Micro- inertia effects in material flow P. Masafu Mwasame, Norman J. Wagner, Antony N. Beris a) Center for Molecular and Engineering Thermodynamics, Deartment of Chemical and Biomolecular Engineering, University of Delaware, Newark, DE 19716 a) Corresonding Author. Tel. 1-30- 831-8018, E- mail address: beris@udel.edu Physical Review Fluids (PRF) Raid Communications Abstract A new alication of Non- Equilibrium Thermodynamics is resented that allows for the incororation of microstructural inertia effects within conformation tensor- based constitutive models of macroscoic material behavior. In general, this requires the introduction of an additional evolution equation for a structural momentum tensor hysically identified with the uer convected time derivative of the conformation tensor. However, to the lowest order in article Reynolds number micro- inertia effects can be simly accounted for through a modification of the time evolution equation for the conformation tensor derived in the inertialess limit by introducing a new non- affine term that coules the conformation and the vorticity tensors. When alied to the shear flow of dilute emulsions, the reduced inertial flow model rovides redictions consistent with literature- available microscoically- based asymtotic results. Contravariant (and their inverses, covariant) conformation tensors are a useful adjunct in modeling of both the elasticity of solid materials and the rheology of fluids which comrise the main toics of continuum mechanics [1,]. Contravariant conformation tensor models have similarly been used in viscoelastic flow models such the classic Johnson- Segalman model to describe olymer dynamics [3]. Emulsion are another class of comlex fluids that have also been well described using conformation tensor models [4,5], both contravariant and covariant, artly motivated from a consideration of the micromechanics of single drolets [6-9]. In all these cases highlighted, an imortant and common assumtion is that microstructural inertia effects, characterized by a article Reynolds number, Re, are negligible. Physically, this means that the extra stress is indeendent of the acceleration of the system; mathematically, this requires the equations to be invariant to acceleration, including solid body rotation. In the flow of comlex fluids with suermolecular structures, such as rigid sheres, Re can become O(1) or larger and microstructural inertia effects may be imortant [10,11] giving rise to new rheological behaviors. In articular, a number of recent studies [1-14] on emulsion drolets in the 1
resence of micro- inertia have revealed emerging rheological and microstructural behaviors, that contrast those seen in zero article Reynolds number flows [15,16]. Most notable are a negative first normal stress difference ( N 1 ) accomanied by the orientation of emulsion drolets at angles, q, greater than 45 o with resect to the flow direction in simle shear flows [1]. To date and to our knowledge, no general macroscoic model, alicable to all flows, has been able to cature these unique behaviors, revealing the need for material flow models that account for micro- inertia effects. Although there exists a broad literature on microstructural inertia, with the most relevant to this work being develoments in liquid crystalline systems, as resented in a research monograh in [17], these are incomlete and inadequate for alications such as emulsion rheology and, in general, the rheology of comlex fluids. Therefore, the goal of this work is to show how the framework of Non- Equilibrium Thermodynamics (NET) can be harnessed to generically modify the most often emloyed, conformation- based non- inertia material flow models so that they can rigorously account for micro- inertia effects. In addition to being rigorous, this framework consistently reduces to these revious models when microinertia effects are negligible. Imortantly, we make use of asymtotic microscoic theory, in the limit of small micro inertia, to validate the corresonding macroscoic model equations develoed through the NET framework for secific examles where such asymtotic results are available. Furthermore, analysis of this framework within the context of microscoic theory identifies cases where conformational tensor inertia can be successfully described without the need of an additional, structure momentum tensor. A major result of this work is to demonstrate that there exists a regime for which the lowest order microstructural inertia effects can simly be handled within existing conformation tensor models through the extension of the Gordon- Showalter derivative, used to describe the evolution of the conformation tensor, by adding a new non- affine term, without the need of an additional structural momentum tensor. The key to success in this endeavor is the develoment of a systematic rocedure to introduce micro- inertial effects into existing conformation tensor- based models. We achieve this goal through the use of the one- generator (Hamiltonian) descrition of dynamic equations in NET [18-0] following its imlementation as detailed in reference [17]. The one- generator aroach is a articular realization of the General Equation for Non- Equilibrium Reversible- Irreversible Couling (GENERIC) [1-3] two- generator (Hamiltonian lus Entroy) formulation for macroscoic systems for which, through a local thermal equilibrium, a unique temerature can be described [4]. This formalism has so not only been able to describe many of the most imortant constitutive equations in the modeling of comlex (such as
viscoelastic) fluids in the absence of inertia, but it has also heled to develo many others with good stability characteristics emerging from their thermodynamic admissibility [17]. In the one generator Hamiltonian descrition, the starting oint is that the time evolution for any arbitrary functional F of the state sace is generated by a linear suerosition of the Poisson{, } and dissiative [, ] brackets, dictating the reversible and irreversible dynamics, resectively 1, as df { FH, } [ FH, ] dt = +, (1) where H is the Hamiltonian functional, which hysically describes the extended, nonequilibrium, internal energy of the system exressed in terms of the system s state variables. In general, this involves adding nonequilibrium contributions to the thermodynamic Helmholtz free energy, such as as kinetic energy and any elastic or interfacial energy, deending on the hysics of the articular system being reresented [17]. Each one of the brackets is defined as a functional of the Volterra derivatives of F and H obeying secific symmetry/antisymmetry roerties as dictated by reversible Hamiltonian dynamics (for the Poisson bracket) and irreversible NET (for the dissiation bracket) [17]. For further rogress, one needs to select/determine a) the dynamic field variables (like the momentum density u º r v, with r the mass density and v the velocity, and the conformation tensor C in conformation- based theories); these define an N- vector field over time and sace, (, ) ztx (in the revious examle ztx (, ) ( utx (, ), Ctx (, )) = ), b) the Hamiltonian, H, defined as a functional over z, c) the Poisson and d) the dissiation brackets. Then, the evolution equations for the z field variables are obtained by forcing the equivalence of Eq. (1) to the direct evaluation of the time derivative for any arbitrary functional Fz (), following the standard rule of chain differentiation, as df d F z = ò å dw, () N k dt W k = 1d zk t where Ω denotes the bulk volume sace. Introducing micro- inertial effects into a second order conformation tensor- based continuum formulation amounts to introducing changes to the inertialess bracket descrition. To motivate the form 1 A simle criterion to distinguish between reversible and irreversible dynamics contributions is based on examining all the terms through the Jacobi identity. 3
of these changes it is instructive to go through a simle relevant examle of a one- dimensional damed oscillator in the context of the bracket formalism. In this examle, the field sace z reduces to a vector sace, the functionals to functions and the Volterra derivatives to artial derivatives. The reversible dynamics of the oscillator are driven by a linear restoring sring force, F k( x x ) s =- -, around an equilibrium location x 0, while the irreversible dynamics are modeled by a linear friction daming force, Fd =- µ x, where a suerscrit dot is used to denote the time derivative. A straightforward alication of Newton s second law for an oscillator of mass m then leads to a second order evolution equation for its osition x as a function of time, ( ) mx + µ x + k x - x 0 = 0, (3) which is recognizable as the classical one- dimensional equation for a linear damed oscillator. The same result can also be obtained with the one generator NET aroach based on the choices indicated in the first row of Table 1. In this case, Eq. (3) emerges as the second of two first order equations, dx H 1 = = dt m ( (), ()) and d =- H - µ H, within a two dimensional state sace dt x z= x t t, after substitution for = mx, the linear momentum. On the other hand, Eq. (3) reduces In the inertialess limit, m 0, to a first order equation µ x+ k( x- x 0 ) = 0, (4) reflecting a comletely damed oscillator. Alternatively, the same inertialess dynamics can be directly obtained within the Hamiltonian/dissiation formalism through model coarsening, following the choices aearing in the second row of Table 1. Table 1: The Hamiltonian/dissiation structure of the inertial vs. inertialess formulations for the damed Harmonic Oscillator Poisson Bracket Dissiation bracket Formulation State Sace z(t) Hamiltonian H ({F, H}) ([F, H]) 1 k Inertial ( xt (), t ()) ( x x ) m + - F H F H - 0 x x 0 F H - µ In rincile a thermodynamic correction to the Hamiltonian due to thermal effects also needs to be added [0]. This introduces entroy corrections in the Dissiative bracket but can be ignored, justifiably, if one considers isothermal systems. 4
Inertialess ( xt ()) ( x ) k 1 F H x- 0 0 - µ x x Of more interest is to summarize from Table 1 the four changes necessary to transform the second row (inertialess) choices into the first row (inertial) ones: i) A momentum variable, t (), is added to the state sace; ii) A kinetic energy term is added to the Hamiltonian; iii) A classic, canonical, Poisson contribution, couling the structural variable, x, to its momentum,, is added to the Poisson bracket and iv) the inertial dissiation bracket is obtained from the inertialess one through the formal substitution, «- µ x. These four changes rovide a guidance for elevating any inertialess formulation to an inertial one. We need however to caution the reader that although the first three changes can be rigorously roven, the last one is heuristically develoed given the nature of the dissiation and the fact that we are moving from a coarser (inertialess) to a finer (inertial) descrition. The intrinsic value and utility of the Bracket formalism, beyond the intuition and understanding offered by alying it to this simle illustrative examle, is that its basic structure is maintained over different levels of descrition- - - see [17] for some additional examles. Therefore, as the system variables change from scalar to conformation tensor- based variables, we believe that the insights gained from the illustrative, harmonic oscillator examle can still be alied as long as the additional comlexity introduced by the higher dimensionality of the conformation tensor is roerly taken into account. As a guidance that this additional comlexity is roerly handled, we aly the theory to a targeted alication (dilute emulsion) where a validation against available, microscoically- based, inertia theories is ossible in the limit of low inertia shear flows. The Poisson bracket for a second order conformation tensor- based inertialess model is fully determined from its contravariant or covariant character. The standard Poisson Bracket for a second order contravariant conformation tensor C is described in [17] or alternatively Eq..49 in [5]. Alied by itself, this Poisson bracket results simly to C Ñ = 0, where C Ñ is another contravariant tensor, the uer convected time derivative, C Ñ, defined, for incomressible flow considered here, as 3 3 For comressible flows, additional corrections emerge for the convected derivative [17]. 5
Ñ DC C º - ( Ca Db + Da Cb ) - ( W a Cb - Ca W b ), (5) Dt 1 T 1 T where Dº ( Ñ v +Ñ v) is the (symmetric) rate of strain tensor, ( v v) (antisymmetric) vorticity tensor and Ñ v is the gradient of the velocity vector field v. The corresonding dissiative bracket in its most general form is given by [17] as W º Ñ - Ñ is the 3 [ FH, ] =- ò L g e - - W ò W x ò Q d F d C d H d C g e dr d F æ æ d H d H ö æ d H d H ö ö Ca Ñ +Ñ b + Cb Ñ +Ñ a d C d ub d u d ua d u è è ø è ø ø d H æ æ d F d F ö æ d F d F ö ö - Cg Ñ +Ñ e + Ce Ñ +Ñ g d C g e d ue d u d ug d u è è ø è ø ø g e æ d F ö æ d H ö 3 Ñ a Ñ dr g è d u b ø è d ue ø 3 dr, (6) where u b is the momentum density vector field equal to r v b (where r is the mass density), and L g e [ = ]s - 1 and Q g e [ = ]Pas are ositive definite henomenological fourth order tensors symmetric with resect to the following interchanges of indices: a b, g e and ( a, b ) ( g, e ) «««, reresenting the relaxation and viscous dissiation effects, resectively, and the scalar x weights non- affine effects arising from the mismatch in the motion between a artially rigid microstructure (e.g. olymer chain or emulsion drolet) and the matrix fluid. Corresonding to the Poisson and dissiation brackets discussed above, the following general evolution equation for C is obtained by forcing equality of Eqs. (1) and () for any functional deendence on the conformation tensor ( x ) D C Ñ d H º C + x ( Ca Db + Da Cb ) =- L g e, (7) Dt d C g e where ( x ) D C Dt is the Gordon- Schowalter derivative [6]. The emergence of the L tensor in Eq. g e (7) is due to the fact that it weights a quadratic term in the dissiation couling the conformation tensor field C with itself (first term in the left- hand- side of Eq. (6)). As a result, it only aears in that structure evolution equation with the hysical meaning of a relaxation term. Conversely, the absence of Q g e in 6
Eq. (7) is due to the fact that it weights a quadratic term in the dissiation couling the gradient of the velocity field Ñ v with itself (last term in the left- hand- side of Eq. (6)). Therefore it can only aear in a b the stress exression where it has the hysical meaning of a viscous stress contribution. A first observation that can be made by comaring the definition of the Gordon- Schowalter derivative, Eq. (7), against that of the uer convected time derivative, Eq. (5), is that x only weights corrections roortional to the symmetric rate of strain tensor, D. However, with the inclusion of (micro)inertia effects, an additional non- affine term involving the antisymmetric vorticity tensor W, weighted by a new non- affine arameter, z, becomes ossible. Postulating such a term leads to a modified evolution equation ( x V, ) D C Ñ d H º C + x ( Ca Db + Da Cb ) + z ( W a Cb - Ca W b ) =- L g e, (8) Dt d C g e where ( x z, ) D C dt is defined as an extension to the Gordon- Schowalter derivative that takes into account additional, inertial, non- affine motion effects. From material objectivity grounds, the extra, z - weighted term in Eq. (8) is only allowed in resence of inertia [7] and z ¹ 0 reresents our first micro- inertia correction. In the bracket formalism, Eq. (8), it leads to a direct modification of the inertialess dissiation bracket [17], Eq. (6), by adding an extra term [ FH ], extra, defined as d F æ æ d H d H ö æ d H d H ö ö Ca Ñ - Ñ b - Cb Ñ +Ñ a d C d u d u d u d u z è è ø è ø ø ò dr. (9) W d H æ æ d F d F ö æ d F d F ö ö - Cg Ñ - Ñ e - Ce Ñ +Ñ g d C g e d ue d u d ug d u è è ø è ø ø b a 3 [ FH, ] º - extra Note that, due to the antisymmetry of [ FH ], extra, this roosed modification leads to no additional changes to the exression for the extra stress tensor, s, which maintains the same form as in the inertialess formulation, s d H = Q Ñ v + 1 ( - x ) C. (10) d C g e g e b g Note that in this stress exression, in addition to the term roortional to the velocity gradient, which accounts for (in general anisotroic) viscous dissiation effects, a second term aears accounting for a a g 7
structural contribution to the stress driven through the deendence of the extended free energy H on the contravariant conformation tensor C. To comrehensively model micro- inertia we now need to comlete the transition to the full inertial model, which is erformed following the aforementioned transformations. In addition to involving Volterra derivatives instead of artial derivatives, the changes required to transform the inertialess brackets to inertial brackets involve: i) the addition of a generalized contravariant internal structural momentum tensor, w, roortional to an objective evaluation of the rate of change of the conformation field C, into the state sace ztx (, ) ( utx (, ), Ctx (, ), wtx (, )) =, ii) the addition of a structural kinetic energy density (defined as 1 : Z w w with Z an inertial arameter analogous to mass) to the Hamiltonian exression used in the inertialess limit iii) the addition to the Poisson bracket of an anti- symmetric couling term, { FH }, extra that between and x ), as, that establishes the symlectic structure between w and C (analogous to 3 { FH} =- -, extra and iv) a new inertial dissiation bracket, [ FH ] aragrah. ò é d F d H d H d F ù ê ú dr ê d w d C d w d C, (11) ë ú û, inertial, the details of which are described in the following Aart from the case x z, 0, where there is a direct analogy to the harmonic oscillator examle, the transformation from the inertialess to the inertial brackets requires an ad- hoc modification to the rule used there to accommodate the resence of the non- standard non- affine couling terms. In articular, it is the second order nature of the conformation field that introduces that comlexity in dissiation through the introduction of couling terms between various comonents of the velocity gradient and the conformation tensor in an analogous fashion to the introduction of similar couling terms in order to obtain an objective definition of the time derivative, the uer convected derivative, defined in Eq. (5). Thus, guided by a comarison to the definition of the extended Johnson- Segalman derivative, Eq. (8), we ostulate (to be checked afterwards for consistency) that the equivalence between the Volterra derivatives in the two bracket descritions, inertial and inertialess, is obtained through the extended transformation 8
æ x æ d d ö x æ d d ö ö d + C C d w g Ñ +Ñ e + e Ñ +Ñ g g e d ue d u d ug d u d è ø è ø «- R d C g e, (1) z æ d d ö z æ d d ö - Cg Ñ - Ñ e + Ce Ñ - Ñ g d ue d u d ug d u è è ø è ø ø where the fourth order relaxation tensor R is the new structural relaxation arameter aearing in the inertial dissiation. Following the damed harmonic oscillator examle, it is defined as the inverse of L, i.e. 1 L = +. - 1 R g e =L g e, imlying (using all available symmetries) that mnrmng e ( d a g d b e d a e d b g ) Following that transformation rule, the new, symmetric, inertial dissiation bracket can now be develoed from the symmetric first and third terms of the inertialess dissiation bracket, Eq. (6) as [ F, H] in º - ò R g e æ æ d F x æ d F d F ö x æ d F d F ö ö ö + Ca Ñ +Ñ b + Cb Ñ +Ñ a d w d ub d u d ua d u è ø è ø z æ d F d F ö z æ d F d F ö - Ca Ñ - Ñ b + Cb Ñ - Ñ a d ub d u d ua d u è è ø è ø ø æ d H x æ d H d H ö x æ d H d H ö ö + Cg Ñ +Ñ e + Ce Ñ +Ñ d wg e d ue d u g è ø d ug d u è ø z æ d H d H ö z æ d H d H ö - Cg Ñ - Ñ e + Ce Ñ - Ñ g d ue d u d ug d u è è è ø è ø ø ø æ d F ö æ d H ö - ò Qg e Ñ a Ñ g d u è b ø è d ue ø 3 d r (13) 3 dr. The form of the dissiation bracket in Eq. (13) can then be interreted as involving a quadratic term of the extended Voltera derivative, indicated by the right- hand side of Eq. (1). This extended derivative is justified as introducing in dissiation analogous changes to d d w to those already seen in forming the uer convected derivative, Eq. (5), in modifying the material derivative DC. The only Dt difference is that now those changes are mediated by general material arameters ( x z, ) instead of having secified values ( - 1, - 1). It is in fact material objectivity that restrains those values, and the Jacobi 9
identity (as the correction terms aear in the Poisson bracket) with only one alternative set ( 1,1 ) alicable for the case of lower convected derivative, aroriate to exress a materially objective form of the time derivative for a covariant second order tensor. In fact, as will see below, d H = w= C is d w a time derivative that is now modified in the dissiation bracket through the non- affine corrections mediated through the x z, arameters. In this way we can most clearly see why those non- affine corrections should be seen as art of the dissiation even when the inertialess formalism is considered (at least the x - deendent term) giving rise to the antisymmetric contribution to the dissiation, which is the second term in the right- hand- side of Eq. (6). Inertial model equations- Emloying the inertial Hamiltonian/dissiation structure mentioned in the revious aragrah leads to the following micro- inertial model equations for a general conformation tensor variable: First, an evolution equation for the momentum variable, wº w/ Z, is develoed as ( x ( ) z ( )) Ñ d H Zw =- - Rg e wg e + Cg De + Dg Ce + W g Ce - Cg W. (14) e d C Second, the evolution equation for the conformation tensor now simly reads as Ñ C = w. (15) This relationshi guarantees that w is also contravariant, and its emergence demonstrates the strict restrictions imosed by the inertial Poisson- dissiation bracket structure, consistent with rior studies []. Finally, the exression for the symmetric extra stress tensor that results is d H d H Ñ s = Qg e Ñ g ve + 1 ( - x ) Cb g + Zwb g - x Cb g Zwa g. (16) d C d w In this exression, the last two terms, roortional to Z, are additional contributions to the stress that are related to microstructural inertia. However, they are not the exclusive mechanism through which microstructural inertia effects influence the stress tensor as the configuration of the conformation tensor is also altered by an additional nonaffine term roortional to z which, as we shall demonstrate later on in an examle, also accounts for some of the effects of microinertia. While it may be desirable to relate the terms roortional to Z directly to Reynolds and acceleration stress contributions, given the level coarsening effect that tyically accomanies the transition from microscoic to macroscoic levels of descrition, such a direct relationshi is not always ossible. Instead, what we roose is the use of a g a g Ñ 10
comlementary, system- secific, asymtotic microscoic theory redictions for the system s rheology as the basis of examining and interreting Eq. (16), as shown below in connection to the dilute emulsion alication. The resultant inertial macroscoic model (Eqs. (14)- (16)) is similar to that used in reviously efforts to model micro- inertia effects, for examle in liquid crystalline systems [17]. However, in addition to the traditional inertial arameters Z (associated with the structural momentum variable w ), an additional, second non- affine arameter z, not aearing in [17] is introduced. Finally, note that, consistently, one can recover the inertialess equations, Eqs. (7) and (10), from the inertial ones, Eqs. (14) - (16), in the limit Z 0, z 0. Alication to dilute emulsions- The ractical relevance of the inertial equations, esecially the role of the new non- affine term scaled by z, is now demonstrated in an alication to dilute emulsion in shear flow where micro- inertia effects have been recently shown to be imortant [11, 1]. In this alication and for this secific limit of simle shear flow, the second order contravariant tensor field, C( t ), considered as homogeneous, reresents an ellisoidal emulsion drolet, and its eigenvalues and eigenvectors corresond to the square of the magnitude and orientation, resectively, of the semi- axes of an ellisoid, exactly as in the inertialess limit [5, 7]. The full exressions in the inertialess limit for the evolution equation for C and the extra stress s have been reviously described in [7]. In that work, the Hamiltonian H is taken to comrise of a sum of contributions from the kinetic energy, the interfacial energy and a mixing entroy. At the same time, the constraint det ( C ) = 1is enforced in the model equations to emulate volume conservation for incomressible emulsion drolets [7,8]. The key ste in develoing the final model equations for C and s, as they have been resented, is the selection of the henomenological matrices L g e and Q g e such that the resultant macroscoic stress exression for s exactly matches existing asymtotic theory for the stress tensor [14,15] in the inertialess limit. By extending the inertialess model to its inertial variant following the framework develoed here, recent asymtotic theory that incororates first order inertia effects [1] (moving beyond asymtotic results in the inertialess limit described in [14,15]) can now also be matched through the Z and z arameter- mediated terms if those arameter values are defined as and ( 7l + 30l + 10)( l + 3) ( 4l + 16)( l + 1) * Z= Re µ f, (17) 11
1 z = Oh ( l + l + ) 160 3 3 1 ( l + ) 9 19 16, (18) while all other arameters retain the values reviously identified in [8] in the inertialess limit. In Eqs. (17) and (18) f is the volume fraction, l º µ / µ is the viscosity ratio (with d µ the disersed hase viscosity d and µ the matrix viscosity), and º µ g (with º a the characteristic drolet radius, g the * Ca a / surface tension and r the mass density) and Re a r µ º characteristic time scales that, when scaled * 1 by a time scale for the flow, g -, lead to the Caillary and article Reynolds numbers, Ca º µg g a and Re º a r g µ, resectively. In addition, the Ohnesorge number (Oh) naturally aears and is defined 1/ * * as Oh º µ /( r at) = Ca / Re [30]. The emergence of Z and z as related to * Re rovides further justification for the inertial brackets used and the generality of the aroach discussed here. Furthermore, note that in the infinitely dilute emulsion limit, f 0, Z 0 but z does not necessarily vanish, even for small Re, if Ca is sufficiently small. Furthermore, in this limit w is no longer an indeendent state variable as Eqs. (14) and (15) can be combined to a single evolution equation for the conformation tensor. In this way, it is clear that the effects reresented by z are distinct from those entering through w in traditional microstructural inertia theories, e.g., in liquid crystalline systems [17]. Thus the final model equations under these constraints ( f 0, Z 0 ) simly comrise of the emulsion model equations in the inertialess limit described in [7], but with ( x ) D C Dt relaced by ( x z, ) D C Dt from this work. The resultant model redictions are in agreement with simulation results in the resence of micro- inertia [11] for simle shear flow and a revious literature correlation given by 1 1 11 1 ( ) N s º ( s - s ) s = cot q [30]. These results are summarized in Fig. 1 where all three methods show a change in the sign of the first normal stress difference, N 1 s, from ositive to 1 negative, as q increases above 45 0. The hysical origin for the shifting orientation of the ellisoids towards the velocity gradient direction with micro- inertia (i.e. q > 45 ) is related to the fore- aft asymmetry of streamlines near the emulsion drolets [13,31]. Clearly z is an imortant model arameter towards caturing micro- inertia effects. The extra contributions to the first and second normal stress differences arising in the asymtotic limit of stress equation of the model for small Re are N µ - z g and 1, extra 1
N µ z g resectively. These contributions dominate at increasingly larger values of, etra Re, since z µ Re, resulting in a change of the normal stress differences relative to those seen in the inertialess limit where z = 0. The increasing disagreement between the model and simulation results in Fig. 1 at larger Re numbers is attributable to the model arameterization based on asymtotic theory which is strictly valid only for small Re and Ca numbers. Although it is ossible to obtain better quantitative agreement by non- linearly extending the deendence of z on Oh by fitting the model to the data at larger Re (whilst still reserving the asymtotic features of the model), this is beyond the scoe of the resent work. Indeed, higher order corrections are known from susension theory to involve nonlinear deendencies to the article Reynolds number [33]. A word of caution is warranted here. Although the macroscoic inertial theory based on an internal conformation tensor is solidly based on theory that is exected to remain valid for general flow fields, i.e., always thermodynamically consistent and materially objective, the microscoic interretation of the conformation field is not. In articular, micro- inertia effects introduce boundary layers and nonlocal effects that hide the direct influence of the article geometry that is so rominent in the inertialess limit. As a result, under those conditions, the conformation tensor is reresenting a more abstract, effective field quantity, and the material arameters change to acquire effective values that are not easy to directly relate to the underlying microstructure and material roerties, as identified in this work in the articular examle of dilute emulsions in the limit of simle shear flows. Still, the observable macroscoic results (i.e., the stress tensor redictions) are exected to be valid, rovided that the material arameters are aroriately adjusted (i.e., they become adjustable henomenological coefficients). This seems to reduce the theory to one of the many available henomenological theories. However, by continuity, it is exected that the material arameters reduce to their asymtotic values in the limit of small inertial shear flows. Therefore, only a few adjustments to the arameters are necessary to use the model more broadly, which can be determined through comarison against targeted exeriments, thus making the fitting rocess feasible and robust, and demonstrating the ower of the modeling aroach illustrated here. In summary, this alication to emulsions demonstrates the key, and more general, innovations introduced in this work: how to move from an inertial to inertialess formalism, and vice versa, in a self- consistent and generic fashion through the bracket aroach of NET, and how a new non- affine term mediated by a new arameter z, which varies inversely roortional to the Ohnesorge number, allows 13
for the lowest order micro- inertia effects (in a manner that does not necessitate introduction of any additional structural momentum variable w ) to be directly incororated into inertialess flow models. We anticiate that this framework will find broader alication in modeling microinertia in comlex fluid flow. FIG. 1. The relationshi between the magnitude and sign of the ratio of the first normal stress difference N and shear stress 1 s, and the orientation of the ellisoid q relative to the flow direction for simulation 1 results [1] (filled symbols) and model redictions (oen symbols) direction as a function of Reynolds number at two different caillary numbers. The solid line reresents the correlation of Jannseune et al. [31] (see text). Viscosity ratio ( l ) in simulation and model calculations is 1. Note that viscous contribution to the stress is not included in these calculations. This material is based uon work suorted by the National Science Foundation under Grant No. CBET 31146. Any oinions, findings, and conclusions or recommendations exressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The authors are indebted to the editor John Hinch for his hel in clarifying the effects of inertia in microscoic multihase systems. References [1] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon Press, 1959). [] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids. Volume 1. Fluid Mechanics (Wiley, New York 1987). [3] M.W. Johnson and D. Segalman, A model for viscoelastic fluid behavior which allows non- affine deformation, J. Non- Newtonian Fluid Mech., 55 (1977). [4] M. Doi and T. Ohta, Dynamics and rheology of comlex interfaces. I, J. Chem. Phys. 95, 14 (1991). 14
[5] P. Maffettone and M. Minale, Equation of change for ellisoidal dros in viscous flow, Non- Newtonian Fluid Mech. 78, 7 (1998). [6] G. I. Taylor, The formation of emulsions in definable fields of flow, Proc. Roy. Soc. Lond. A. 146, 501 (1934). [7] G. I. Taylor, The viscosity of a fluid containing small dros of another fluid, Proc. Roy. Soc. Lond. A. 138, 41 (193). [8] J. Oldroyd, The elastic and viscous roerties of emulsions and susensions, Proc. Roy. Soc. Lond. A. 18, 1 (1953). [9] G. Batchelor, The stress system in a susension of force- free articles, J. Fluid Mech. 41, 545 (1970). [10] C- J Lin, J. H. Peery and W. R. Schowalter, Simle shear flow round a rigid shere: inertial effects and susension rheology, J. Fluid Mech. 44, 1 (1970). [11] H. Haddadi and J. F. Morris, Microstructure and rheology of finite inertia neutrally buoyant susensions, J. Fluid Mech. 749, 431 (014). [1] X. Li and K. Sarkar, Effects of inertia on the rheology of a dilute emulsion of dros in shear, J. Rheol. 49, 1377 (005). [13] R. V. Raja, G. Subramanian and D. L. Koch, Inertial effects on the rheology of a dilute emulsion, J. Fluid Mech. 646, 55 (010). [14] R.K. Singh and K. Sarkar, Inertial effects on the dynamics, streamline toology and interfacial stresses due to a dro in shear, J. Fluid Mech. 683, 149 (011). [15] W. Schowalter, C. Chaffey and H. Brenner, Rheological behavior of a dilute emulsion, J. Colloid Interf. Sci. 6, 15 (1968). [16] N. Frankel and A. Acrivos, The constitutive equation for a dilute emulsion, J. Fluid Mech. 44, 65 (1970). [17] A. N. Beris and B. J. Edwards, Thermodynamics of flowing systems: with internal microstructure (Oxford University Press, New York, 1994). [18] A. N. Kaufman, Dissiative Hamiltonian systems: a unifying rincile, Phys. Lett. A 100, 419 (1984). [19] P. J. Morrison, Bracket formulation for irreversible classical fields, Phys. Lett. A 100, 43 (1984). [0] M. Grmela, Bracket formulation of dissiative fluid mechanics equations, Phys. Lett. A 10, 355 (1984). [1] M. Grmela and H.C. Öttinger, Dynamics and thermodynamics of comlex fluids. I. Develoment of a general formalism, Phys. Rev. E 56, 660 (1997). [] H. C. Öttinger and M. Grmela, Dynamics and thermodynamics of comlex fluids. II. Illustrations of a general formalism, Phys. Rev. E 56, 6633 (1997). [3] H. C. Öttinger, Beyond Equilibrium Thermodynamics (Wiley, Hoboken, 004). [4] B. J. Edwards, An analysis of single and double generator thermodynamic formalisms for the macroscoic descrition of comlex fluids, J. Non- Equilib. Thermodyn. 3, 300 (1998). [5] B. J. Edwards and A. N. Beris, Non- canonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity, J. Phys. A- Math. Gen. 4, 461 (1991). [6] R. J. Gordon and W.R. Schowalter, Anisotroic fluid theory: a different aroach to the dumbbell theory of dilute olymer solutions, Trans. Soc. Rheol. 16, 79 (197). 15
[7] G. Ryskin, The extensional viscosity of a dilute susension of sherical articles at intermediate microscale Reynolds numbers, J. Fluid Mech 99, 513 (1980). [8] P. M. Mwasame, N. J. Wagner and A. N. Beris, On the Macroscoic Modeling of Dilute Emulsions Under Flow, J. Fluid Mech. (submitted), (017). [9] B. J. Edwards, M. Dressler, M. Grmela and A. Ait- Kadi, Rheological models with microstructural constraints, Rheol. Acta 4, 64 (003). [30] G. H. McKinley and M. Renardy, Wolfgang von ohnesorge, Phys. Fluids 3, 17101 (011). [31] T. Jansseune, J. Mewis, P. Moldenaers, M. Minale and P. Maffettone, Rheology and rheological morhology determination in immiscible two- hase olymer model blends, J. Non- Newtonian Fluid Mech. 93, 153 (000). [3] P. M. Kulkarni and J. F. Morris, Susension roerties at finite Reynolds number from simulated shear flow, Phys. Fluids 0, 04060 (008). [33] G. Subramanian, D. L. Koch, J. Zhang and C. Yang, The influence of the inertially dominated outer region on the rheology of a dilute disersion of low- Reynolds- number dros or rigid articles, J. Fluid Mech. 674 307 (011). 16