A general formulation of the reversible stress tensor for a nonlocal fluid

International Journal of Engineering Science 70 (2013) 124–134

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A general formulation of the reversible stress tensor for a nonlocal fluid Jiujiang Zhu a,⇑, John W. Crawford b, John W. Palfreyman a a b

SIMBIOS Centre, University of Abertay Dundee, Dundee DD1 1HG, UK Faculty of Agriculture, Food and Natural Resources, University of Sydney, NSW 2006, Australia

a r t i c l e

i n f o

Article history: Received 5 September 2011 Received in revised form 28 September 2012 Accepted 25 March 2013 Available online 29 May 2013 Keywords: Nonlocal fluid dynamics Generalized stress tensor Nonlocal functional variational principle Intermolecular interaction

a b s t r a c t The nonlocal stress tensor is an indispensable constitutive equation required to close the thermodynamic system of nonlocal fluid dynamics. A nonlocal functional variational principle is employed to derive a general expression for the thermodynamically reversible stress tensor for a two-phase, single component, nonlocal fluid. The Euler–Lagrange equation and Noether’s current are used to obtain the general form of the stress tensor, which is then used to derive a wide range of functional forms found in the literature. We also clarify some existing ambiguities. The general form of the nonlocal stress tensor is able to represent micro scale intermolecular interactions, and provides an efficient mesoscale numerical tool for multi-scale analysis using Lattice Boltzmann simulation. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction Nonlocal intermolecular interactions play a crucial role in the nucleation of phase transition and in the formation of interfaces (Drossinos & Kevrekidis, 2003; Llovell, Galindo, Blas, & Jackson, 2010; Roy, Rickman, Gunton, & Elder, 1998; Wertheim, 1976; Wu & Li, 2007). In the last few decades, these interactions have attracted significant attention not only from the point of view of theoretical studies aimed at understanding the mechanisms underlining phase transformation, but also in relation to a wide array of engineering problems, including fluid behavior in confining geometries (Tarazona, Marconi, & Evans, 1987), droplet and bubble formation (Khatavkar, Anderson, Duineveld, & Meijer, 2007a; van Giessen, Bukman, & Widom, 1997; Zhang & Kwok, 2005), contact line and wetting dynamics (Zhang & Kwok, 2004b; Zhang & Kwok, 2006), stability of capillary waves (Tarazona, Checa, & Chacon, 2007), and most recently, nanochannel flow and microfluidics (Li & Kwok, 2003). The challenges of dealing with nonlocal interactions are considerable and a wide range of different methods including continuum and particle-based approaches have been applied. the diffuse-interface model (Anderson, McFadden, & Wheeler, 1998; Anderson, Cermelli, Fried, Gurtin, & McFadden, 2007; Khatavkar et al., 2007a; Khatavkar, Anderson, & Meijer, 2007b); diffuse-interface model incorporating contact line motion (Ding & Spelt, 2007); Monte Carlo and molecular dynamics simulations (Galliero, 2010; Galliero et al., 2009; Yang, Fleming, & Gibbs, 1976); thermodynamic perturbation theory that incorporates molecular dynamics simulations (Zhou & Solana, 2009); density functional theory (Frink, Salinger, Sears, Weinhold, & Frischknecht, 2002; Li & Wu, 2006; Lovett & Baus, 1999; Patra, 2007; Reguera & Reiss, 2004; Tarazona et al., 2007; Tarazona et al., 1987; Wu & Li, 2007); Statistical associating fluid theory for variable range potential functions (Llovell et al., 2010); Van der Waals model dealing with Contact line and wetting dynamics (Sullivan, 1981); Dynamic van der Waals theory (Onuki, 2007); Integral equation method (Iatsevitch & Forstmann, 1997; Iatsevitch & Forstmann, 2000; Lovett, Mou, & Buff, 1976; Wertheim, 1976); Phase-field model based on order parameter (Chakraborty, 2007; Roy et al., 1998); Continuum ⇑ Corresponding author. Tel.: +01 2483828358. E-mail address: [email protected] (J. Zhu). 0020-7225/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2013.03.014

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hydrodynamic model (Qian, Wu, Lei, Wang, & Sheng, 2009); Experimental observation with laser scanning confocal microscopy (Aarts, Schmidt, & Lekkerkerker, 2004); Observation using grazing-incidence X-ray scattering experiments (Fradin et al., 2000); along with others. The advantage of micro scale models is that they can explicitly incorporate molecular scale interfacial details and study the larger-scale properties that emerge as a consequence. However micro scale models are computationally demanding and, as such, they can only model a limited range of spatial scales. Coarse graining is required to model larger scales. Traditional macro scale continuum models use a constitutive equation for the local stress based purely on heuristic functions of strain and their derivatives. This ignores micro scale information, such as intermolecular interactions within the interface. There are extended continuum models that can take the changes in bulk material behavior near material interface into consideration (Slattery, Oh, & Fu, 2004; Slattery, Sagis, & Oh, 2007). However in order to incorporate subscale effects in the continuum mechanics level, these models have to be extended either through adjustable parameters or correction for long-range intermolecular forces (Slattery et al., 2004). Recently the Lattice Boltzmann method has emerged as a potential compromise for the numerical modelling of mesoscale processes to deal with interface and contact line dynamics involving phase transitions (Chang & Alexander, 2005; Davies, Summers, & Wilson, 2006; Gonnella, Lamura, & Sofonea, 2007; Li & Tafti, 2007, 2009; Nourgaliev, Dinh, Theofanous, & Joseph, 2003; van der Graaf, Nisisako, Schroen, van der Sman, & Boom, 2006; Zhang & Kwok, 2005; Zhang & Kwok, 2006). The Lattice Boltzmann method has also been applied to the numerical simulation of nanochannel flow and microfluidics (Li & Kwok, 2003; Zhang, 2010). In order to carry out a Lattice Boltzmann simulation in a fluid including a phase interface, the stress tensor of fluid has to be explicitly set out. The pressure tensor of complex fluids has been previously investigated by a number of groups. For example, Maurits, Zvelindovsky, and Fraaije (1998), Todd, Evans, and Daivis (1995), Todd and Hansen (2008) and Zhang and Kwok (2004a) calculated the pressure tensor from first principles using methods of statistical mechanics. Varea and Robledo (1996) derived a smooth nonlocal stress tensor, which was related to the local second order density gradient term. Romero-Rochin and Percus (1996) and Percus (1996) proposed an expression for the stress tensor and proved that its divergence may correspond to intermolecular forces. However, as discussed in detail in Section 4.7, some formulations of the nonlocal stress tensor in the literature have important limitations (e.g. in the formulation of the nonlocal stress tensor by Zhang, Li, & Kwok (2004) and Zhang & Kwok (2004b) one term was missing). In order to explicitly connect the properties of intermolecular interactions within the phase interface to macroscopic flow properties using the Lattice Boltzmann method, a multiscale approach has to be employed. Ideally the stress tensor should be able to abstract microscopic nonlocal correlations and map these onto a mesoscale continuum model. Li and Tafti (2007) explicitly derived a nonlocal pressure equation using mean-field free energy theory, however the proposed formulation is only suitable for that case. A generalized expression for the stress tensor for a nonlocal fluid is still missing and this is an important impediment to practical engineering application. A nonlocal continuum field theory has been established by Eringen (1972) and Eringen (2002), who decomposed the constitutive dependent variables into two parts, namely a static (equilibrium or reversible) part and a dynamic (dissipative) part. However Eringen (2002) made an additional simplifying assumption that the static part of stress is a local tensor. Thus the mechanism for calculating the static part of stress tensor of a nonlocal fluid is still an open problem. The aim of this paper is to derive a general expression for the reversible part of the stress tensor corresponding to a nonlocal fluid by means of the variational principle, so that the full constitutive equation of the nonlocal continuum field is addressed. Since the part the stress tensor corresponding to dissipation is not considered here, we can restrict ourselves to the study of isothermal thermodynamic equilibrium processes under Eringen’s nonlocal continuum field framework. In spite of the numerical efficiency of the Lattice Boltzmann approach, it still places significant computational demands. Because of this, attention has focused on models for single component multiphase flow, because it minimizes the demands on computer memory. The present paper considers the simple but important case, of single component two phase flow. Section 2 first provides a brief review of Eringen’s framework, and then the nonlocal variational principle is used to determine the extremum of the generalized action functional to derive the explicit form of the generalized stress tensor. Section 3 uses the generalized form to derive the special case corresponding to a local fluid. Section 4 then investigates a number of cases as validation, and demonstrates the application of the new formulation. 2. Nonlocal variational principle 2.1. Nonlocal fluid dynamics framework The framework for generalized nonlocal fluid dynamics can be found elsewhere in the literature, for example the book by Eringen (2002) includes details of the constitutive equations of memory-dependent nonlocal thermoviscous fluids. In summary the nonlocal stress tensor could be written as

TðxÞ ¼ R TðxÞ þ D TðxÞ

ð1Þ

in which RT is the reversible element of the stress and DT is the dissipation (or viscous) element of stress, the component form of DT may be written as (Eringen, 2002) D T kl ðxÞ ¼

Z

v

3

½kðjy  xjÞdmm ðyÞdkl þ gðjy  xjÞdkl ðyÞd y

ð2Þ

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J. Zhu et al. / International Journal of Engineering Science 70 (2013) 124–134

where the deformation-rate tensor

dkl ðyÞ ¼

  1 1 @ v k ðyÞ @ v l ðyÞ ½v k;l þ v l;k  ¼ þ 2 2 @yl @yk

ð3Þ

The viscosity coefficients k(jy  xj) and g (jy  xj) represent the nonlocal effects of the deformation-rate tensor at location y on the viscous stress tensor at location x. The total stress tensor at location x is then includes the integral of nonlocal contributions over the whole material space. If the stress at location x does not depend on any neighbourhood y but only on x, then the viscosity coefficients take a local form such that



kðjy  xjÞ ¼ k0 dðy  xÞ

ð4Þ

gðjy  xjÞ ¼ g0 dðy  xÞ in which d() is the Dirac delta function, Eq. (2) then becomes D T kl ðxÞ

¼ k0 dmm ðxÞdkl þ g0 dkl ðxÞ

ð5Þ

which is the constitution equation corresponding to a local fluid. Thus the viscous stress in Eringen’s framework was fully expressed in terms of nonlocal form. However in Eringen’s theory (Eringen, 2002; Eringen, 1972) the reversible part of the stress RT was assumed to take local form only. The aim here is to extend reversible stress to include nonlocal interaction. 2.2. Dual variational principle Consider the total nonlocal free energy or dual action functional

W½q ¼

Z Z v

v

3

3

L½y; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞd yd x

ð6Þ

In which, the dual grand potential L½y; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞ is the generalized nonlocal Lagrangian, where q (x) and q(y) are the density function evaluated at positions x and y respectively, and both x and y are integrated over whole body under consideration. The variation of the dual action functional Eq. (6) then reads as:

dW½q ¼

Z Z  v

 @L @L @L @L dqðyÞ þ ry dqðyÞ þ dqðxÞ þ rx dqðxÞ d3 yd3 x @ðry qðyÞÞ @ qðxÞ @ðrx qðxÞÞ v @ qðyÞ

ð7Þ

Eq. (7) may be rewritten in the form

dW½q ¼

Z Z " v

v

# @Ls @Ls dqðxÞ þ rx dqðxÞ d3 yd3 x @ qðxÞ @ðrx qÞ

ð8Þ

in which

Ls ¼ Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ ¼ L½y; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞ þ L½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ

ð9Þ

and is equal to two times the symmetrical part of L. For simplicity, in the remainder of the paper, Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ is denoted as Ls only. Obviously Ls satisfies

Ls ½y; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞ ¼ Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ

ð10Þ

Then the Lagrangian functional is defined as

L½x; qðxÞ; rx qðxÞ ¼

Z 1 3 Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞd y 2 v

ð11Þ

where the density function satisfies the mass conservation equation i.e.

Z

qðxÞd3 x ¼ const

ð12Þ

v

Finally, the total Lagrangian functional L is defined as

LT ½x; qðxÞ; rx qðxÞ ¼ L½x; qðxÞ; rx qðxÞ  lqðxÞ

ð13Þ

in which l is the Lagrange multiplier relating to Eq. (12), and the physical interpretation of l is the chemical potential. The nonlocal variational principle then reads

Z  3 d LT ½x; qðxÞ; rx qðxÞd x ¼ 0

ð14Þ

v

Note that rxdq(x) and dq (x) are not independent variables. Through manipulation of the partial integration, Eq. (14) leads to the Euler–Lagrange equation

J. Zhu et al. / International Journal of Engineering Science 70 (2013) 124–134

rx 

  dL½x; qðxÞ; rx qðxÞ dL½x; qðxÞ; rx qðxÞ  þl¼0 dðrx qðxÞÞ dqðxÞ

127

ð15Þ

in which,

Z dL½x; qðxÞ; rx qðxÞ @Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ 3 d y ¼ dðrx qðxÞÞ @ðrx qðxÞÞ v

ð16Þ

Z dL½x; qðxÞ; rx qðxÞ @Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ 3 d y ¼ dqðxÞ @ qðxÞ v

ð17Þ

and

For a process in thermodynamic equilibrium, Noether’s theorem (Noether, 1971; Tavel, 1971) may be extended to the nonlocal case. Noether’s current then gives the nonlocal reversible stress tensor RT

¼ LT I  rx qðxÞ 

@LT @ðrx qðxÞÞ

ð18Þ

Substituting Eq. (15) into Eq. (13)

  LT ½x; qðxÞ; rx qðxÞ ¼ L þ qðxÞ rx 

dL



dðrx qðxÞÞ



 dL dqðxÞ

ð19Þ

Substituting Eq. (19) into Eq. (18)

  T ¼ LI þ q ðxÞ r  R x

dL



dðrx qðxÞÞ



 dL @L I  rx qðxÞ  dqðxÞ @ðrx qðxÞÞ

ð20Þ

Substituting Eqs. (16), (17) and (11) into Eq. (20) R TðxÞ

¼

1 2

Z v

  Z 3 Ls d y I þ qðxÞ rx 

 Z  Z  @Ls @Ls 3 1 @ 3 3 d y  d y I  rx qðxÞ  Ls d y 2 @ðrx qðxÞÞ v v @ðrx qðxÞÞ v @ qðxÞ

ð21Þ

It should be emphasized again that the brief notation Ls in Eq. (21) stands for Ls ½x; y; qðxÞ; qðyÞ; rx qðxÞ; ry qðyÞ, defined in Eq. (9). 2.3. Commentary To investigate a smooth attenuated neighbourhood fluid, it may be of interested to extend the constitutive variables set to include up to the nth order of the density gradient, i.e. fy; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞ;    ; rny qðyÞ; rnx qðxÞg. From a mathematical point of view, this extension does not introduce additional complexity, since the density and its gradients are not constitutive independent variables in the variation process. The nth order density gradient can be converted to a density through n partial integration steps, which leads to the nth order Euler–Lagrange equation. A special case of this kind of fluid is the so-called smooth attenuated nonlocal fluid, or more simply, the smooth nonlocal fluid, where the free energy at location x depends on fx; qðxÞ; rx qðxÞ;    ; rnx qðxÞg only. Eq. (21) sets out a general formulation of the reversible part of the stress tensor for nonlocal fluids. This is as same as the dissipation component of the nonlocal stress tensor (or viscous stress) in Eringen’s constitutive theory, where the stress tensor at each point may interact globally with all points of the body. This nonlocal interaction is expressed as an integration over whole body. The great advantage of this integral constitutive equation is that the integration is carried out in real physical space, which is convenient for practical engineering applications. Comparing Eq. (21) with earlier forms of the integral constitutive equations for nonlocal fluids (Romero-Rochin & Percus, 1996; Percus, 1996), the previous stress tensors were expressed in terms of the integration of a perturbation parameter k over an interval (0,1). The physical interpretation of the perturbation parameter k is unclear, which presents challenges for practical applications, and in addition the interpretation of the divergence of the previous stress tensor in terms of intermolecular force may not be unique (; Romero-Rochin & Percus, 1996). Section 4 will demonstrate that all stress tensors found in the current literature are special cases of the general formula Eq. (21). In contrast to the constitutive equations proposed by Romero-Rochin and Percus (1996) and Percus (1996) which apply only to smooth attenuate nonlocal fluids, the form in Eq. (21) applies to local, smooth attenuate nonlocal (refer to the next section) and real nonlocal fluids. 3. Local fluid Here, a special case that of the local fluid with density gradient effects, the smooth nonlocal fluid, is considered. For smooth nonlocal fluids, the dual grand potential is assumed to take the form

L½y; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞ ¼ L½x; qðxÞ; rx qðxÞdðy  xÞ

ð22Þ

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J. Zhu et al. / International Journal of Engineering Science 70 (2013) 124–134

in which d() is the delta function, Eq. (11) then reduces to

L½x; qðxÞ; rqðxÞ ¼ L½x; qðxÞ; rqðxÞ

ð23Þ

i.e. L½x; qðxÞ; rx qðxÞ is the local Lagrangian. Also Eq. (6) can be reduced to a local action function of the form

Z

W½q ¼

v

3

L½x; qðxÞ; rqðxÞd x

ð24Þ

The Euler–Lagrange equation Eq. (15) then becomes

r



 dL½x; qðxÞ; rqðxÞ dL½x; qðxÞ; rqðxÞ  þk¼0 dðrqðxÞÞ dqðxÞ

ð25Þ

and Eq. (21) becomes

    @L½x; qðxÞ; rqðxÞ @L½x; qðxÞ; rqðxÞ 1 @L½x; qðxÞ; rqðxÞ  I  rqðxÞ  T ¼ L½x; q ðxÞ; r q ðxÞI þ q ðxÞ r  R @ðrqðxÞÞ @ qðxÞ 2 @ðrqðxÞÞ ð26Þ 4. Some special cases In this section, the general stress tensor formulation obtained in section 2 is used to study certain special cases found in the literature. 4.1. Ideal gas An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas obeys the ideal gas law, a simplified equation of state. For ideal gas, assume that

L ¼ Lid ½qdðy  xÞ ¼ qðxÞfid ½qðxÞdðy  xÞ

ð27Þ

Lid ½q ¼ qkT ln q

ð28Þ

fid ½qðxÞ ¼ kT ln q

ð29Þ

where

and

And k is the specific gas constant, such that

k¼

R R kB ¼ ¼ M mNA m

ð30Þ

in which R is the universal gas constant, Na is Avogadro’s constant, kB is Boltzmann’s constant, and m is the mass of single molecule. Substituting Eq. (28) into Eq. (26) gives



R TðxÞ

¼

Lid ½qðxÞ  qðxÞ

 @Lid ½qðxÞ I @ qðxÞ

ð31Þ

Denoting

Pid ¼ q

@Lid ½q @fid ½q  Lid ½q ¼ q2 ¼ qkT @q @q

ð32Þ

Introducing the number of moles n ¼ mN , where N is total number of molecules, and noting that q ¼ mN , then using Eq. (30), M V Eq. (32) may be rewritten as

Pid V ¼ nRT

ð33Þ

Thus RT

¼ Pid I ¼ qkTI

ð34Þ

4.2. Van der Waal’s Gas A Van der Waal’s Gas is assumed to satisfy Van der Waal’s Gas equation, which is based on a modification of the ideal gas law and approximates the behavior of real fluids, taking into account the nonzero size of molecules and the attraction between them. For Van der Waal’s Gas taking

L ¼ Lv dw ½qdðy  xÞ ¼ qðxÞfv dw ½qðxÞdðy  xÞ

ð35Þ

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J. Zhu et al. / International Journal of Engineering Science 70 (2013) 124–134

where

Lv dw ½q ¼ qkT ln

q 1  bq

 aq2

ð36Þ

 aq

ð37Þ

and

fv dw ½qðxÞ ¼ kT ln

q 1  bq

where a and b are Van der Waal’s constants. Then substituting Eq. (35) into Eq. (26) gives RT

¼ Pv dw I

Pv dw ¼ kT

ð38Þ

q 1  bq

 aq2

ð39Þ

4.3. Carnahan–Starling fluid The Van der Waal’s equation Eq. (37) may be generalized as

f ½q ¼ kT ln q þ bkT

Z

vðqÞdq  aq

ð40Þ

which gives rise to the relationship

(

P ¼ phs ðqÞ  aq2

ð41Þ

phs ðqÞ ¼ qkT½1 þ bqvðqÞ where phs(q) is the pressure of a hard sphere reference fluid. For example, for a Carnahan–Starling fluid

v¼

1  bq=8

phs ðqÞ ¼ qkT (

ð42Þ

ð1  bq=4Þ3 " # 1 þ h þ h2  h3

ð43Þ

ð1  hÞ3

h ¼ bq=4

ð44Þ

3

b ¼ 2pd =3m

in which d and m are the diameter and mass of the molecules, respectively. The free energy corresponding to Eq. (42) or Eq. (43) is consistent with the result given by van Giessen et al. (1997)

(

L ¼ Lcs ½qdðy  xÞ ¼

"

qkT ln q þ

#

3  2h ð1  hÞ

)

 1  lq  aq2 þ pbulk dðy  xÞ 2

ð45Þ

Eq. (40) may also be used to describe other fluids given the form of the function v(q). For example, for a dense gas, v was given by Chapman and Cowling (1970)

5 8

v ¼ 1 þ ðbqÞ þ 0:2869ðbqÞ2 þ 0:1103ðbqÞ3 þ 0:0386ðbqÞ4 þ   

ð46Þ

4.4. Local fluid with density gradient effects (smooth nonlocal fluid) To illustrate the results of Section 3, and similar to Abraham (1979), Nadiga and Zaleski (1996), Swift, Osborn, and Yeomans (1995), Anderson et al. (1998),ee and Lin (2003, 2005) and Chang and Alexander (2005), we assume Eq. (22) takes the form

L½qðyÞ; qðxÞ ¼



1 2



qðxÞf ½qðxÞ þ j½rx qðxÞ2 dðy  xÞ

ð47Þ

in which j is a constant capillary coefficient. Substituting Eq. (47) into Eq. (26) gives RT

  1 ¼ P 0 þ jðrqÞ2 þ jqr2 q I  jrq  rq 2

ð48Þ

In which P0 is given by

P0 ¼ q2 ðxÞ

@f ½qðxÞ @q

ð49Þ

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J. Zhu et al. / International Journal of Engineering Science 70 (2013) 124–134

and is the thermodynamic pressure. In practice this could take a variety of forms depending on the structure of the Helmholtz free energy f[q]. Eq. (48) is known as Kerteweg’s stress tensor. This may be rewritten in the form of a pure pressure part plus a pure shear stress part as RT

  1 ¼ P0  jðrqÞ2 þ jqr2 q I þ j½ðrqÞ2 I  rq  rq 2

ð50Þ

The matrix form of the second part of the right hand side of Eq. (50) is then

0

 @@xq

0

B

@q @q j½ðrqÞ2 I  rq  rq ¼ jB @  @y @x

@q @y

0

 @@zq @@xq

 @@zq @@yq

 @@xq

@q 1 @z

 @@yq

@q C @z A

C

ð51Þ

0

which represents a pure shear stress produced by the density gradient. For a plane interface, all variables depend only on z and not x or y. Under this particular condition Eq. (51) equals zero. However in the most general cases, the shear stress component is non-trivial. The stress tensor, Eq. (50), was introduced to the Lattice Boltzmann scheme by Swift et al. (1995) to model the hydrodynamics of phase separation in two phase flow. In their model, a nonlocal term was added to the equilibrium distribution, so that the Navier–Stokes–Kerteweg equation could be recovered from the Lattice Boltzmann equation directly by performing a Chapman–Enskog expansion. Recently Lee and Lin (2003, 2005) and Chang and Alexander (2005) provided a standard Lattice Boltzmann model to simulate the diffusion interface in two phase flow. It can be proved using the Chapman–Enskog scheme that the standard Lattice Boltzmann procedure recovers the ideal gas Navier–Stokes equation. Thus in their Lattice Bhatnagar–Gross–Krook equation in the Lattice Boltzmann model, the intermolecular interaction force

F ¼ rðqc2s Þ þ r  ðR TÞ

ð52Þ

was introduced to modify the momentum equation from the ideal gas Navier–Stokes equation to the Navier–Stokes–Kerteweg equation, where qc2s is the pressure of an ideal gas. Substituting Eq. (50) into Eq. (52), the intermolecular interaction force may be rewritten as

F ¼ rðqc2s  P 0 Þ þ jqrðr2 qÞ

ð53Þ

in which P0 is given by Eq. (49). 4.5. Nonlocal fluid without gradient effects A particular case is a nonlocal fluid in the absence of a density gradient. In this case, Eq. (6) is reduced to

W½q ¼

Z Z v

v

3

3

L½y; x; qðyÞ; qðxÞd yd x

ð54Þ

Eq. (9) then reads

Ls ¼ L½y; x; qðyÞ; qðxÞ þ L½x; y; qðxÞ; qðyÞ

ð55Þ

and Eq. (21) becomes RT

¼

 Z  Z 1 @Ls 3 3 Ls d y  qðxÞ d y I 2 v v @ qðxÞ

ð56Þ

The reversible stress now reduces to a pure pressure part. Introducing the thermodynamic pressure

P ¼ qðxÞ

Z

Z @Ls 3 1 3 d y Ls d y 2 v v @ qðxÞ

ð57Þ

Eq. (56) may then be rewritten as RT

¼ PI

ð58Þ

4.6. Van-Kampen nonlocal fluid As a case of Section 4.5, following Li and Tafti (2007), Sullivan (1981) and van Kampen (1964), we assume the dual grand potential takes the special form

1 L½y; x; qðyÞ; qðxÞ ¼ L0 ½qðxÞdðy  xÞ þ wðjy  xjÞ½qðyÞ  qðxÞ2 4

ð59Þ

J. Zhu et al. / International Journal of Engineering Science 70 (2013) 124–134

131

Substituting the first part of the right hand side of Eq. (59) into Eq. (57) gives

P0 ¼ qðxÞ

@L0 ½qðxÞ  L0 ½qðxÞ @q

ð60Þ

which is the same as Eq. (27) of Li and Tafti (2007). Noting that for the second part of Eq. (59) (denote as L0 )

L0s ¼ 2L0 ¼

1 wðjy  xjÞ½qðyÞ  qðxÞ2 2

ð61Þ

Substituting Eq. (61) into Eq. (57) we find 0 RT

 Z  Z 1 @L0s 3 3 ½ L0s d yI þ qðxÞ  d y I 2 v v @ qðxÞ  Z    Z 1 3 3 ¼ wðjy  xjÞ½qðyÞ  qðxÞ2 d y I þ qðxÞ wðjy  xjÞ½qðyÞ  qðxÞd y I 4 v v ¼

ð62Þ

Eq. (60) together with Eq. (62) gives RT

   Z  @L0 ½qðxÞ 1 3 ¼  qðxÞ  L0 ½qðxÞ I þ wðjy  xjÞ½qðyÞ  qðxÞ2 d y I @q 4 v   Z 3 þ qðxÞ wðjy  xjÞ½qðyÞ  qðxÞd y I

ð63Þ

v

or

P ¼ qðxÞ

Z Z @L0 ½qðxÞ 1 3 3  L0 ½qðxÞ  wðjy  xjÞ½qðyÞ  qðxÞ2 d y  qðxÞ wðjy  xjÞ½qðyÞ  qðxÞd y @q 4 v v

ð64Þ

which is the same as that given by Li and Tafti (2007). If we assume L0 ½qðxÞ is expressed as

L0 ½qðxÞ ¼ qðxÞf ½qðxÞ

ð65Þ

Substituting Eq. (65) into Eq. (60) we have

P0 ¼ q2 ðxÞ

@f ½qðxÞ @q

ð66Þ

This is as same as Eq. (49). Li and Tafti (2007) built a Lattice Boltzmann model by generalizing the thermodynamic pressure method. In order to validate their pressure formula, they carried out the following Taylor expansion

1 2

qðyÞ  qðxÞ  ðy  xÞ  rqðxÞ þ ½ðy  xÞ  r2 qðxÞ

ð67Þ

Li and Tafti (2007) took w(jy  xj) in Eq. (64) as the Lennard–Jones potential. Substituting Eq. (67) into Eq. (64) and ignoring the terms of order higher than jy  xj2, they proved that the pressure given in Eq. (64) is consistent with the pressure component in Eq. (50), i.e. the first term in the right hand side of Eq. (50). It is worth emphasizing that the first term in the right hand side of Eq. (48) is not a pure pressure component of stress tensor, as the second term is not pure shear stress. Comparing Eq. (59) with Eq. (1) in Li and Tafti (2007), one may find that Li and Tafti chose the special case of Van-Kampen Nonlocal fluid. 4.7. Zhang and Kwok model A mean field free energy model had been proposed by Zhang et al. (2004) and Zhang and Kwok (2004b). In their model the dual grand potential took the form

1 L½y; x; qðyÞ; qðxÞ ¼ fw0 ½qðxÞ þ qðxÞVðxÞgdðy  xÞ þ qðxÞ/ff ðjy  xjÞ½qðyÞ  qðxÞ 2

ð68Þ

Substituting the first part of the right hand side of Eq. (68) into Eq. (57) gives

P0 ¼ qðxÞ

@fw0 ½qðxÞ þ qðxÞVðxÞg  fw0 ½qðxÞ þ qðxÞVðxÞg @q

ð69Þ

P0 ¼ qðxÞ

@w0 ½qðxÞ  w0 ½qðxÞ @q

ð70Þ

or

Thus the external potential V(x) makes no contribution to the thermodynamic pressure. Substituting the second part of the right hand side of Eq. (68) (denoted as L0 ) into Eq. (55) results in

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1 L0s ¼  /ff ðjy  xjÞ½qðxÞ  qðyÞ2 2

ð71Þ

and substituting Eq. (71) into Eq. (57) gives

P0 ¼ qðxÞ

Z v

3

/ff ðjy  xjÞ½qðyÞ  qðxÞd y þ

1 4

Z v

3

/ff ðjy  xjÞ½qðyÞ  qðxÞ2 d y

 ð72Þ

Adding Eqs. (72) and (70) gives

P ¼ qðxÞ

Z  Z @w0 ½qðxÞ 1 3 3  w0 ½qðxÞ þ qðxÞ /ff ðjy  xjÞ½qðyÞ  qðxÞd y þ /ff ðjy  xjÞ½qðyÞ  qðxÞ2 d y @q 4 v v

ð73Þ

Zhang et al. (2004) and Zhang and Kwok (2004b) derived the pressure formula following Yang et al. (1976). Comparing Eq. (73) with Eq. (2) in Zhang et al. (2004) and Zhang and Kwok (2004b), it is found that the last term in the right hand side of Eq. (73) is missing in their Eq (2). Li and Tafti (2007) also found that by substituting the Taylor expansion into the Eq. (2) of Zhang et al. (2004) and Zhang and Kwok (2004b), they were unable to recover the classical square-gradient theory in Li and Tafti’s model. However, they did not point out that this was due to an incorrect formulation of the pressure. As pointed out by Li and Tafti (2007) an incorrect total nonlocal stress tensor formulation may cause the system to move away from the critical point. Following the same approach as Li and Tafti (2007) by substituting Eq. (67) into Eq. (73) and ignoring higher order terms, it is found that resulting pressure agrees with the pressure component in Eq. (50), which is the stress tensor for the classical square gradient theory. 4.8. Fluid with both density gradient and nonlocal effects In some complex fluids, such as polymers or copolymers, the dual grand potential may result from both density gradient effects and general nonlocal interactions. Following Maurits and Fraaije (Maurits & Fraaije, 1997) it is assumed that

L¼

  1 2 1 1 1 qðxÞqðyÞ aq ðxÞ þ bq4 ðxÞ þ jjrx qðxÞj2 dðy  xÞ  c2 2 2 2 2 jy  xj

ð74Þ

where a,b,j,c are material constants. According to Helfand’s penalty on high density fluctuation (Uneyama & Doi, 2005; van Vlimmeren, Maurits, Zvelindovsky, Sevink, & Fraaije, 1999), the first term is a short range interaction. The second term represents the long range cohesive interaction (see also Drossinos & Kevrekidis (2003), Drossinos, Kevrekidis, Lazaridis, & Georgopoulos (2000) and Roy et al. (1998)). Substituting Eq. (74) into Eq. (21) results in RT

    Z 1 1 1 qðyÞ 3 ¼  q2 ðq  1Þða þ bq2 Þ þ jðrqÞ2 þ jqr2 q I  jrq  rq  c2 qðxÞ d y I 2 2 2 v jy  xj

ð75Þ

With Eq. (75), one may model the mesoscale phase separation kinetics of long range interaction copolymer melts in terms of an elegant nonlocal fluid dynamics framework. 5. Conclusion An explicit generalized formulation of the reversible part of the stress tensor for nonlocal fluids is obtained by means of the functional variational principle. It is well known that the equation of state of an ideal gas is required to close the set of equations for classical gas dynamics, similarly the generalized nonlocal stress tensor is an indispensable constitutive equation required to close the thermodynamic system of nonlocal fluid dynamics. Unlike previous pressure or stress tensor formulations, which are only suitable for a particular fluid, the stress tensor obtained in this paper is general, and is suitable for any nonlocal fluid, where the dual grand potential L½y; x; qðyÞ; qðxÞ; ry qðyÞ; rx qðxÞ could be an arbitrary function. We demonstrate that different forms of the non-local fluid stress tensors that can be found in the literature may be produced from the general formulation, and indeed that some of them can only be generated in this way. Using this approach we were also able to clarify some ambiguous formulations of pressure/stress tensors found in the literature using the general formulation. In a similar manner to the dynamic part of nonlocal stress tensor given by Eringen (1972, 2002), the form of the static part of the stress tensor obtained in this paper is also expressed in terms of nonlocal integration over the entire 3 dimension spatial domain occupied by the fluid,, which is essential for practical engineering application. The derived form of the nonlocal stress tensor is able to represent micro scale intermolecular interactions, and provides an efficient mesoscale numerical tool using Lattice Boltzmann simulation to carry out a multi scale analysis. Acknowledgment Authors would like thank Prof. David Bradley for his constructive discussing and correcting of the manuscript. JJZ would like thank RCUK for the support of fellowship.

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