Substructural interactions and transport in polymer flows

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International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

Substructural interactions and transport in polymer %ows P.M. Marianoa;∗ , C.M. Casciolab , E. De Angelisb a Dipartimento

di Ingegneria Strutturale e Geotecnica, Universita di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy di Meccanica ed Aereonautica, Universita di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy

b Dipartimento

Received 12 June 2002; accepted 19 November 2002

Abstract A model for semidilute polymer %ows is developed within the setting of multi3eld theories describing material substructures. We associate a coarse grained order parameter to the family of polymer chains in each material element and account for substructural interactions which develop power in the rate of the order parameter and are balanced. A measure of substructural interactions occurring between neighboring families of polymeric chains is prescribed 3rst; then we 3nd the need of the existence of self-interactions in each family by means of a requirement of invariance of the power. We obtain balance equations that involve terms that stabilize numerical algorithms in turbulent regime. Versions of the standard dumbbell model, that 3t experimental data, seem to fall within our modeling. Moreover, we obtain evolution equations which are su9ciently %exible to be applied to di:erent (even non-standard) cases. In fact, we make distinction between the balance of (standard and substructural) interactions and their representation. Then, such a representation is a consequence of the prescription of two ingredients: the explicit form of the bulk free energy density and appropriate ‘viscous’ coe9cients. The transport of polymer chains between neighboring material elements is also discussed. ? 2003 Elsevier Ltd. All rights reserved.

1. Introduction Polymeric %uids are described variously, depending on the physical situation envisaged (see the review in [1]; see also [2,3]). For dilute polymer solutions, the interactions in the %uid due to the polymeric substructure are commonly considered to be purely local, in the sense that they originate within the family of molecules occupying each material element, while neighboring

∗ Corresponding author. Tel.: +39-06-44585276; fax: +3906-4884852. E-mail addresses: [email protected] (P.M. Mariano), [email protected] (C.M. Casciola), [email protected] (E. De Angelis).

material elements do not interact. On the contrary, in the case of concentrated solutions, the entanglements between molecules and the formation of star polymers may render non-local the interactions. In the case of semidilute solutions (see [1]), interactions may occur between neighboring material elements. They have a weak non-local character in the sense of the gradient. Such kind of interactions may also be generated by the turbulence even in dilute polymer solutions and can be the source of terms involving spatial gradients of the substructural descriptors in the evolution equations describing the substructural changes of shape of the families of polymer chains (see [4,5]). Various approaches have been followed to describe the interactions generated by the polymeric substructure and their balance (a comprehensive critical review is in [1]). Each approach starts from di:erent points

0020-7462/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0020-7462(02)00212-3

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P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

of view and considers prominent some aspects of the underlining physics instead of others. As a 3rst step, we associate to each material element a coarse grained descriptor (order parameter) of the family of chains within it. Let r be for each chain an end-to-end stretchable vector. We choose as order parameter some average R of the dyad r ⊗ r on the family of chains. To develop a model of interactions in semidilute polymeric %ows, we associate an appropriate measure of substructural interaction to the rate of the order parameter R and deduce balance equations from a requirement of invariance under changes of observer of the power of all interactions in the %uid. Then we obtain constitutive restrictions for the measures of interactions from a mechanical dissipation inequality. Explicit constitutive equations can be obtained either from some microrehological model through averages (and various examples are in the scienti3c literature; see, e.g., [6]) or starting from general basic concepts as the one of multiple natural con3gurations [7–9]. Versions of the standard dumbbell model seem to be special cases of our modeling. We do not refer to any speci3c rehologic model and make distinction between the balance of interactions and their constitutive representation. This allows us to underline clearly the class of interactions that can be envisaged in semidilute polymer %ows. The choice of any speci3c constitutive model permits (1) to distinguish what kind of interactions is considered and what kind is neglected, and (2) to evaluate the magnitude of the various interactions through experiments. The developments of these two points are outside the aim of the present paper because they are related to the speci3c modeling of any given special situation. In the balance equations obtained, terms involving the spatial derivatives of R appear as a natural consequence of the presence of ‘contact’ interactions between material elements containing families of chains. We then discuss the case of migration of chains from one material element to another: an evolution equation for the number density of chains is then obtained. The kinematics of polymeric %ows and the rules of changes of observers are described in Section 2. Balance equations for standard and substructural interactions are deduced in Section 3. Section 4 deals with constitutive equations. Transport phenomena are

discussed in Section 5. Concluding remarks and comparisons with some existing models are collected in Section 6. 1

2. An order parameter for the polymer uid Let B0 be the regular region of the Euclidean point space E3 occupied by the %uid in a con3guration taken as reference. Point of B0 are indicated with X. The standard deformation of B0 is described by continuous and piecewise continuously di:erentiable bijections xˆ : B0 → E3 such that ˆ 0 ) indicated with B is • the current con3guration x(B a regular region too; ˆ • the mapping x(·) is orientation preserving in the sense that the gradient ∇x of x (indicated in the following with F) is such that det F ¿ 0. The point x(X) shows the present placement of the material element placed at X in the reference con3guration. As usual, 3elds de3ned on B0 are called material while 3elds on B are called spatial. Any regular subset b of B is called part.

1 Notations. Let a, b and c be vectors. The vector product a × b determines a vector whose ith component is given by (a × b)i = eijk aj bk , where e is the Ricci’s alternating tensor and the summation over repeated indices is understood. Let A and B be second-order tensors. The product Aa is a vector such that (Aa)i = Aij aj ; AB is a second-order tensor and we have (AB)ij = Aik Bkj ; A · B is a scalar Aij Bij as well as a · b is a scalar ai bi too. Moreover tr A denotes the trace of A. The product eA is a vector such that (eA)i =eijk Ajk while ea is a second-order tensor such that (ea)ij = eijk ak . The tensor product a ⊗ b is a second-order tensor de3ned by (a ⊗ b)c = (b · c)a, i.e. (a ⊗ b)ij = ai bj . Two di:erent sets are considered in the following: B0 and B with geometrical properties described in the section below. X denotes points in B0 while x denotes points in B. The di:erential operators grad (gradient) and div (divergence) involve derivatives with respect to coordinates x while ∇ (gradient) with respect to X. We denote as usual with the operator divgrad and with Sym(R3 ; R3 ) the space of symmetric linear mappings of R3 into itself. Let C be any third-order tensor; Ca is a second-order tensor whose components are (Ca)ij = Cijk ak , and CT denotes the tensor whose components are Ckij . If C and D are third-order tensors, in our notations CD is a second-order tensor and we have (CD)ij = Cikl Dklj ; moreover C · D is the scalar product and furnishes a scalar. No distinction between covariant and contravariant components has been made so far in this footnote.

P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

In the standard kinematical picture of %uids just summarized no geometric information on the substructure generated by the polymeric chains has been assigned. To introduce such information, we imagine each material element P placed at X in the reference con3guration to be a box containing a population of chains. We do not treat problems of possible entanglement of polymeric chains. Each chain is described only (and perhaps roughly) by an end-to-end stretchable vector (see [1], and references therein) r. Then, at each point X of B0 we have a distribution function fP(X) (r) of r representing the population of chains within the material element P placed at X. The traditional use of the second-order tensor R[ (X) = r ⊗ rfP(X) (r)

(1)

as coarse grained geometrical descriptor (order parameter) of the population itself is suggested by the need to be indi:erent to the transformation r → −r. In (1), the parenthesis · denotes ensemble average on the family of chains described by fP(X) (r). We associate then to each material element not only its placement in E3 but also the descriptor R[ of the polymeric substructure. The mapping Rˆ [ : B0 → Sym(R3 ; R3 ) is assumed to be continuous and piecewise continuously di:erentiable. A spatial representation R of R[ is obviously available and is given by the mapping Rˆ : B0 → Sym(R3 ; R3 ), so that R = ˆ R(x(X)) is the value of the order parameter at x. The physical requirement that each polymer chain cannot be stretched in3nitely implies that R must belong to some convex subset M of Sym(R3 ; R3 ), as, e.g., a bounded ball centered at zero. Then, motions are time-parametrized families (twice di:erentiable in time t ∈ [0; d]) of placements and order parameter. Rates in the material ˙ representation are indicated by x(X; t) and R˙ [ (X; t), while in the spatial representation by v(x; t) and ˙ t). Obviously, we have v = x˙ and, by chain rule, R(x; R˙ = @t R + (grad R)v, where @t R is time derivative of R holding x 3xed. We consider the order parameter an observable object in the sense that it senses a change of observer: two external spatial observers O and O
, seeing the same con3guration, evaluate two di:erent pairs of rates, ˙ and (v
; R˙
), respectively. We consider namely (v; R) classical observers related by rigid body motions

459

described by time-parametrized di:erentiable families of mappings Qˆ : [0; d] → SO(3) (i.e., at each instant t, ˆ ∈ SO(3), where SO(3) is the orthogonal group). Q(t) ˙ The observer O evaluates x, R and the rates v, R.

The observer O evaluates a new placement x , a new value R
of the order parameter and the correspond
ing rates v
, R˙ . Since the relation between the two observers is ruled by the orthogonal group SO(3), we have x
= w(t) + Q(t)(x − x0 );

R
= QT (t)RQ(t); (2)

where w(t) is an arbitrary C 1 -mapping from [0; d] into E3 . Relation (2a) de3nes the isometry induced by SO(3) on E3 ; while (2b) is the standard transformation of second-order tensors induced by SO(3). By time di:erentiation, one obtains ˙ ˙ + Q(t)(x v
= w(t) − x0 ) + Q(t)v (3)
T ˙ ˙ (t)RQ(t): R˙ =Q˙ T (t)RQ(t)+QT (t)RQ(t)+Q

(4)


We now pull back the rates v
and R˙ in the observer O by the inverse transformation induced by QT and de3ne v∗ = QT v
and R∗ = QR
QT so as, putting ˙ c(t) = QT (t)w(t) and taking into account that QT Q˙ is a skew-symmetric tensor, we obtain the following rules of changes of observers: ˙ × (x − x0 ); v∗ = v + c(t) + q(t)

(5)

˙ R˙ ∗ = R˙ + Aq(t):

(6)

In (5) and (6), c(t) is the rigid translation velocity, ˙ the rotational velocity (i.e. the characteristic vecq(t) ˙ and the linear operator A is the in3nitesitor of QT Q) mal generator of the action of SO(3) on Sym(R3 ; R3 ), i.e. a third-order tensor whose covariant components are given by Aijk = eilj Rlk − Ril ejlk . While (5) is quite standard, the transformation (6) is peculiar of the choice of the order parameter assumed here. A treatment dealing with changes of observers ruled by the group of di:eomorphisms Di: can be also developed and will be matter of future work. 3. Balance of interactions Interactions occur between families of polymeric chains occupying adjacent material elements in

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P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

semidilute solutions. In fact, when the family of chains of a given material element su:ers a rearrangement of its topological distribution, it interacts with the neighboring families. The consequent ‘contact’ interactions develop power in the rate of con3guration change of each family, thus in the rate of R. In principle, we could consider even bulk external interactions on the families of chains, but, for the sake of simplicity, we neglect them. In the following, the word substructural quali3es the interactions generated between neighboring families and within each of them. A basic question is thus the explicit representation of these interactions. Following general issues of multi3eld theories ([10– 14], and references therein), we represent the ‘contact’ interactions between neighboring material elements through an appropriate third-order tensor S called microstress; then we 3nd the need of considering the existence of interactions within each family of chains as a consequence of the invariance of the power: these interactions are measured through a second-order tensor z. For any arbitrary part b of B, we then represent the power Pbext developed on b by all external standard and substructural interactions as

ext ˙ ˙ (Tn · v + Sn · R); (7) Pb (v; R) = b · v + b

@b

where b is the vector of standard bulk forces, T the Cauchy stress, n the outward unit normal to the boundary @b of b. S represents the interactions exchanged by neighboring material elements across the surface @b as a consequence of the rearrangement of the polymeric chains in at least one of them. To obtain balance equations we require that Pbext is invariant under changes of observers described by (5) and (6), i.e. ˙ Pbext (v∗ ; R˙ ∗ ) = Pbext (v; R)

(8)

˙ and b. for any choice of c(t), q(t) The immediate consequence of this axiom is that  


b+ Tn + q˙ · (x − x0 ) × b c· b


+

@b

@b

b

 ((x − x0 ) × Tn + AT Sn) = 0

(9)

˙ and b. The arbitrariness of for any choice of c(t), q(t) ˙ implies the integral balance laws c(t) and q(t)

b+ Tn = 0; (10)


b

b

@b

(x − x0 ) × b
+

@b

((x − x0 ) × Tn + AT Sn) = 0

(11)

Basically, (10) is the standard integral balance of forces, while (11) is a non-standard balance of couples. The arbitrariness of b implies the standard Cauchy balance b + div T = 0

in B:

(12)

To account for standard inertial e:ects, we require that the vector of bulk forces is the sum of inertial (in) a non-inertial (ni) contributions, namely b = bin + bni , and identify bin with −v˙ ( is the density of mass), so as to write (12) in the standard form ˙ bni + div T = v:

(13)

The arbitrariness of b and the validity of (12) imply also from (11) that eT = AT div S + (grad AT )S:

(14)

To explicit the nature of (14), we 3nd the need of the existence of an element z
of the cotangent space of M at each R such that (see [10,11,14]) AT z
= eT − (grad AT )S:

(15)

This allows us to write (14) as div S − z
= z



(16)

with z

another arbitrary element of the cotangent space of M at R such that AT z

= 0:

(17)

Note that z

can vanish identically: this happens when, at each R, the range of A covers the whole tangent space of M at R; on the contrary, some indetermination on z arises and may be eliminated case by case depending on M. The sum z
+ z

is indicated with z and called self-force. Since the microstress S represents the interactions between neighboring material

P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

elements, the self-force accounts for the interactions between chains in the same material element. In absence of bulk motion, the interactions exchanged between the members of the population at a given material element are balanced by the interactions exchanged with the populations of the neighboring elements. 4. Restrictions on the constitutive equations Following a common procedure in mechanics, we use a mechanical dissipation inequality which states that, for any part b and for any choice of the velocity 8elds, the rate of the free energy minus the external power is lesser or equal to zero; in symbols
d ˙ 6 0; − Pbext (v; R) (18) dt b where is the bulk density of the free energy. By making use of the balance equations (12) and (16), we may rewrite (18) as

d ˙ − (T grad v + z · R˙ + S grad R) dt b b 6 0:

(19)

The free energy is assumed to have at the thermodynamic equilibrium a constitutive structure of the type = ˆ (F; R; grad R): (20) However, since we are dealing with %uids, must remain unchanged under the action of the unimodular group (i.e. the group of all second-order tensors H with |det H| = 1) which is the fundamental group of material symmetries of %uids [15]. The invariance implies that must remain unchanged under the transformation F → FH. Since the tensor H = |det F|F−1 is obviously unimodular, we reduce (20) to ˆ (F; R; grad R) = ˆ (–; R; grad R) (21) where – = |det F| is the speci8c volume of the %uid. By considering b stationary and developing the time derivative of (21), the arbitrariness of b allows to reduce (19) to (–@– I − T + (grad R)T S) · grad v + (@grad R − S) · grad˙ R + (@R − z) · R˙ 6 0;

(22)

461

where @y means partial derivative with respect to the argument y and we have used the identities –˙=–I·grad v and grad˙ R = grad R˙ + (grad R) grad v. Since the inequality (22) is linear in the rates and their gradients, and must hold indeed for any choice of them, the following constitutive restrictions hold: T = –@– I − (grad R)T S;

(23)

z = @R ˆ (–; R; grad R);

(24)

S = @grad R ˆ (–; R; grad R):

(25)

Note that the term (grad R)T S plays an analogous rˆole of Ericksen’s stress in liquid crystals. Remark 1. Some special cases are of prominent interest. Namely, when we select a decomposed free energy of the form ˆ (–; R; grad R)= ˆ 1 (–)+ ˆ 2 (R)+ 1  grad R 2 2

(26)

with  some constant to adjust physical dimensions, the balance of substructural interactions (16) reduces to a Ginzburg–Landau equation for R: PR − @R ˆ 2 (R) = 0:

(27)

Moreover, a partially coupled form of the free energy like ˆ (–; R; grad R) = ˆ 3 (–; R) + 1  grad R 2 2

(28)

implies the balance PR − @R ˆ 3 (–; R) = 0:

(29)

Dissipation e:ects may be accounted for by assuming that some (or all) measures of interaction are the sum of a rate-dependent non-equilibrium part (ne) and an ‘equilibrium’ part (eq). This is, indeed, the standard procedure used to obtain Navier–Stokes equations: the dissipation inequality implies, in fact, that cannot depend neither on the velocity nor on its gradient. Consequently, viscous e:ects can be considered by assuming the existence of a non-equilibrium part of the stress which is intrinsically dissipative. We start

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P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

by considering the decompositions eq ˙ T=Tne (–; R; grad R; grad v; R)+T (–; R; grad R); (30)

˙ z=z (–; R; grad R; grad v; R)+z (–; R; grad R); ne

eq

(31) where Teq and zeq are given by (23) and (24). The free energy provides constitutive restrictions only on the rate-independent parts Teq and zeq . Then, we assume that Tne and zne satisfy the ‘strong’ dissipation inequality Tne · grad v + zne · R˙ ¿ 0:

(32)

Since the self-force z represents the self-interaction of each family of polymeric chains on itself at any given material element, the dissipative part of zne , namely zne , develops power in the relative motion of the population due to the %ow of the %uid. In other words, zne may account for possible ‘friction’ between each family of chains and the surrounding %uid (as in the basic assumptions of the standard dumbbell model). For this reason, we can write 

Tne · grad v + zne · R ¿ 0

(33)



instead of (32), where R is the Lie derivative of the 

tensor Rij following the velocity 3eld v: i.e. R = ˙ R−(grad v)R−R(grad v). Common rehologic micromechanical models are in general based on the assumption leading to (33) (see, e.g., [6]). A solution of the inequality (33) is given by 

Tne =a1 (–; R; grad R)grad v+a2 (–; R; grad R)R; (34) 

zne =a3 (–; R; grad R) grad v+a4 (–; R; grad R)R; (35) where ai , i = 1; 2; 3; 4, are appropriate positive de3nite functions of the state. In principle, ai may be fourth-order tensor valued or scalar functions. Here we consider only the case in which ai ’s (i = 1; 2; 3; 4) are scalars, being conscious that complicated situations may require the tensor case.

Remark 2. When a2 and a3 vanish identically, we may change (27) into the time-dependent Ginzburg– Landau equation 

a4 (–; R; grad R)R = PR − @R ˆ 2 (R)

(36)

and the balance of momentum reduces to the standard Navier–Stokes equation. An equation analogous to (36) can be obtained by assuming that the distribution function fP(X) (r) satis3es a Boltzmann equation (see, e.g., [16,17]) and calculating directly the average r ⊗ rfP(X) (r) . However, this statistical procedure does not allow us to obtain the term PR or more articulated terms involving spatial derivatives unless we consider some special Brownian transport motion of the polymeric chains (e.g. [18]). Our results point out that these type of terms are natural (often necessary) consequences of a rather detailed picture of substructural interactions because they arise from the weak non-locality induced by the ‘contact’ interactions measured by the microstress S. Remark 3. The constitutive restrictions (23)–(25) permit to clarify the role played by the unusual equation (15). In fact, if we consider (23)–(25) in analyzing (15), we 3nd that (15) reduces to the condition that the free energy be indi:erent when its arguments are subjected to the action of SO(3). Remark 4. Cases in which the families of polymeric chains induce a macroscopic pressure may occur. Let us consider, for example, a free energy of the form = ˆ (–; ), with  = –I + R. By developing the time derivative of this special form of , inserting it in (19) and considering the arbitrariness of b, taken as stationary, with the same procedure used to derive (23)–(25) from (22), we obtain T = (–@– + tr z)I; S = 0; z = @ . In this case, we note the occurrence of a macroscopic pressure induced by the polymers. Obviously, the choice of any special expression of the free energy should be suggested by the physics of special circumstances. 5. Transport of polymeric chains In the picture of the %uid presented above, we have implicitly assumed that there is no migration, i.e. the

P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

passage of a polymeric chain from one material element to another. We now consider migration phenomena since they occur in standard polymeric %uids. We indicate with n(x; t) the number density of polymeric chains, so that, for any part b, the integral  n(x; t) is the number of chains in b at the instant t. b We have the following equation of continuity:

d n(x; t) = !(x; t) · n: (37) dt b @b

463

Since (43) is linear in the rates which can be chosen arbitrarily, we deduce 3rst that @grad n = 0

(44)

i.e. we 3nd that the free energy must be independent of the gradient of the density of chains and must be of the form ˆ (–; R; grad R; n). Then, we obtain relations formally analogous to (23)–(24), but with the explicit dependence on n and

If we take b as stationary, its arbitrariness implies

Q = @n ˆ (–; R; grad R; n);

(45)

n˙ = div !:

! · grad Q ¿ 0:

(46)

(38)

In (37), the vector ! is the
(40)

where Q is the chemical potential per unit mass of polymeric chains. The use of (40) and the arbitrariness of b (taken as stationary) allow us to write a local version of (39) prescribing that

A possible solution of the inequality (46) is the following: ! = (–; R; grad R; n; ) Q grad ; Q

(47)

where (·) is a positive de3nite scalar function of state variables. As a consequence, the equation of continuity (37) changes as n˙ = div((–; R; grad R; n; ) Q grad(@n ˆ (–; R; grad R; n))):

(48)

When  is a constant Q and is interpreted as a coe9cient of migration (or, say, di:usivity), Eq. (48) reduces to n˙ = P Q Q

(49)

i.e. ! and grad Q are parallel. Cases in which  is a second-order tensor in (47) may occur in principle.

˙ − Qn˙ − ! · grad Q − T · grad v − S · grad R˙ − z · R˙ 6 0

(41)

for any choice of the rates involved. We now assume a constitutive structure for the free energy of the form = ˆ (–; R; grad R; n; grad n)

(42)

(instead of (21)), which allows us to write (41) as (–@– I − T + (grad R)T S) · grad v + (@grad R − S) · grad˙ R Q · n˙ + (@R − z) · R˙ + (@n − ) ˙ n − ! · grad Q 6 0: + @grad n · grad

(43)

6. Additional remarks and comparisons with previous models Terms stabilizing the evolution equation of the order parameter are a natural consequence of the modeling of ‘contact’ interactions between material elements. These interactions (measured by the microstress S) are weakly non-local because they are generated by the spatial variation of R and may occur even in absence of migration of polymeric chains from one material element to another. We note that standard versions of the dumbbell model fall within our modeling. For example, equations used in [16,19] imply a decomposition of Cauchy’s stress T in the sum of an

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P.M. Mariano et al. / International Journal of Non-Linear Mechanics 39 (2004) 457 – 465

equilibrium part (pressure) Teq , a Newtonian (viscous) part Tv and a non-Newtonian non-equilibrium part Tp (vanishing at equilibrium) due only to polymers in a way such that T = Teq + Tv + Tp . The evolution equation of R is then given by 

R = −sTp + p I;

(50)

where s is a constitutive constant and p a pressure depending on tr R. Eq. (50) and the associated modi3ed Navier–Stokes equation seem to be compatible with the choice of the free energy discussed in Remark 4 and with (34) and (35) in the case in which a3 = 0. Another example is given by the equations used in [17]. The decomposition T = Teq + Tv + TQ p is still assumed (with the polymeric extra stress TQ p vanishing at the equilibrium) but now TQ p is slightly di:erent from Tp . Eq. (50) changes as 

Q · R)TQ p : R = −s(R

(51)

One may verify that this case corresponds to a free energy of the form = ˆ (–; R), to the use of (34) and (35), and to the identity a2 = s(R Q · R)−1 . Equations more complicated than (36) can be deduced in the framework presented here. For instance, if we assume that even the microstress S has a non-equilibrium part (which corresponds to consider friction between neighboring families of polymeric chains), we may obtain evolution equations involving 

the spatial gradient of R. Boundary conditions on the order parameter can be of the type Q R(X) = R;

X ∈ @BR ;

S(X)n(X) = ;

X ∈ @B ;

and the external environment and endow it with surface energy. In this case we could derive variational boundary conditions by minimizing, e.g., the total energy of the %uid imagined at rest (perhaps with the help of some null-Lagrangians). If we decide to not describe directly the polymeric substructure of the %uid and to make use of the concept of multiple natural con3gurations to account for their in%uence on the macroscopic behavior indirectly, we need only to assign standard boundary conditions because the resulting model has the characteristic features of a Cauchy’s continuum (see the discussions in [20]). However, this point of view and the one in the present paper can be reconciled by means of Capriz’s concept of latence [10]. In fact, if we assume the existence of an internal constraint linking R with F, by using the general procedure in [11], we note that the balance of substructural interactions (16) disappears and the presence of substructural interactions between neighboring families of polymeric chains is measured through modi3cations of the constitutive structure of the Cauchy stress (and a possible appearance of a hyperstress in the balance of forces). When the free energy is assumed to be independent of grad R (in other words when we neglect interactions between neighboring families of polymeric chains) and latence occurs, in principle we may link directly the point of view presented in this paper with the one of Rajagopal and Srinivasa [20]. In this case, the boundary conditions are given only in the standard manner of Cauchy’s continua.

(52)

Acknowledgements

(53)

We thank Prof. Renzo Piva for many profound and useful suggestions. P.M.M. acknowledges also the support of the Italian National Group of Mathematical Physics (GNFM-INDAM).

where RQ and are prescribed values and when a mixed problem is considered we need to have @BR ∪ @B = @B and @BR ∩@B =∅. Although from a mathematical point of view (52) and (53) are natural, the basic problem is whether it is possible to prescribe through some experiments the values of RQ and . Basically, (52) and (53) have constitutive nature (above all (52)). However, we may also prescribe boundary conditions that have in part constitutive nature and in part procedural nature. This is the case in which we consider the boundary of the body as an interface between the body

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