided g has the right value. This result might have a bearing on the experimental results on nonNewtonian ¯uids that suppose the model (1) in a stable experimental situation. It is worthwhile to reevaluate the experimental data and tests for non-Newtonian ¯uids on the basis of the tripolar ¯uid of grade 3.

8. Asymptotic stability of the rest state Proposition 8.1. Suppose that m1r0, m2r0 with one of these inequalities strict, that f>0 and either a1r0 or a1R0 and g>3a21/r. Then there exist constants c>0, c1>0 such that for every solution v(
,
) of the equations and the boundary condition on O  [0, 1) one has jjv…
; t†jjW2;2 …O† Rc1 jjv…
; 0†jjW2;2 …O† eÿct

…80†

for every t r0. Proof. Denote by I(t) the second integral in (64) if m1r0, m2r0 and if one of these inequalities is strict, one has I…t†rc3 jjv…
; t†jj2W2;2 …O†

…81†

with a positive constant independent of the solution. On the other hand, clearly c4 jjv…
; t†jj2W2;2 …O† RE‰v…
†ŠRc5 jjv…
; t†jj2W2;2 …O†

…82†

for another two constants c4>0, c5>0 independent of the solution. Combining (81) with (82), then E(t) Rc6I9t) and using this together with (64) gives _ ‡ c2 E…t†R0: E…t† Then integration gives 2

E…t†RE…0†eÿc t and (82) then implies (80). Q.E.D.

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9. Conclusions Estimation of the value of g and the characteristic length. The basic inequality for the positiveness of the total energy 3 g > a21 r enables one to estimate the least possible value of g. g0

a21 r

…83†

from the known value of a1. Furthermore, the terms a1

and g

in the expression of energy  … 1 2 1 2 2 rv ‡ a1 jDj ‡ gjgradDj dx 2 2 show that the ratio g/a1 has the physical dimension of the square of length, L, g L2 ˆ ; a1

…84†

where L is the characteristic length. In the dynamical equation, the constants a1 and g enter the constitutive equation for T through, qualitatively, T0a1 A_ 1 ‡ gDA_ 1 : Hence a1, g are important primarily in dynamical processes, AÇ 1$0 and the characteristic length is important in the dynamical processes. Hence, the departures from the classical theory, with g = 0, are observed in processes in which AÇ 1 changes considerably at intervals of length L. From (83) and (84) a1 L2 ˆ : r Taking a10ÿ 1 g/cm, r = 1 g/cm3, we obtain the characteristic length L01 cm:

Acknowledgements Research supported, in part, by ONR Grant N00014-95-1-0438.

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References [1] W. Noll, On the foundations of the mechanics of continuous media. Carnegie Institute of Technology, Dept of Mathematics, Report 17, 1957. [2] A.E. Green, R.S. Rivlin, Simple force and stress multipoles, Arch Rat Mech Anal 16 (1964) 325±53. [3] A.E. Green, R.S. Rivlin, Multipolar continuum mechanics, Arch Rat Mech Anal 17 (1964) 113±47. [4] A.C. Eringen, Theory of micro polar ¯uids, J math Mech 16 (1966) 1±18. [5] C. Truesdell, W. Noll, The non-linear ®eld theoreies of mechanics. Flugge's Handbuch der Physik 3. Springer, Berlin, 1965. [6] J.E. Dunn, K.R. Rajagopal, Fluids of the Di€erential type: Critical review and thermodynamic analysis, Int J Engng Sci () (1994in) press . [7] B.D. Coleman, W. Noll, An approximation theorem for functionals, with applications in continuum mechanics, Arch Rat Mech Anal 6 (1960) 355. [8] R.S. Rivlin, J.L. Ericksen, Stress-deformations for isotropic materials, J Rat Mech Anal 4 (1955) 323. [9] R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of ¯uids of third grade, Proc R Soc Lond A339 (1980) 351. [10] M.C. Patria, Stability questions for a third-grade ¯uid in exterior domains, Int J Non-Linear Mech 24 (5) (1989) 451±7. [11] G.P. Galdi, M. Padula, K.R. Rajagopal, On the conditional stability of the rest state of a ¯uid of second grade in unbounded domains, Arch Rat mech Anal 109 (2) (1990) 173±82. [12] J.L. Bleustein, A.E. Green, Dipolar ¯uids, Int J Engng Sci 5 (1967) 323±40. [13] J. NecÏas, M. Silhavy, Multipolar viscous ¯uids, Q Appl Math XLIX (2) (1991) 247±66. [14] H. Bellout, F. Bloom, J. NecÏas, Phenomenological behavior of multipolar viscous ¯uids, Q Appl Math L (3) (1992) 559±83. [15] M. Silhavy, Private Communication, 1994. [16] J.L. Lions, Quelques methodes de resolution des problem au x limites non-lineaires. Dunod, Paris, 1969.