Digital Object Identifier (DOI) 10.1007/s00161-004-0197-xContinuum Mech. Thermodyn. (2005) 17: 183–200

Original article

Conservation laws and constitutive relationsfor density-gradient-dependent viscous fluids

M.M. Mehrabadi1, S.C. Cowin2, M. Massoudi3

1 Department of Mechanical Engineering, Tulane University, New Orleans, LA 70118, USA2 The Center for Biomedical Engineering and The Department of Mechanical Engineering, The School of Engineering of the City

College and The Graduate School of the City University of New York, New York, NY 10031, USA3 U.S. Department of Energy, National Energy Technology Laboratory, P. O. Box 10940, Pittsburgh, PA 15236, USA

Received April 3, 2004 / Accepted December 6, 2004Published online March 4, 2005 – © Springer-Verlag 2005Communicated by K.R. Rajagopal

Abstract. Conservation laws and constitutive relations for a density-gradient-dependent viscousfluid as a multipolar continuum obeying an entropy inequality with generalized entropy flux andsupply density are considered in this paper. A decomposition of the rate of work of dipolar stress,which reveals the contribution of various parts of this stress to the energy equation, is used todiscuss the inconsistencies between the results obtained here and those obtained by Bluestein andGreen [1] on the basis of the pioneering work of Green and Rivlin [8]. Furthermore, we discussthe connection between the model presented here and the materials of Korteweg type consideredby Dunn and Serrin [6]. In particular, we relate the rate of work of dipolar stress and the interstitialworking introduced by Dunn and Serrin [6].

Key words: conservation laws, constitutive relations, density gradient, dipolar stress, interstitialworking

1 Introduction

In studying a variety of problems in fluid mechanics, and in the theory of convective heat and mass transfer intwo-phase systems with a mobile interphase boundary, one often encounters the so-called capillary phenomenon,which is caused by the existence of surface tension on the interphase boundary. Examples such as the retardationof droplets and bubbles moving in a liquid containing dissolved surface-active substances and the motion ofliquids in thin capillaries are to name but a few [11]. Capillary phenomenon in general occurs in two cases: (1)when the surface of separation possesses considerable curvature, and (2) when the surface tension varies frompoint to point on the surface. It is customary to represent this phase interface as a singular surface, rather thanas a three-dimensional region of some thickness and, in general, in order to consider the effects of capillarity,jump conditions are introduced. For example, if we consider the phase interface between two Newtonian liquidshaving different temperatures and densities, we can be certain that the stress tensor cannot be represented bythe Newtonian model in the interfacial region. Another class of problems where density variations may play asignificant role is in the theory of interacting continua, i.e., mixture theory. Within the context of this theory,constitutive relations are needed for stress tensors, interaction forces, heat flux vectors, etc. Density gradient can

Correspondence to: M.M. Mehrabadi (e-mail: mmm@tulane.edu)

184 M.M. Mehrabadi et al.

appear as an independent constitutive parameter in these relationships. The reason for including the gradient ofdensities in the diffusive body force, for example, is that in a fluid-solid system, or in a mixture of two gases,the particles need not be uniformly distributed. This point was established and elaborated by Muller [17], whoshowed that the omission of the density gradients would lead to models that are too restrictive.

The ‘gradient-type’ theories for fluids have appeared and have existed in continuum mechanics, at least, sincethe time of Stokes [22] and Maxwell [14]. Truesdell (see, Truesdell and Noll [25, p. 551] gave a generalizationof ‘Stokesian’ and ‘Maxwellian’ fluids. Rivlin and Ericksen [21] provided a scheme for grade-type fluids ofthe differential type, while Oldroyd [19] did the same for the rate-type fluids. A recent review of the fluidsof differential type is given by Dunn and Rajagopal [5]. To replace the classical theory of capillarity, whichspecifies a jump condition at the surface separating homogeneous fluids possessing different densities, Korteweg[10] proposed smooth constitutive equations for the Cauchy stress that included density gradients. His equationin modern notation is (cf. [25, p. 514]):

Tij = (−p + αρ,kρ,k + βρ,kk)δij + γρ,iρ,j + νρ,ij + λDkkδij + 2µDij, (1.1)

where p,α,β,γ, ν,λ, andµ, are functions of the density,ρ, and the temperature, θ, only. It is interesting to note thatif the density ρ is replaced by the temperature θ, then one obtains equations that are very similar to the constitutiveequations proposed by Maxwell [14] for stresses in rarified gases that arise from inequalities of temperature.Korteweg’s model, however, is incompatible with conventional thermodynamics unless the coefficients α, β, γ,and ν vanish identically (see, e.g., [6]).

In [1], Bluestein and Green considered a dipolar fluid model as an application of the multipolar continuummodel presented by Green and Rivlin [8]. Motivated by the generalized elasticity theories, to resolve the difficultyin formulating the boundary conditions in “Maxwellian” type theories, Bluestein and Green [1] considered stresstensors of higher rank than the second to be present in their dipolar theory since they had included density gradientsin the free energy function. In the Maxwellian type theories, a single second rank stress tensor of the classicaltype is used with second and higher gradients of velocity in the constitutive equations. These higher gradientsincrease the order of the differential equations of motion with no indication as to the appropriate form of theboundary conditions. Multipolar theories, on the other hand, include a derivation of the appropriate boundaryconditions. In the present work, we return to the starting point of Bluestein and Green [1] and consider thedensity-gradient dependent viscous fluids in the context of the dipolar continuum model of Green and Rivlin [8]and Green and Naghdi [7]. We introduce an equivalent (symmetric) stress which consists of the symmetric partof the Cauchy stress and a part of the couple-stress tensor. The significance of this equivalent stress is that it isderivable from the free energy function and that one can write the conservation laws in their classical forms.

Dunn and Serrin [6] and Dunn [4] have also considered the thermodynamics of gradient materials includingthe elastic materials of grade three and a subclass of materials they refer to as the “materials of Korteweg type”.The theory of density gradient dependent viscous fluids considered in this paper is closely related to the theoryof materials of Korteweg type. As described in the following two paragraphs, however, there are two distinctionsbetween the two theories.

First, in the work of Dunn and Serrin [6] and Dunn [4], no higher order stresses are explicitly present and theobjectivity of the free energy function leads to the symmetry of Cauchy stress. Dunn and Serrin [6] and Dunn[4] intentionally confine their attention to developing a thermodynamic structure which retains the classicalforms of purely mechanical principles of linear and angular momentum balance, as well as the purely thermalClausius-Duhem inequality. The local net force and the net local torque actions exerted on the boundaries arealso presumed to be delivered in the standard way by the symmetric Cauchy stress tensor. The only modificationthey make is to introduce in the energy equation an additional term called the interstitial working, u, as the rateof supply of mechanical energy across every material surface of the body. The interstitial working is, in addition,to the working of the tractions on the surface and the flow of heat through the surface of the body and is supposedto represent longer range mechanical interactions similarly to the long-range radiation heat supply to the body,r, in the classical form of the energy equation.

The second distinction between the class of materials considered here and the materials of Korteweg type isthat we employ, instead of the Clausius-Duhem inequality, the entropy inequality proposed by Muller [16,18]and Liu [12,13] in order to relate the working of the dipolar body force and the radiation over temperature to theentropy supply (see Sect. 5). The entropy flux is also more general than the classical heat flux over temperatureand it includes, in addition, the working of the dipolar stresses. We should point out, however, that except for thisinterpretation of the entropy flux and the entropy supply, all other results reported here, including the restrictions

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 185

on the response functions, could be inferred by using the classical Clausius-Duhem inequality along with thestandard arguments of Coleman and Noll [3]. Employing the Clausius-Duhem inequality, Dunn and Serrin [6]show that under the conditions of local equilibrium, the Cauchy stress has the representation

Tij|E = −ρ[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ,k

),k

]δij − ρ

∂ψ

∂ρ ,iρ,j . (1.2)

As discussed by Dunn and Serrin [6] and Dunn [4], this equation has several interesting features. For example,although thermodynamic considerations prevent the free energy from depending on the second gradient ofdensity, the stress tensor depends on this parameter through the divergence term in the spherical part of stress.Another important feature of the stress given by (1.2) is that it admits a potential function so that the threethird order partial differential equations of equilibrium, which usually lead to an overdetermination of density,can be replaced by a single partial differential equation of second order. Comparing (1.2) and (1.1), Dunn andSerrin [6] observe that Korteweg’s model is incompatible with their thermodynamics considerations unless thecoefficientsα , β, and ν are related in a specific manner and γ vanishes identically. As mentioned earlier, we findit very useful to introduce a symmetric second-rank tensor, which we refer to as the equivalent stress tensor,Te. This quantity is obtained by summing the Cauchy stress and the divergence of the couple stress part of thedipolar stress. The significance of the equivalent stress tensor is that it is symmetric and it appears in place of theasymmetric Cauchy stress, in the linear momentum equations. We show that under the conditions of equilibrium,the equivalent stress can be obtained from the free energy function using the relation

Teij|E = −ρ

[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ ,k

),k

]δij − ρ

∂ψ

∂ρ ,iρ,j . (1.3)

The right-hand side of (1.3) coincides with (1.2). Furthermore, the equivalent stress tensor given by (1.3) hasall the desirable properties of the Caushy stress tensor given by (1.2). In particular, it is derivable from the samepotential function and it helps reduce the number and the order of the differential equations of equilibrium.

The results presented here are mostly based on an unpublished work that was first collected in a NSF report[15] with the intention of eventually applying it to granular materials. We have decided to publish it at this timebecause of the recent renewed interest in gradient theories for describing the inelastic behavior of materialsincluding granular materials. As mentioned earlier, a general shortcoming of the gradient theories when thestresses of higher order than the second are not included is the difficulty in specifying the relevant boundaryconditions such that they can be measured and quantified. On the other hand, a shortcoming of theories thatinclude stress tensors of higher order than the second is the plethora of material moduli that are needed torepresent, in addition to the usual stress tensor, the third-rank dipolar stress tensor. The skew-symmetric part ofthe Cauchy stress and the completely skew-symmetric part of the dipolar stress should only satisfy the angularmomentum equation and the boundary conditions. The only thermodynamic restriction on the remaining part ofthe dipolar stress that we find in this paper is a relationship between the rate of working of this stress, and theflux of heat and entropy across the surface of the body. In closing this paragraph, we should also mention that analternative and promising theory called the multiple natural configuration theory has recently been developed byRajagopal and his co-workers and has been successfully employed in many applications [20]. The basic premiseof this theory is that materials which possess a certain microstructure, in general, have numerous evolving naturalconfigurations associated with the deformed states of the body.

The outline of the paper is as follows: In Sect. 2 we present a summary of the basic definitions of the kinematicvariables. In Sect. 3, the conservation laws are presented for a general dipolar continuum. In Sect. 4, we discussthe decomposition of the rate of work of dipolar stress, introduce the equivalent stress tensor, and summarize thefield equations and the boundary conditions of the theory. In Sect. 5, the entropy inequality is briefly discussedand an expression for the entropy supply is derived assuming that the entropy supply is linearly related to otherbody supplies. We find that in addition to the radiation heat supply, the working of the symmetric part of thedipolar body force can also be a source of entropy supply. In Sect. 6, we discuss the constitutive relations andderive restrictions imposed by the entropy principle on various response functions. The equilibrium conditionsare discussed in Sect. 7, where the similarities between the results obtained here and those obtained by Dunnand Serrin [6] are discussed. Section 8 is devoted to Summary and Conclusions.

186 M.M. Mehrabadi et al.

2 Preliminaries

The motion of the continuum is referred to a fixed set of rectangular Cartesian axes. The coordinates of a typicalparticle of the continuum at time t are denoted by xi (i=1,2,3). The components of the velocity of the particle attime t are denoted by vi . The symmetric and the skew symmetric (spin tensor) part of the velocity gradient aredenoted by Dij and Wij , respectively, and are defined in the usual manner, i.e.,

Dij =12(vi,j + vj,i), Wij =

12(vi,j − vj,i), (2.1)

where in (2.1) and in the sequel, a comma denotes differentiation with respect to xi. Material differentiation withrespect to time will be denoted either by d/dt, or by a superposed dot. The invariants of Dij are defined as

ID = trD, IID =12[(trD)2 − trD2], IIID = detD. (2.2)

As mentioned in the introduction, the mass density of the body is denoted by ρ and the invariants of the seconddensity gradient, ρ,ij, are denoted by I ρ, II ρ, and III ρ, and are defined similarly to (2.2) by (1.4)3.

Throughout the paper, lower case Latin indices are associated with coordinates xi (i=1,2,3). A square bracketenclosing a number of free and dummy indices indicates that the tensor is completely antisymmetric with respectto the enclosed free indices. Parentheses enclosing a number of free and dummy indices indicate that the tensoris completely symmetric with respect to the enclosed free indices.

3 Conservation laws

Consider an arbitrary material volume V of the continuum bounded by a surface A at time t. We assume that thecontinuum admits monopolar and simple dipolar body forces and monopolar and simple dipolar stresses. Thenas a special case of the general energy balance postulated by Green and Rivlin [8,9] for a multipolar continuum,we have for the present case

ddt

∫Vρ

(ε+

12vivi

)dV =

∫A(tijvj,i + tivi − q)dA+

∫Vρ(bijvj,i + bivi + r)dV, (3.1)

where ε is the specific internal energy, dV is an element of volume, tij are the components of the dipolar traction,ti are the components of the monopolar traction, q is the heat flux across the boundary, per unit area per unittime, dA is an element of area of the boundary, bij are the dipolar body forces per unit mass, bi are monopolarbody forces per unit mass, and r is the heat supply function per unit mass per unit time.

Taking the volume V in (3.1) to be a tetrahedral element and employing arguments similar to those of Greenand Rivlin [8], we find the boundary conditions

ti = nj Tji on A, (3.2)

t[ij] = nk Tk[ij] on A, (3.3)

[ t(ij) − nk Tk(ij) ] Dij = q − qi ni on A, (3.4)

where n is the unit normal to a surface whose monopolar and dipolar tractions are ti and tij . Tij are the monopolarstress components, Tijk are the dipolar stress components, and qi are heat fluxes across the coordinate planes,per unit area of the planes per unit time.

Requiring invariance of (3.1) under rigid-body motions as in Green and Rivlin [8], assuming sufficientsmoothness, one obtains the following field equations

ρ + ρ vi,i = 0 in V, (3.5)

Tji,j + ρ bi = ρ vi in V, (3.6)

Tk[ij],k + T[ij] + ρ b[ij] = 0 in V, (3.7)

ρε = T(ij) Dij + Tk(ij),k Dij + T(ij)k vk,ji − qi,i+ ρb(ij) Dij + ρr in V, (3.8)

where (3.5-8) are the local forms of the equations of conservation of mass, conservation of linear momentum,conservation of angular momentum, and the conservation of energy principles, respectively.

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 187

Equations (3.2)-(3.8) which constitute the boundary conditions and the field equations for a dipolar continuaare similar to those originally derived by Green and Rivlin [8] with two seemingly minor differences. First, inthe work of Green and Rivlin, the boundary condition (3.4) is assumed to be decoupled so that

q = qi ni on A, (3.9)

andt(ij) = nk Tk(ij) on A. (3.10)

Second, they combine the distinct terms T(ij) Dij and Tk(ij),kDij in the energy equation (3.8) by introducing asymmetric quantity τij ,

τij = Tkij,k + Tij + ρbij = τji , (3.11)

which they use to write the energy equation (3.8) as:

ρε = τij Dij + T(ij)kvk,ji − qi,i + ρr. (3.12)

As mentioned in the introduction, when (3.12) is used in the entropy inequality to obtain restrictions on theconstitutive responses, it is important to take into account the divergence of the dipolar stress, namely, Tk(ij),kthat is hidden in the quantity τij through its definition (3.11). Assuming a constitutive equation for τij andignoring the divergence term Tk(ij),k in the energy equation (3.8), as is customarily done in multipolar theories,would result in severe restrictions on constitutive equations. In particular, we will show that the working of thisdivergence term, namely, Tk(ij),kDij, constitutes a part of the interstitial working of Dunn and Serrin [6]. This isdiscussed further in the next section in connection with the decomposition of the working of the dipolar stressesTi(jk),i Djk, and T(ij)kvk,ji that appears in the energy equation (3.8).

4 Decomposition of the rate of work of dipolar stress

As mentioned earlier, the equations of the (simple) multipolar theory of Green and Rivlin [8] were specializedto the case of a dipolar fluid by Bluestein and Green [1] who studied the problem of isothermal Poiseuille flowin a capillary tube. The properties of the dipolar stress, Tijk , were investigated by Green and Naghdi [7] whocompared the theory of Green and Rivlin [8] with that of Toupin [23,24]. Green and Naghdi [7] emphasized thesignificance of the contribution of the antisymmetric part of the dipolar stress, T[ij]k , to the surface boundaryconditions. Consistent with the earlier remarks of Green and Rivlin [8], Green and Naghdi [7] suggest that onlythe part T(ij)k (with 18 independent components) of the dipolar stress Tijk (with 27 independent components)contributes to the energy equation while T[ij]k (with nine independent component) makes no contribution to it.This suggestion was based on the energy equation (3.12). As noted before, however, the term τij Dij in (3.12)contains a contribution by the dipolar stress. In fact, the only part of the dipolar stress that does not contributeto the energy equation is its completely antisymmetric part T[ijk]. To show this, first note that since the spaceof completely skew-symmetric tensors is one dimensional, T[ijk] has only one independent component and canthus be represented as [23]

T[ijk] = m eijk , (4.1)

where eijk are the components of the permutation tensor, and where m is a scalar. Next, define

Hijk = Ti(jk) , Mijk = Ti[jk] , (4.2)

so thatTijk = Hijk + Mijk . (4.3)

The tensor Mijk is similar to Toupin’s [23] couple-stress tensor1 and is given by the relation (see, Appendix A;also [23], Eqs. 4.17 and 4.20)

Mijk = meijk +23(M(ij)k − M(ik)j). (4.4)

This shows that the 27 components of Tijk are equivalent to the 18 independent components of Hijk, the eightindependent components of M(ij)k, and the one independent component, m, of T[ijk] . Next, note that

T(ij)k vk,ji = ( Hijk − 2 M(kj)i) Dkj,i , (4.5)

1 Toupin’s couple-stress tensor is skew-symmetric with respect to the first two indices. The quantity Hijk defined by (4.2) is alsosimilar to Toupin’s [23, p. 404], Hijk .

188 M.M. Mehrabadi et al.

where the identityWkj,i = Dki,j − Dji,k , (4.6)

which follows from (2.1) has been used. Finally, substituting from (4.2) and (4.5) into the energy equation (3.8)and rearranging, we find

ρε = T(ij) Dij + (Hijk Dkj), i − 2 M(kj)iDkj,i − qi,i + ρb(ij)Dij + ρr . (4.7)It can be seen from (4.7) that the completely antisymmetric part of the dipolar stress, T[ijk], and the antisymmetricpart of the Cauchy stress make no contributions to the energy equation. Later in this section, we find that thesequantities make no contributions to the linear momentum equation either but they do play a significant role inthe angular momentum equation and the traction boundary conditions.

Note that (4.7) can also be written asρε = [T(ij) + 2 M(ij)k,k] Dij + [(Hijk − 2 M(kj)i)Dkj], i − qi,i+ ρb(ij)Dij + ρr , (4.8)

orρε = Te

ij Dij + uei, i − qi, i + ρb(ij)Dij + ρr , (4.9)

whereTe

ij = T(ij) + 2 M(ij)k,k , (4.10)

andue

i = (Hijk − 2 M(kj)i) Djk . (4.11)1We refer to Te as the equivalent (symmetric) stress and to ue as the equivalent interstitial work flux because aswe will see shortly, the equivalent stress appears in the momentum equation in place of the asymmetric Cauchystress and the equivalent interstitial work flux is the counterpart of the interstitial work flux of Dunn and Serrin[6] within the framework of multipolar continuum mechanics. When the couple-stress is not incorporated intothe theory as is the case in the theory of Dunn and Serrin [6], we obtain for the interstitial work flux of Dunnand Serrin, u,

ui = Hijk Djk . (4.11)2In terms of the couple-stress tensor, M ijk , the conservation of angular momentum equation (3.7) takes the

formMkij,k + T[ij] + ρb[ij] = 0 . (4.12)

Substituting from (4.12) into (3.6), using the identity,M(kj)i + M(ki)j + M(ij)k = 0 , (4.13)

we find that[T(ij) + 2M(ij)k,k],j + ρbi + (ρb[ij]),j = ρvi . (4.14)

Defining an equivalent body force bei , where

ρbei = ρbi + (ρb[ij]),j , (4.15)

and employing definitions (4.10) and (4.15), the equations of motion (4.14) will take the classical formTe

ij,j + ρbei = ρvi . (4.16)

An expression for the equivalent stress tensor, Te , in terms of the free energy function under equilibriumconditions will be derived in Sect. 7.

To recapitulate, we can rewrite the field equations of the theory for a dipolar continuum asρ + ρ vi,i = 0 , (3.5)R

Teij,j + ρbe

i = ρvi , (4.16)RMkij,k + T[ij] + ρb[ij] = 0 , (4.12)R

ρε = Teij Dij + ue

i, i − qi,i + ρb(ij)Dij + ρr , (4.9)Rwhile using (3.7), (4.2), (4.4), (4.10), and (4.12), the boundary conditions (3.2-4) are written as

ti = Te njij + [ m ekij +

43

( M(ik)j − M(ij)k)],k nj + ρ b[ij]nj , (4.17)

t[ij] = nk Mkij , (4.18)

t(ij) Dij − q = ni ( Hijk Dkj − qi ) . (4.19)

Note that in (4.17), the integral over the surface of the quantity in brackets vanishes.

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 189

5 Entropy inequality

To infer thermodynamic restrictions on the constitutive equations, we postulate an entropy production inequalityof the form proposed and developed by Muller [16,18]), and Liu [12,13]∫

Vρη dV +

∫AφdA −

∫V

s dV ≥ 0 , (5.1)

where η is the specific entropy, φ is the surface entropy flux out of the volume per unit area, per unit time, and sis the entropy supply density. The inequality (5.1) reduces to Clausius-Duhem inequality when the entropy fluxis set equal to the heat flux divided by temperature and when the entropy supply density is set equal to the heatsupply divided by temperature, i.e., when

φ =qθ

and s =ρrθ

, (5.2)

where θ � 0 is the absolute temperature. If we let pi denote the entropy flux across the coordinate planes, onecan then employ an argument similar to that of Green and Rivlin [8] to show that

φ = pi ni , (5.3)

and thatρη + pi,i − s ≥ 0 . (5.4)

As mentioned in the introduction, we can employ the Clausius-Duhem inequality, in place of (5.1), and obtainthe same restrictions on constitutive response functions for the class of materials we consider in this paper. Weprefer (5.1), however, because it allows us to establish the relationship between the conventional thermodynamicquantities such as heat flux and the non-traditional parameters such as the working of dipolar stresses, or theinterstitial working of Dunn and Serrin [6]. In particular, we prefer to use the arguments of Muller [16,18] andLiu [12,13] concerning the external supplies and the entropy supply. Unlike the specific entropy, η, and theentropy flux, pi, appearing in (5.4), the entropy supply s is not usually a constitutive quantity but it depends onthe external supplies of the body. To get around this difficulty, Muller considers supply-free bodies and assumesthat the entropy supply vanishes when the external supplies do, i.e.,

s = 0, when bi = 0 , bij = 0 , and r = 0 . (5.5)

The idea is that since the response functions are independent of external supplies, the thermodynamic restrictionson such functions may be based on the consideration of supply-free bodies. For classical and relativistic fluids,Liu [13] has shown that once these thermodynamic restrictions are found, restrictions are found, it is then possibleto derive an expression for s, if this quantity is assumed to be linear in the external supplies. For the class ofproblems considered here this assumption implies that

s = λij bij + λi bi + λ r . (5.6)

where λij , λi , and λ are functions of the constitutive variables (to be specified in the next section). Employing(5.6) , the inequality (5.4) takes the form

ρη + pi,i − λij bij − λi bi − λr ≥ 0. (5.7)

Introducing the free energy ψ asψ = ε − θη , (5.8)

(5.7) becomes

−ρ(ψ + ηθ) + T(ij)Dij − 2M(kj)iDkj,i + ai,i − piθ,i

+(ρDij − θλij)bij − θλibi + (ρ− θλ)r ≥ 0, (5.9)

where (4.7) and (5.6) have been employed, and where

ai = HijkDkj + θpi − qi. (5.10)

Assuming that the response functions and their derivatives appearing in (5.9) are independent of the externalsupplies and requiring that (5.9) holds for arbitrary bij, b i , and r, we should have

λij =ρ

θDij , λi = o , λ =

ρ

θ. (5.11)

190 M.M. Mehrabadi et al.

Substituting for λij, λi, and λ , from these equations into (5.6), we find that

s =ρ

θb(ij) Dij +

ρrθ, (5.12)

which shows that the working (of the symmetric part) of the dipolar body force contributes to the entropy supplyjust as the long range radiation source, r. This is similar to the interstitial working of Dunn and Serrin [6], whichis supposed to represent longer range mechanical interactions and is in addition to the flow of heat through thesurface of the body.

Now employing (5.11) in (5.9), this inequality reduces to:

− ρ(ψ + η θ) + T(ij)Dij − 2 M(kj)iDkj,i + ai,i − pi θ,i ≥ 0 . (5.13)

We will use this inequality in the next section to infer restrictions on the constitutive equations. Note that (5.13)is derived using the energy equation (4.7) rather than (4.9) because to find the restrictions on the constitutiveresponses, it is easier to deal with the divergence of the vector a than the divergence of the couple stress M.

6 Materials of Korteweg type and dipolar stresses

Within the framework of the thermodynamics postulated in Sect. 5, we reconsider the materials of Kortewegtype of Dunn and Serrin [6] and discuss the formulation of constitutive relations when dipolar stresses arepresent. Among other things, we will show that although the governing equations are very similar for a materialof Korteweg type and a dipolar fluid, the presence of dipolar stresses result in a less restrictive form for theinterstitial work flux.

We consider constitutive response functions for the free energy, the entropy, the heat flux, the entropy flux,the parts of Cauchy stress and the dipolar stress that contribute to the energy equation (4.9) and appear in theinequality (5.13). Thus we consider the class of materials for which, at any point and time, the set of responsefunctions

Σ′ = { ψ ; η ; T(ij) ; M(ij)k ; qi ; pi ; Hijk} , (6.1)

or alternatively, from (5.10), the set

Σ = { ψ ; η ; T(ij) ; M(ij)k ; qi ; pi ; ai}, (6.2)

depend on the following set of constitutive parameters:

{ ρ ; θ ; ρ,i ; θ,i ; ρ,ij ; vi,j } , (6.3)

the dependence being single-valued and sufficiently smooth. The above set is identical to the set of constitutiveparameters of the materials of Korteweg type of Dunn and Serrin [6]. The response functions (6.2) will alsobecome similar to theirs if we delete the dipolar stress, M, and interpret a as the interstitial work flux.

Requiring invariance under rigid body motions it then follows that Σ or Σ′ can depend only on the objectiveset

Z = { ρ ; θ ; ρ,i ; θ,i ; ρ,ij ; Dij } , (6.4)

and that the functionsΣ = Σ ( Z ) or Σ′ = Σ′ ( Z ) , (6.5)

are hemitropic in the set of independent variables Z. Assuming further that the material possesses a center ofsymmetry, the functions Σ and Σ′ must then be isotropic in their tensor arguments. Note that by employing theentropy inequality (5.13), we can infer restrictions only on the constitutive equations for the symmetric part ofstress, T(ij) and the part M(kj)i of the couple-stress tensor Mijk .

With the constitutive equations (6.5), and the relations (2.1), (3.5), and

(ρ,i)· = −(ρ,iDkk + ρ,kvk,i + ρDkk,i), (6.6)

(ρ,ij)· = −(Dkkρ,ij + ρ,iDkk,j + Dkk,iρ,j + ρDkk,ij+vk,jρ,ik + vk,iρ,kj + vk,ijρ,k), (6.7)

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 191

the inequality (5.13) becomes

−ρ(∂ψ∂θ

+ η)θ +

[T(ij) +

(ρ2 ∂ψ

∂ρ+ ρρ,k

∂ψ

∂ρ,k+ ρρ,kl

∂ψ

∂ρ,kl

)δij

+ ρρ,j∂ψ

∂ρ,i+ 2ρρ,jk

∂ψ

∂ρ,ki

]Dij+ +

[(ρ2 ∂ψ

∂ρ,i+ 2ρρ,k

∂ψ

∂ρ,ki

)δkj

+ ρ( ∂ψ

∂ρ,ijρ,k + +

∂ψ

∂ρ,ikρ,j − ∂ψ

∂ρ,kjρ,i

)+

∂ai

∂Djk− 2M(kj)i

]Dkj,i

+ρ2 ∂ψ

∂ρ,ijDmm,ij + ρ(

∂ψ

∂ρ,iρ,j + 2

∂ψ

∂ρ,ikρ,kj)Wij − ρ

∂ψ

∂DijDij

−ρ ∂ψ∂θ,i

(θ,i)· +∂ai

∂θ,jθ,ij +

∂ai

∂ρ,jkρ,ijk +

∂ai

∂ρρ,i

+∂ai

∂ρ,jρ,ij +

(∂ai

∂θ− pi

)θ,i ≥ 0. (6.8)

Since the values of θ , Dij , Dmm,ij , (θ,i)· , and Wij , can be specified independently of any other term in theabove inequality, we must have the following relations

∂ψ

∂θ+ η = 0 ,

∂ψ

∂Dij= 0,

∂ψ

∂ρ,kj= 0 ,

∂ψ

∂θ,i= 0,

∂ψ

∂ρ,[iρ,j] + 2

∂ψ

∂ρ,[ikρ,kj] = 0. (6.9)

From (6.9)1−4 it follows thatψ = ψ(ρ, θ, ρ,i) (6.10)1

Note that since ψ given by (6.10) must be an isotropic function of ρ,i, i.e.,

ψ = ψ(ρ, θ, I), (6.10)2where I is defined by (1.4)1, (6.9)5 is identically satisfied. Similarly, one can argue that the values of Dkj,i , θ,ijand ρ,ijk can be chosen independently of any other term in (6.8). Hence[

ρ2 ∂ψ

∂ρ,iδkj +

∂ai

∂Djk− 2M(kj)i

]Dkj,i = 0 (6.11)

∂ai

∂ρ,jkρ,ijk = 0. (6.12)

∂ai

∂θ,jθ,ij = 0 , (6.13)

Employing (6.9) through (6.13), the inequality (6.8) now becomes[T(ij) +

(ρ2 ∂ψ

∂ρ+ ρρ,k

∂ψ

∂ρ,k

)δij + ρ

∂ψ

∂ρ,iρ,j

]Dij+

∂ai

∂ρρ,i +

∂ai

∂ρ,jρ,ij +

(∂ai

∂θ− pi

)θ,i ≥ 0. (6.14)

The residual inequality (6.14) and the restrictions (6.9) through (6.13) become identical to those for the materialsof Korteweg type when a is interpreted as the interstitial work flux and the part M(kj)i of the couple-stress tensorMijk appearing in (6.11) vanishes. We will see later in this section that the presence of M(kj)i in (6.11) allows a todepend on the rate of deformation, D, in a rather general manner. From (6.11), the part M(kj)i of the couple-stresstensor satisfies the following equation

2M(kj)i =∂ai

∂Djk+ ρ2 ∂ψ

∂ρ,iδkj . (6.15)

Since a depends on the set of constitutive variables (6.4), using (6.12) and (6.13), it follows that

∂ai

∂ρρ,i +

∂ai

∂ρ,jρ,ij +

∂ai

∂θθ,i = ai,i − ∂ai

∂DjkDjk,i = ai,i − 2M(kj)iDkj,i + ρ2 ∂ψ

∂ρ,iDkk,i , (6.16)

192 M.M. Mehrabadi et al.

where (6.15) has been used to write (6.16)2. Substituting from this equation into the inequality(6.14), makinguse of the definition of the equivalent stress tensor (4.10), the residual inequality takes the form{

Teij + ρ

[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ,k

),k

]δij + ρ

∂ψ

∂ρ,iρ,j

}Dij +

[ai − 2M(kj)iDkj + ρ2 ∂ψ

∂ρ,iDkk

],i

− piθ,i ≥ 0. (6.17)

Recalling the identity (4.13), the completely symmetric part of the right hand side of (6.15) should vanish, thus

∂a(i

∂Djk)= 0 , (6.18)

where

ai = ai + ρ2 ∂ψ

∂ρ,iDmm . (6.19)

Equations (6.18), (6.12), and (6.13) must be solved for a. Substitution of a in (6.15) would then yield an expressionfor M(kj)i. The solution of (6.18) and (6.12) can be readily found by employing a result obtained by Dunn andSerrin [6] (see their Appendix C) who show that the vector a satisfying an equation of the type (6.18) should beof the form

ai = a(1)i + a(2)

ijk Djk + D∗ik a(3)

k , (6.20)

where a(Γ), (Γ = 1,2,3), are independent of D, D∗ is the adjucate of D given by

D∗ij = Dik Dkj − ID Dij + IID δij , (6.21)

and wherea(2)(ijk) = 0 . (6.22)

Substituting from (6.19) into (6.20) and solving for a yields:

ai = a(1)i + a(2)

ijk Djk + D∗ik a(3)

k − ρ2 ∂ψ

∂ρ,iDmm . (6.23)

The solution of (6.12) is of a form similar to (6.18) with D replaced by the second gradient of density. Substituting(6.23) into (6.12) and (6.13), using the representation theorems for isotropic functions [26,27] and the repre-sentation theorem of Dunn and Serrin [6] mentioned above, we find that (6.23) will satisfy all three equations(6.18), (6.12), and (6.13) if

a(1)i = r(1)i + r(2)ijk ρ,jk + ρ

∗ r(3)kik , (6.24)

r(2)(ijk) = 0 , (6.25)

whereρ∗ij = ρ,ik ρ,kj − Iρ ρ,ij + IIρ δij , (6.26)

where Iρ and IIρ are the invariants of the second gradient of density, and where a(2), a(3) of (6.23), and r(1), r(2),and r(3) of (6.24) are all independent of ρ,ij , D, and θ,i ,

r(Γ) = r(Γ) (ρ , θ , ρ,i ) , (Γ = 1, 2, 3) ,

a(Γ) = a(Γ) (ρ , θ , ρ,i ) , (Γ = 1, 2) .(6.27)

Hence, the solution (6.23) along with (6.24-27) satisfies all the restrictions posed by the inequality (6.8) on theform of vector a originally defined by (5.10).

An expression for M(kj)i is obtained by differentiating (6.23) with respect to D, using (6.24-27), and substi-tuting the result into (6.15). The result is

2M(kj)i = a(2)ijk + a(4)

ijkrs Drs , (6.28)

wherea(4)ijkrs = Iikrs a(3)

j + Iijrs a(3)k + Iirkj a(3)

s + Iiskj a(3)r − 2Ikjrs a(3)

i , (6.29)

and where

Iijrs =14( δir δjs + δis δjr − 2δij δrs ) . (6.30)

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 193

It is clear from (6.28-30) that the principal part of the couple-stress tensor, M(kj)i, is linear in Dij and that (6.28)satisfies the identity (4.15). It is also clear from (6.28) that the equilibrium value of M(kj)i is given by

2M(kj)i|E = a(2)ijk , (6.31)

and that2M(kj)i = 0 , when a(2)

ijk = 0 and a(3)i = 0 , (6.32)

Noting from (6.29-30) and (6.21) that

a(4)ijkrs Djk Drs = 2D∗

ik a(3)k , (6.33)

we find from (6.28) that2M(kj)i Djk = a(2)

ijk Djk + 2D∗ik a(3)

k . (6.34)

In terms of M(kj)i , the vector a has the representation

ai = 2M(kj)iDkj + a(1)i − D∗

ik a(3)k − ρ2 ∂ψ

∂ρ,iDkk , (6.35)

where (6.23) and (6.34) have been employed.Withψ, ai and M(kj)i given by (6.10), (6.23), and (6.28), respectively,all the thermodynamic restrictions, namely, (6.9) and (6.11-13) are now satisfied.

We now return to the residual inequality (6.17) and eliminate a using (6.35) to find{Te

ij + ρ

[ρ∂ψ

∂ρ− (ρ

∂ψ

∂ρ,k),k

]δij + ρ

∂ψ

∂ρ,iρ,j

}Dij +

(a(1)i − D∗

ika(3)k

),i

− piθ,i ≥ 0. (6.36)

This inequality becomes identical to the corresponding inequality obtained by Dunn and Serrin ([6], see theirinequality (4.5)) when (6.32) is satisfied, i.e., when the principal part of the couple-stress tensor, M(kj)i, vanishes(so that the equivalent stress reduces to the Cauchy stress and a(3) equals zero) and when, in addition, the entropyflux equals the heat flux over temperature. The residual inequality (6.36) places the same restrictions on the staticpart of the vector a, i.e., a(1), that the residual inequality of Dunn and Serrin ([6], Appendix B and C) places onthe static part of the interstitial work flux, u. If we set D and θ,i equal to zero in (6.36), we find that

a(1)i, i ≥ 0. (6.37)

Noting that a(1) is given by (6.24) and as such satisfies a relation of type (6.13), it follows from (6.37) that

∂a(1)i

∂ρρ,i +

∂a(1)i

∂ρ,jρ,ij ≥ 0 , (6.38)

for all the values of temperature, density, and its gradients. Dunn and Serrin ([6], Appendix B and C) haveshown that the restriction (6.38) along with objectivity and the assumption that the material possesses a centerof symmetry implies that2

a(1) = a(1)(ρ , θ , ρ,i , ρ,ij ) = 0 . (6.39)

Hence, the static part of a and r(1), r(2), and r(3) of (6.24) all vanish. Consequently, the expression (6.35) for areduces to

ai = 2M(kj)iDkj − D∗ik a(3)

k − ρ2 ∂ψ

∂ρ,iDkk , (6.40)

while the residual inequality (6.36) simplifies to{Te

ij + ρ

[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ,k

),k

]δij + ρ

∂ψ

∂ρ,iρ,j

}Dij −

(D∗

ika(3)k

),i

− piθ,i ≥ 0. (6.41)

Recalling the definition of a according to (5.10) and the definition of u according to (4.11)2, we have thusobtained a relationship between the interstitial work flux u and the extra entropy flux,

ui + θpi − qi ≡ Hijk Dkj + θpi − qi = 2M(kj)i Dkj − D∗ik a(3)

k − ρ2 ∂ψ

∂ρ,iDkk . (6.42)

2 Note that when the material does not have a center of symmetry (see Dunn and Serrin’s Eq. (4.7)):

a(1)i = eijk(r1ρ,jρ,lρ,kl + r2ρ,jθ,k + r3ρ,lθ,kρ,jl),

where r1, r1, r1, are all functions of ρ, θ, and I.

194 M.M. Mehrabadi et al.

Dunn and Serrin [6] show that for any material of Korteweg type which possesses a center of symmetry, theinterstitial work flux u has the form

ui = ρρ∂ψ

∂ρ,i. (6.43)

Employing the conservation of mass equation (3.5), we can clearly see that (6.42) reduces to (6.43) when theextra entropy flux is zero and (6.32) is satisfied. The interstitial work flux, u, given by (6.42) depends on thequadratic term in D because the principal part of the couple-stress tensor, M(kj)i, is incorporated in the theory.This can also be verified by setting M(kj)i equal to zero in (6.15).An integration of (6.15) will then yield (6.43).

The expression for the couple-stress tensor can be obtained by substituting for M(kj)i from (6.28) into (4.4).The result is

Mijk = meijk +23

(a(2)[kj]i + a(4)

[kj]irsDrs

). (6.44)

Using this equation and (4.12) we obtain the skew-symmetric part of Cauchy stress as

T[ij] = m,kekji +23

(a(2)[ij]k + a(4)

[ij]krsDrs

),k

− ρb[ij]. (6.45)

The asymmetric Cauchy stress can also be obtained by combining (4.12) and the definition of the equivalentstress (4.10). We obtain after some manipulations using (4.13) that

Tji = Teij + ρb[ij] − (2Mijk − 3meijk),k ,

= Teij + ρb[ij] −

[mekji +

43

(a(2)[kj]i + a(4)

[kj]irsDrs

)],k. (6.46)

Note that the quantity in parentheses in (6.46)1 and the quantity in brackets in (6.46)2 are divergence-free anddo not contribute to the linear momentum equation.

We will return to the residual inequality (6.41) in the next section to derive expressions for various quantitiesunder the conditions of equilibrium.

7 Equilibrium

Equilibrium is defined as a process in which the temperature is uniform and no motion occurs. Thus defining

XA = {Dij, θ,i}, (A = 1, .., 9) (7.1)

equilibrium process will then correspond to X A = 0. Let σ denote the left hand side of the residual inequality(6.41). Then at equilibrium σ attains its minimum value, namely zero. The necessary conditions for this minimumare that

∂σ

∂XA|E = 0 , (7.2)

and ∣∣∣∣ ∂2σ

∂XA∂XB

∣∣∣∣E

= 0, (7.3)

where the index E refers to equilibrium.As mentioned in Sect. 6, the residual inequality (6.41) will become identical to the corresponding inequality

obtained by Dunn and Serrin [6] when the principal part of the couple-stress tensor, M(kj)i, vanishes and theentropy flux equals the heat flux over temperature. Under these conditions, (6.41) and (7.2) will then yield theequilibrium stress (1.2) obtained by Dunn and Serrin [6],

Tij|E = −ρ[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ,k

),k

]δij − ρ

∂ψ

∂ρ ,iρ,j , (1.2)R

which admits a potential function, i.e.,(Tij|E),j = ρµ,i , (7.4)

where the potential, µ , is given by

µ = −∂(ρψ)∂ρ

+(∂(ρψ)∂ρ,k

),k. (7.5)

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 195

This is an attractive feature of this theory because it enables one to replace the three third order partial differentialequations of equilibrium, which usually lead to an overdetermination of density, by a single partial differentialequation of second order. Comparing (1.2) and (1.1) and noting that the free energy is an isotropic function ofthe density gradient as indicated by (6.10)2, Dunn and Serrin [6] show that Korteweg’s equation for stress underequilibrium conditions (see (1.1)), i.e.,

Tij|E = (−p + αρ,kρ,k + βρ,kk)δij + γρ,iρ,j + νρ,ij, (7.6)

is incompatible with (1.2) unless the coefficients α , β, γ, and ν satisfy the relations

α =12∂(ρc)∂ρ

, β = ρc, γ = −c, ν = 0, (7.7)

where

c = 2ρ∂ψ

∂I, (7.8)

which is usually referred to as the surface tension coefficient, is assumed to be independent of I and a functionof ρ and θ, alone.

When the principal part of the couple-stress tensor, M(kj)i, is non-zero, (6.36) and (7.2) yield an expressionfor the equivalent stress tensor at equilibrium,

Teij|E = −ρ

[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ,k

),k

]δij − ρ

∂ψ

∂ρ ,iρ,j, (7.9)

which is identical to the expression (1.2) obtained by Dunn and Serrin for the Cauchy stress, T. Hence, theequivalent stress, Te, also admits the potential function (7.5) and we have

(Teij|E),j = ρµ,i . (7.10)

This is significant because as we have shown (see (4.16)), the equivalent stress appears in place of the Cauchystress in the linear momentum equation and the three third order partial differential equations of equilibrium canbe replaced by a single partial differential equation of second order even in the presence of the couple-stresses.Since objectivity requires that ψ be an isotropic function according to (6.10)2, (7.9) implies that

Teij|E =

[−ρ2 ∂ψ

∂ρ+ ρ

∂c∂ρρ,kρ,k + ρcρ,kk +2ρ

∂c∂Iρ,kρ,lρ,kl

]δij − cρ,jρ,j, (7.11)

where “c” which is the surface tension coefficient defined by (7.8) is now a function of ρ, θ, and I. Similarly, thepotential function, µ, given by (7.5) has the explicit form

µ = −ψ − ρ∂ψ

∂ρ+∂c∂ρ

I + cρ,kk + 2∂c∂Iρ,kρ,lρ,kl. (7.12)

Employing (7.11) and (7.12), it is easy to show that the equivalent stress has the same representation as theCauchy stress of Dunn and Serrin ([6], Eq. (5.5)),

Teij|E = ρ(µ+ ψ)δij − cρ,iρ,j, (7.13)

As we emphasized earlier, when couple stresses are taken into account the equivalent stress tensor playspractically the same role in the momentum equations as the Cauchy stress. For completeness, however, we willderive expressions for the symmetric part and the total asymmetric part of Cauchy stress. It is interesting perhapsthat the expression for the symmetric part of the Cauchy stress at equilibrium for the theory presented here issomewhat more similar to Korteweg’s equation (7.6) than Eq. (1.2) of Dunn and Serrin [6] mentioned earlier. Toobtain such an expression, we may use (7.9) directly with (4.10) and (6.28), or we may first write the residualinequality (6.41) in terms of the Cauchy stress using (4.10), (6.28), and (6.33), and then employ (7.2). Choosingthe latter approach, (6.41) becomes after some calculation,{

Tij + a(2)kij,k + ρ

[ρ∂ψ

∂ρ−

(ρ∂ψ

∂ρ,k

),k

]δij + ρ

∂ψ

∂ρ,iρ,j

}Dij + D∗

ija(3)j,i − piθ,i ≥ 0. (7.14)

Recalling that (i) a(2) and a(3) satisfy (6.22) and (6.27)2, (ii) the material is assumed to possess a centerof symmetry, and (iii) the response functions (6.27)2 for a(2) and a(3) must be objective, the representationtheorems for isotropic functions can be used to show that:

a(2)kij =

a(2)

2(ρ,jδik + ρ,iδjk − 2ρ,kδij),

a(3)i = a(3)ρ,i,

(7.15)

196 M.M. Mehrabadi et al.

where the coefficients a(2) and a(3) are functions of ρ, θ, and I. Considering this fact, we calculate the divergenceof a(2) and a(3) and substitute the result in (7.14) to obtain, from the equilibrium condition (7.2),

T(ij)|E =[−ρ2 ∂ψ

∂ρ+

(ρ∂c∂ρ

+∂a(2)

∂ρ

)ρ,kρ,k + (ρc + a(2))ρ,kk

+2(ρ∂c∂I

+∂a(2)

∂I

)ρ,kρ,lρ,kl

]δij −

(c +

∂a(2)

∂ρ

)ρ,jρ,j

−a(2)ρ,ij − ∂a(2)

∂Iρ,k (ρ,ikρ,j + ρ,jkρ,i) , (7.16)

where (6.10)2 has also been used. The symmetric part of the Cauchy stress at equilibrium for the present theoryis therefore given by (7.16). The terms involving the coefficient a(2) in this equation are due to the presence ofthe principal part of the couple-stress tensor and as we can see when a(2) vanishes, (7.16) reduces to (7.11) andthe symmetric part of the Cauchy stress will then be the sameas the equivalent stress. Defining the parameter pby the relation

p = ρ2 ∂ψ(ρ, θ, I)∂ρ

|I=0, (7.17)

we may write (7.16) as

T(ij)|E =

[−p +

12

∫ I

0

(c − ρ

∂c∂ρ

)dI +

(ρ∂c∂ρ

+∂a(2)

∂ρ

)ρ,kρ,k + (ρc + a(2))ρ,kk+

+2(ρ∂c∂I

+∂a(2)

∂I

)ρ,kρ,lρ,kl

]δij −

(c +

∂a(2)

∂ρ

)ρ,jρ,j

−a(2)ρ,ij − ∂a(2)

∂Iρ,k(ρ,ikρ,j + ρ,jkρ,i). (7.18)

A comparison of (7.18) and Korteweg’s equation (7.6) indicates that if we let

α =12∂(ρc)∂ρ

+∂a(2)

∂ρ, β = ρc + a(2), γ = −c − ∂a(2)

∂ρ, ν = −a(2), (7.19)

and assume further that both c and ν are independent of I and are functions of ρ and θ alone then (7.18) and (7.6)will coincide.

The equilibrium value of M(kj)i is given by (6.31), i.e.,

2M(kj)i|E = a(2)ijk . (6.31)R

The values of the couple-stress tensor, the skew symmetric part of the Cauchy stress, and the asymmetric Cauchystress are found from (6.44), (6.45), and (6.46), respectively, by setting D to be zero. We find

Mijk|E = meijk +23a(2)[kj]i = meijk +

a(2)

2(ρ,jδik − ρ,kδij), (7.20)

T[ij] =[mekji +

a(2)

2(ρ,jδik − ρ,iδjk)

],k

− ρb[ij], (7.21)

and

Tji = Teij + ρb[ij] −

(mekji +

43a(2)[kj]i

),k

= ρ(µ+ ψ)δij − cρ,iρ,j + ρb[ij] −[mekji + a(2)(ρ,jδik − ρ,kδij)

],k, (7.22)

where (7.15)1 has been used to write (7.20)2, (7.21), and (7.22)2. For the latter equation, we have also used(7.13). Note that the potential function µ in (7.22)2 is explicitly given in terms of the free energy function andthe density gradients by (7.12). Again, we note that the divergence of the quantity in brackets is zero.

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 197

Turning our attention to the entropy flux, it follows from the residual inequality (6.41) and the condition (7.2)that the entropy flux vanishes at equilibrium. From this and (6.42)2,3 it then follows that the equilibrium valueof heat flux is also zero3, i.e.,

θpi|E = qi|E = 0. (7.23)

Further restrictions on the conductivity and the viscosity tensors can be obtained from the residual inequality(6.41) in a manner similar to Dunn and Serrin [6].

As seen from the balance of angular momentum equation (4.12), the divergence of the couple-stress tensorMkji, is balanced by the skew-symmetric parts of the stress and the dipolar body force. The boundary conditionsare obtained either from (4.17) and (4.18) using (6.31), (7.13), (7.15) and (7.20), or directly from (3.2) and (3.2)using (4.2), (7.22), and (7.20). The results are

ti =

[Te

ij −(

mekji +43a(2)[kj]i

),k

+ ρb[ij]

]nj

= ρ(µ+ ψ)ni − cρ,iρ,jnj −[mekji + a(2)(ρ,jδik − ρ,kδij)

],k

nj + ρb[ij]nj (7.24)

and

t[ij] =(

mekij +23a(2)[ji]k

)nk= mekijnk +

a(2)

2(ρ,inj − ρ,jni) (7.25)

It is important to note, from (7.19) and (4.14), that when m and ψ are sufficiently smooth and when the boundaryof the region has no edges, we have for a supply-free body∫

AtidA = 0 (7.26)

However, when the boundary of the region consists of a finite collection of smooth surfaces, we obtain∫A

tidA = −∮

c[a(2)eijkρ,jsk + msi]dl (7.27)

where [ ] appearing in the integral over the edges C of the boundary denotes the difference in values of theenclosed quantity as a given point on an edge is approached from either side, and where si is the unit tangent tothe curve C. In deriving (7.27), use has been made of the relation ([23]; Eq. (B), pp. 27)∫

ADi f nj dA =

∫A

( bkk ni nj − bij ) f dA +∮

c[mi nj f] dl , (7.28)

where Di f is the surface gradient of a smooth field f, bij = − Di nj is the second fundamental form of the surface,and where m = s x n .

8 Summary and conclusions

We have shown that we can recast the conservation laws of a dipolar continuum as

ρ+ ρvi,i = 0, (3.5)R

Teij,j + ρbe

i = ρvi, (4.16)R

Mkij,k + T[ij] + ρb[ij] = 0, (4.12)R

ρε = TeijDij + ue

i,i − qi,i+ρre, (8.1)

3 As mentioned in footnote c, when the material does not have a center of symmetry, the static part of the interstitial work doesnot vanish. Hence, at equilibrium, instead of (6.49), (5.10) and (6.23) yield

a(1)i = |θpi − qi|E ,

i.e., the extra entropy flux equals the static part of the interstitial work over temperature.

198 M.M. Mehrabadi et al.

where the equivalent stress, Te, and the equivalent body force, be, in the conservation of linear momentum (4.16)are given by (4.10) and (4.15), namely,

Teij = T(ij) + 2M(ij)k,k, (4.10)R

andρbe

i = ρbi + (ρb[ij]),j, (4.15)R

and where the equivalent interstitial work flux, ue, in the energy equation (8.1) is defined by (4.11), namely,

uei = (Hijk − 2 M(kj)i) Djk . (4.11)1R

The energy equation (8.1) is obtained by introducing an equivalent heat supply, re,

re = b(ij)Dij + r, (8.2)

into the energy equation (4.9). Hence, the working of the symmetric part of the dipolar body force contributes tothe body heat supply. Note that, in comparison, the equivalent interstitial working which is associated with theworking of the dipolar stresses through (4.11) is the rate of supply of mechanical energy across every materialsurface. Employing the definition (8.2), the entropy production inequality (5.4) takes the form

ρη + pi,i − ρre

θ≥ 0. (8.3)

Hence, except for the conservation of angular momentum (4.12) which does not appear in the theory of thematerials of Korteweg type, the remaining conservation laws, i.e., (3.5), (4.16), and (4.9) are similar in formto those given for such materials by Dunn and Serrin [6]. This fact along with a comparison of the boundaryconditions (4.17–19) in the present work, i.e.,

ti = Teijnj + [mekij +

43(M(ik)j − M(ij)k)],k nj + ρb[ij]nj, (4.17)R

t[ij] = nkMkij, (4.18)R

t(ij)Dij − q = ni(HijkDkj − qi), (4.19)R

to the boundary conditions in the theory of Dunn and Serrin [6], i.e.,

ti = Tijnj, q = niqi, u = uini, (8.4)

reveals that for the local field equations and the boundary conditions in the two theories to coincide, it is sufficientthat (1) the couple-stress tensor and the skew-symmetric part of the dipolar body force vanish,

Mkij = 0, b[ij] = 0, (8.5)

and, (2) the boundary condition (4.19) is decoupled as in (8.4)2 and (8.4)3. Note that when (8.5) is satisfied thenthe interstitial work flux of Dunn and Serrin [6] takes the form

ui = HijkDjk. (4.11)2R

Within the thermodynamics framework postulated here, we have reconsidered the materials of Korteweg typeof Dunn and Serrin [6] in the context of a dipolar continuum theory in which the dipolar stress and body forceare assumed to be present. The point of departure between the dipolar continuum model developed here and thedipolar continuum model that follows from the work of Green and Rivlin [8] (e.g., Bluestein and Green [1]) isa decomposition of the rate of work of dipolar stress which allows us to reduce the conservation laws of thedipolar continuum model, with the exception of balance of angular momentum, to a form similar to those formaterials of Korteweg type. The conservation laws for materials of Korteweg type have a form similar to those ofa classical continuum except for an additional term in the energy equation which is referred to as the interstitialwork flux by Dunn and Serrin [6]. We have shown that although the governing equations are very similar for amaterial of Korteweg type and a dipolar fluid, the presence of dipolar stress results in a less restrictive form forthe interstitial work flux. Dipolar stress also plays a role in the boundary conditions when the boundary of theregion consists of a finite collection of smooth surfaces. Further implications of the decomposition of the rate ofwork of dipolar stress to dipolar fluids of Bluestein and Green [1] will be discussed in another paper.

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids 199

Appendix: Derivation of (4.4)

Similarly to the decomposition of tensors of rank two into a symmetric and a skew-symmetric part, tensors ofrank three are resolved into four irreducible parts. These are the symmetric part, TS, completely skew-symmetricpart, TA, and parts TP, and TQ with mixed symmetries. They are defined for a general tensor of rank three witharbitrary symmetry as follows:

TSijk =

16(Tijk + Tjki + Tkij + Tikj + Tkji + Tjik),

TAijk =

16(Tijk + Tjki + Tkij − Tikj − Tkji − Tjik),

TPijk =

13(Tijk + Tkji − Tjik − Tkij),

TQijk =

13(Tijk + Tjik − Tkji − Tjki).

(A.1)

For the tensor M defined by (4.2), the irreducible parts are

MSijk = 0,MA

ijk =13(Mijk + Mjki + Mkij),

MPijk =

23(M(kj)i − M(ik)j),M

Qijk =

23(M(ij)k − M(kj)i). (A.2)

Adding these four irreducible parts of M, we obtain,

Mijk = MAijk +

23(M(ij)k − M(ik)j). (A.3)

Recalling (4.1) and noting thatMA

ijk = T[ijk] = meijk, (A.4)

we obtain (4.4).

Acknowledgements. This paper is based upon work initially supported by the National Science Foundation under Grants to TulaneUniversity (including Grant No. MEA-8318967). In addition, MMM’s work in summer of 2001 and 2002 was partially supportedby DOE National Energy Technology Laboratory, Pittsburgh, through the Oak Ridge Institute for Science and Education.

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