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Can Conventional Continuum Mechanics Provide Adequate Simulations for Flows of Complex Fluids? J. R. A. Pearson* Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, U.K.

It is argued in this article that the remarkable successes achieved in simulating the flow of Newtonian fluids in complex geometries cannot currently be achieved for most important flows of complex fluids. The examples chosen to illustrate this thesis are the processing of polymer melts, the flow of wormlike micellar surfactant fluids, the flow of suspensions, and the flow of non-Newtonian fluids in porous media. Despite the efforts of rheologists over the past five decades, fully adequate rheological equations of state cannot be provided for any of these fluids. This can be attributed to the mesoscale structures (within the fluids viewed as continua) underlying their rheological complexity. It is concluded that simulations carried out at several scales, using suitable up-scaling techniques, will be necessary to provide numerical predictions at the process (continuum) level. Empiricism will continue to play a role in process engineering. 1. Introduction The second world war provided a great impetus to science and technology. Rapid advances in nuclear science, electromagnetics, and aerodynamics coincided with the rapid development of large-scale computation. This was continued after the war by the enthusiastic application of these and other developments in material science, chemical engineering, computer science and technology, and mechanics generally. In particular, it was confidently asserted by many that the solution of all problems in hydro- and aerodynamics would be possible using high-speed electronic computers. Computer software suitable for giving numerical solutions to the NavierStokes equations for Newtonian fluids was expected to provide engineers with all the information they needed; there would no longer be any need for repeated experiments (trials) to achieve acceptable designs. Inevitably difficulties would be experienced with turbulent flows: both direct numerical simulation accounting for all the eddy scales of significance and empirical turbulence models (e.g., k, ω or k, ) had their advocates. Commercial software (e.g., FLUENT) provided both options. The same arguments were advanced when the mass and momentum conservation equations were coupled to the energy equation (the heat transfer problem) and even to those for chemical reaction in multicomponent, but still Newtonian, single-phase liquids. Thus combustion problems, often implying dominant effects of gas dynamics, have been tackled hopefully, if not always successfully. This apparently successful computational approach to predicting the single-phase flow of Newtonian fluids has led many to believe that an analogous approach would be possible for rheologically complex, i.e., non-Newtonian, fluids. Whether the complexity comes from dissolved polymers or surfactants, from suspended solids, from dispersed gases or immiscible liquids, or from a combination of some of these, the approach has usually been a continuum mechanical one. (By continuum mechanical,1-3 I mean that all the relevant variables can be defined at every point within the space occupied by the fluid in question and are continuous differentiable functions of position. At phase boundaries, appropriate boundary conditions are * Tel.: +44 1223 315576. Fax: +44 1223 353099. E-mail: [email protected].

applied. Within any fluid, integro-differential equations are assumed to provide a mathematical description of the system.4) In its simplest form, an equivalent single-phase fluid with nonNewtonian properties, e.g., viscoelasticity, replaces the Newtonian fluid; a more elaborate (nonlinear as opposed to the linear Newtonian) relation between stress and strain, known as a rheological equation of state4 (REoS), is used to represent the (Cauchy) stress in the momentum conservation equation. In the formal continuum mechanical theories that retain the identity of the separate phases, the continuum is composed of separate “interpenetrating” or “interacting” continua.5 An example of this approach is provided by the now traditional reservoir flow equations for multiphase flow in porous media.6 Rheology can be defined7 most simply as the study of the mechanical response of deforming continua. In the case of complex fluids, this discipline now includes experimental investigations into structural effects at scales smaller than the dimensions of the sample being deformed as well as full-scale measurements of overall deformation and stress which provide direct continuum data for REoS. Theoreticians also include detailed consideration of structure in some of their models; in all cases the link to a continuum model that does not separately consider structural elements is maintained.8-10 This is usually achieved by defining continuum averagessensemble, time, or spatialsof structural characteristics: evolutionary equations for these additional structural variables, together with stress relations that are functions of both deformation rate and these additional structural variables, replace the previously defined REoS. In our context, the role of the rheologist is thus clear: it is to provide the relevant momentum conservation equations to replace the standard Navier-Stokes equation in engineering calculations on the flow of complex fluids. The rest of this article is concerned with some details of how this can best be achieved and of the limitations on what is currently available to engineers. Four examples are used to develop this thesis: polymer solutions and melts; surfactant micellar dispersions; suspensions of solid particles; multiphase effects in porous media. As an aside, it is worth noting that, at the technological level, knowledge can conveniently be ordered as follows: (a) Phenomenological, that is, what happens at the simplest descriptive level: When a manufacturing process works as expected on the basis of limited expectations, no further investigations or knowledge is thought to be needed;

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(b) Empirical:11 Simple functional relations are sought between “inputs” and “outputs” in processes where improvements in cost, efficiency, or quality are felt necessary; these are often no more than rules of thumb. (c) Model-based and predictive: This is the level at which established scientific theories and knowledge are brought to bear on engineering problems, with the eventual aim of optimizing some process a priori, rather than by trial-and-error methods. We shall concentrate here on the last of these levels, while recognizing that the second is the norm in much of engineering. 2. Flow and Rheology of Polymeric Fluids. Modeling of Polymer Processes Chemically this includes a wide range of materials. Here we are concerned with long-chain flexible polymers as such: these may be melts, in which case they are inevitably entangled, or solutions; solutions may be dilute, in which case individual chains do not interact, or concentrated, in which case entanglement dominates their properties. Chemical physicists have analyzed their response to deformation in statistical mechanical terms for over 60 years.8,9,12,13 This involves consideration of each polymer molecule, and its response to deformation, in a statistical sense. There have been three distinguishable approaches. The first relates to dilute solutions where the typical molecule is treated as a finite number of independent “freely jointed” bonds, either rigid or elastic. Each of these links interacts with its surroundings, this being the interstitial Newtonian solvent. The solvent is assumed to move under a uniform (“affine”) rate of deformation and to apply drag forces to the links according to a linear law, and the links are also subject to Brownian forces. In a statistical (average) sense, the molecules are deformed by the flowing polymer from their (mean) equilibrium configuration. A polymer contribution to the total stress in the fluid can be calculated from their mean deformation. The second is derived directly from the highly successful theory of rubber elasticity,14 and deals with entangled solutions in terms of reversible cross-links that provide a network structure and constrain the chain segments between them. An “affine” deformation assumption is applied to the cross-links. A random breaking and re-forming rule for the cross-links provides a stress/ strain relaxation mechanism. The thirdsmost obviously applicable to meltssreplaces the transient cross-link assumption by a “reptation” argument9 which associates the required relaxation mechanism with the random reorientation of (free) chain ends. Again an “affine” deformation pattern is assumed to apply in a statistical sense. (Some authors treat the “affine” assumption as being the same approximation as “equivalent medium” theory.) The details are not important for our purposes, largely because the final result of the analyses and calculations are expressed as an REoS. Surprisingly, the archetypal Oldroyd eight-constant model REoS that was derived from purely continuum mechanical considerations is remarkably similar to the majority of those derived using statistical mechanical arguments.15 When any one of these REoS is compared to actual rheological data obtained experimentally for a typical polymer system, it has been found that considerable alteration has to be made to its basic form to get acceptable quantitative agreement with even a limited portion of the data.16,17 This has led a growing number of workers to use a purely discrete elemental approach to cover a complete (nonuniform) flow field, to avoid the need to derive an REoS at all.18,19 The separate elements (chains and subchains) are subject to Brownian, mean advective, and local internal

forces. The averaging process (which yields continuum variables) takes place over the individual translating elements lying within a “representative volume” at the end of a step in the simulation. Because of its commercial importance, I shall concentrate on polymer melt processing. The earliest flow analyses of such processes date from the 1940s, and a series of textbooks20-23 subsequently provided frameworks for modeling the most important ones. Two main classes were identified: confined, pressure-driven flowsssuch as screw extrusion through dies and injection moldingsand unconfined flowsssuch as fiber spinning and film blowing. An alternative classification into continuous or cyclic was also introduced, but in mechanical terms (following a particle) all flows are unsteady. In all of these heat transfer plays a significant or dominant role, leading to large (in terms of rheological behavior) temperature gradients and phase changes. In die extrusion, melt slip at metal walls can take place, sometimes leading to unsteady extrusionsvariously called melt fracture or melt flow instability24sand grossly deformed product. In film blowing, similar unsteadiness, or asymmetric bubble geometry, is frequently encountered.44 In terms of the levels of knowledge given in the last section, it is fair to say that much, if not most, processing is done on standard commercial equipment by processors who adopt a largely phenomenological approach to achieving acceptable end products, and who evolve their own empirical methods to control or improve them. A limited number of processors, but a larger proportion of machinery manufacturers, have attempted to adopt a fully model-based approach to process design and development.25,26 It is instructive to analyze why such model-based approaches are not fully established and standardized. The first difficulty is the wide range of raw materialsshomopolymers, copolymers, blends, foaming systems, filler or fiber containing mixturess employed in the industry. Each of these will have different rheological and thermal properties and different end uses. However, successful modeling for a single well-defined process should be sufficient encouragement to extend such an approach. The second difficulty, to which I have alluded earlier, is the provision of adequate equations of state for the material in question. There are now numerous examples of attempts to use rheological data to yield useful REoS and to validate them by simulating complex flows. Ideally the validation should be for an entire and realistic polymer process, using commercially available software (such as POLYFLOW), where the necessary rheological data had been obtained on standard commercially available equipment. The third is that most processes start and end with solid material. To model the process completely, the rheological behavior has to include the many physical statessglassy, crystalline, elastomeric, or liquidsthrough which the material passes. (A crucial complication is that the material cannot be regarded as being continuously in thermodynamic equilibrium, not even locally. The time scales needed to reach local equilibrium under constant environmental conditions overlap the time scales over which the rheological and thermodynamic variables, i.e., the environmental conditions, change during the process.) Given sufficient off-line experimental data, a relationship between current stress and past history of deformation and temperature could, in principle, be derived. Corresponding thermodynamic parameters such as thermal conductivity also depend upon the same deformational and thermal history. It soon becomes clear that any analysis has to be partial and approximate. The full process has to be treated as a series of

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unit operations, though, unlike the normal use of unit operations in the design of chemical plant, the description/definition of the inputs and outputs to and from any unit operation presents huge difficulties. Thus, in continuous screw extrusion, the flow of granules from hopper to rear end of screw, the conveyance and melting of granules in the melting section, the pumping of melt in the transition and pumping sections, the mixing and thermal homogenization in any mixing section, and finally the flow through the die have effectively to be treated as separate processes. Each of these units is intended to achieve a different function, so detailed attention need be concentrated only on the relevant phenomena concerned. Simplified mechanical models for these have been given, usually in terms of a homogeneous melt. Problems arise because the “units” are intimately coupled, particularly as far as control is concerned. When stability of operation is important (e.g., avoidance of surging), it is the behavior of the whole dynamical system that has to be studied, because it is the interactions between the “units” that lead to surging. Similarly, in continuous film blowing, extrusion through a circular annular pipe die is followed by heating of the pipe (this is not always the case) while subjecting it to internal pressure and in-line tension. This precipitates simultaneous radial expansion and axial extension at which stage (air) cooling of the bubble27 is imposed, leading to freezing. This bubble expansion/extension process has been successfully modeled and provides a striking example of what can be achieved in terms of prediction [see (c) in section 1 above]. Success has been based on the assumptions made in the model: locally irrotational distortion; use of variables averaged across the film thickness; steady inlet and wind-up conditions. This effectively simplifies the REoS required and the set of model equations to be solved, which are strictly applicable to this process alone. Simulation is carried out by special-purpose software. 3. Flow and Rheology of Micellar Surfactant Fluids Chemically this can be as wide a class of fluids as polymer solutions. If the concentration of surfactant is below the cmc (critical micelle concentration), the solution is Newtonian and so introduces no complication in flow modeling. However, above the cmc, a wide range of micellar structures can arises spherical, cylindrical or wormlike, lamellar or onion-shapeds conferring gel-like viscoelastic properties on the solution (more properly termed a dispersion). Of these the wormlike systems can display entangled structures similar to polymer solutions. The rheology of these complex fluids has been widely studied, and the REoS proposed for polymeric systems have been adapted to apply to wormlike micellar fluids (WLMF).8 There is one difference, which actually seems to dominate the rheological behavior, caused by the breaking and rejoining of the micellar worms. The kinetics of this particular process, together with that of reptation, provides the most important rheological time scale. One application for these fluids is in hydraulic fracturing (HF) of oil and gas reservoirs,28 which is probably their dominant global application in volume terms. In some cases the WLMF itself is pumped down a well to create a large fracture in a hydrocarbon-bearing formation at the bottom of the well. In others it is pumped as a foam mixed with gas. In the later stages of the fracturing process, sand or ceramic particles (proppant) are added. Once these have entered the fracture they serve to hold it open, to facilitate production of the hydrocarbon (oil or

gas) in the formation. An important advantage of WLMF is that it loses its gel-like properties and behaves like water when mixed with oil; it can then be removed from the fracture and the neighboring formation by the produced fluid. There is a clear need in terms of engineering design for predictive models and simulations of the process. This involves mixing processes, multiphase turbulent flow in the “down” pipe in the well, multiphase nonturbulent flow in the fracture being propagated, and low Reynolds number flow in the porous formation and closed (propped) fracture. The relevant length scales for describing the geometry of the whole system range from 1 µm (micron) for the pores in the formation to 100 m for the extent of the fracture. As in the previous examples in polymer processing, the flow field can be broken down into a series of connected “unit” flow processes, and separate assumptions and approximations are made to derive pressure-drop/flowrate relations in each “unit”. If the fluid (or fluid mixtures) involved could be treated as Newtonian, then detailed simulations could be provided for each “unit” flow process, using whichever CFD (computational fluid dynamics) software was most appropriate, with a reasonable expectation of providing reliable predictions. However, in the case of WLMFs, we do not have a reliable REoS29 that can be used for arbitrarily complex geometries and more importantly we do not know of a means of generating a continuum REoS given a sufficient set of rheological measurements. This difficulty, also encountered for polymeric fluids as discussed in the last section, has led some workers to propose direct simulation at the micellar scale, i.e., to follow the paths of individual (dividing and recombining) micelles, however coarsely the micelles are represented. The bulk of the fluid material is still water molecules, so the mechanics and kinematics of the water flow must still be treated in a largely continuum fashion. Rather than adapt the FE, FD, or FV methods that are used in most CFD, the alternative LB (lattice-Boltzman)30 has been advocated by some, on the basis that the flexible lattice methods of the LB technique might be adapted to accommodate the behavior of wormlike micelles. This can be viewed as a step back from continuum mechanics toward statistical mechanics. Among those currently developing useful predictive models, it is noticeable that empirical [see (b) in section 1] relations connecting continuum-averaged variables are sought and employed. In our HF case, in down-pipe flow this leads to crosssectional averaging for which pressure gradient/flow rate curves are appropriate. If two or three phases are present, then relative slip between the phases is possible so that differences between relative fractions of the phases in the input or output and the instantaneous fractions within the pipe arise. If transient variations in the input fractions are imposed, then axial mixing will take place leading to different transient variations in the output. Direct simulation of these processes is not currently feasible for foamed and/or proppant-laden WLMF. (Even for Newtonian fluids in turbulent flow, it would be difficult and unreliable.) However, simpler one-dimensional models can readily be provided to match experimental observations: such specialized functional relations are what is often deemed empiricism, although analysis of locally steady-state behavior can sometimes allow the functional relations to be derived from a variant of the lubrication approximation, using direct simulation where appropriate. A two-dimensional set of width-averaged point variables can similarly be used to describe flow within the fracture. Again, local quasi-steadiness can simplify the problem to be tackled. The key point in both cases is that the

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simulation problem involved becomes a function of just one space variable and can be validated against a specially designed experiment. However, where the down-pipe flow accelerates and turns orthogonally to pass through the perforations in the pipe into the fracture beyond, a complex flow field that cannot be modeled other than by using the full three-dimensional equations arises. No progress has been made on this. Direct simulation of WLMF sand slurry flow has not been attempted, although it remains a problem of great interest and importance. This is discussed in more detail in the next section. The question of how WLMF leaks into the permeable formation that is being fractured is technologically equally important. This issue will be discussed in section 5. 4. Flow and Rheology of Suspensions This is an example where difficulties can arise even when the suspending fluid is Newtonian, the particle concentration is uniform, and inertia forces can be neglected.31-33 Any such difficulties are compounded when the suspending fluid is complex, where buoyancy and inertial forces are important and where spatial variations in particle concentration occur. To clarify these various issues, it is useful to consider the various characteristic lengths relevant to the problem. The first, d, is that of the suspended particles (for spheres it would be the diameter); the second, l, is the mean separation of particles (measured in the suspending medium); the third, Lmin, is the least dimension of the flow field, with Lmax its greatest significant dimension. If R1 ) d/l , 1, the suspension is very dilute. In general, particles will not then interact with one another. The effect of the particles on the fluid rheology will be small, so the kinematics of the flow field will be largely determined by the mechanics of the suspending fluid alone. The motion of any particle relative to the local suspending fluid will be determined by a balance between the gravity force due to the density difference between particle and fluid, inertial forces, and drag forces caused by the relative motion between fluid and particle. Provided an REoS is known for the fluid, and R2 ) d/Lmin , 1, simulation of the flow of individual particles will be practicable. However, the suspension is not usually dilute. If R1 . 1, then we can expect lubricated granular flow. Unless the individual particles are smooth at the length scale l, then particle-particle contacts will occur and the problem is not normally treated by fluid mechanical techniques. If R1 ) O(1), as is often the case,31,32,34 then particles in a deforming flow will interact, though will not usually touch one another. A range of approaches to analyzing the flows involved is possible, both continuum mechanical and discrete (following each particle or statistical mechanical). If both R2 and R3 ) l/Lmin are ,1, particle-particle and fluid-particle interactions can be expected to take place in a locally uniform mean deformation field, whose gradient length (this is defined as |∇v/ ∇∇v|, where v is the local mean velocity field) is taken to be Lmin (this is the “affine” assumption mentioned earlier). For the present we neglect buoyancy forces and suppose the particle concentration to be constant. To tackle the flow problem on the Lmin and Lmax scales, it is obviously sensible to seek an REoS for the supposedly uniform mixture, i.e., one that applies on a scale LREoS where l , LREoS , Lmin. The mixture is thus treated as a complex fluid, whose properties can be determined, either by experiment or by simulation taking into account each particle in a suitable representative volume. This has not been studied as well as have the phenomenologically more arresting cases

of polymer solutions or WLMF, mainly because buoyancy effects are what lead to interesting and important effects. If gravity cannot be neglected, relative motion between the phases will arise. This motion will now be a balance between buoyancy and inertial forces, migration forces caused by the mean deformational shear flow, and drag forces. Continuum theories for this situation exist using as separate continuum variables the local particle concentration, the local mean fluid velocity, and the local mean particle velocity. At this local LREoS scale, the aim is to express the relative velocity between the phases as a function of the local mean shear gradient in the volume-averaged velocity field, the particle concentration, and the buoyant force on the particles. These can be regarded as empirical functions to be either obtained by experiment or calculable by simulation,35 taking into account every particle (the statistical mechanical problem) and assuming an REoS to be given for the suspending fluid. Either of these represents a very demanding task. Once these functions are obtained, the continuum problem can be tackled on the Lmin scale, and will be determined not only by boundary conditions but also by the overall rheological behavior of the fluid mixture. This means deriving an REoS for the fluid as a whole, different from that for the suspending fluid. As a first approximation, it could be taken to be the same as that for neutrally buoyant particles in the same fluid at the same constant concentration, but that only simplifies the problem slightly. Note that the gradient length scale has initially been taken to be Lmin; however, there are circumstances (e.g., near horizontal boundaries) where self-induced gradients in the particle concentration will arise on an autonomous length scale l < Laut < Lmin and even destroy the affine deformation assumption when Laut approaches l. As explained in the preceding two sections, ad hoc modeling for specific phenomena is the norm. Severe approximations are made before simulation is attempted, whether it is direct, in which case every particle is followed individually (commercial software to do this is available in two and three dimensions), or continuum mechanical, using specialized appropriate constitutive relations, retaining particle concentration as a field variable. From our point of view such restricted continuum approaches are an extension of traditional (pre-electronic digital computer era) methods of applied mechanics, with roots in empiricism. Computation is treated as an alternative to laboratory experiments, and they are mutually validated by requiring successful quantitative comparison. The case of a suspension of sand in fracturing fluid36 (whether it is a standard polymer solution or a micellar viscoelastic surfactant dispersion) flowing in an HF operation is not easily simplified. There is not just one characteristic flow dimension as explained in the preceding section for either pipe flow or fracture flow: Lmin , Lmax. In the fracture R2 ) O(1) while Re > 1. This implies that the local flow can be treated as quasisteady only in a time-averaged sense. A single-phase-fluid model is barely appropriate, even if it were to contain particle concentration as a field variable. Gravity forces will lead to relative motion between sand and suspending fluid which in all but uniformly vertical fractures will have a component (albeit small) normal to the fracture surfaces, and thus to an autonomous length scale smaller than the fracture width. Lastly, the asperities on sand particles are such that particle-particle collisions may occur which have to be accounted for. There are as yet few, if any, extensive simulations of the onedimensional steady mean flow problem for sand-laden fracturing fluids, where the fracture walls are taken to be plane and parallel,

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and the independent variable is the coordinate normal to them, based on individual particle tracking. There have been no investigations of behavior at mean shear rates in the “plateau region” for WLMF. 5. Flow in Porous Media Porous media can be conveniently described here as having complex internal pore geometry, where the characteristic pore size is 0.1 mm or less; in the case of permeable rocks or fine filters the relevant length scale is often no more than 1 µm. This means that the molecules or molecular aggregates, causing rheological complexity in fluids flowing through the porous media, can approach the scale of the pores; certainly wall effects cannot be neglected, while the length scale based on the gradient in local deformation rate will approach that of the “molecules” themselves. Flow velocities are usually so small that inertial forces can be neglected, though the characteristic time scale for change of deformation rate (when moving with a fluid particle) will be no larger, and in some cases much smaller, than the characteristic fluid time scale. In the language of viscoelastic fluid mechanics, the Deborah number will be large. For such flows, continuum mechanicians traditionally and conveniently adopt Darcy’s averaging procedure,37-39 now formalized in many ways including that of homogenization.40 This is so that permeation through rocks, for example, can be treated on a scale so much larger than the pore scale that the details of behavior at the pore scale are completely subsumed in a large-scale relation between mean pressure gradient and associated mean fluid flux, the averages being taken over volumes large compared with the pore scale. Early work with Newtonian fluids introduced the notion of a characteristic permeability (a scalar in the context of isotropic rocks and a second-rank tensor for anisotropic rocks). The linearity of the governing equations ensures a linear relation between pressure gradient and volume flux, and stream lines are independent of flow rate or relatively slow changes in flow rate. Enhanced diffusive effects were early recognized as important because of the complexity of many flow paths linking one point in the pore space with another distant point; these are formalized in terms of a mechanical dispersion tensor,41 having the dimensions of length. For nonuniform miscible fluid flow these continuum quantities allow of simple mathematical models well adapted to CFD simulation. Because oilfield applications involve two or three (typically gas, oil, and water) separate fluid phases, simple permeability is not sufficient to describe multiphase flow. Additional continuum field variables, the relative mean proportions of the phases in the pore space (known as saturations), have to be introduced, in terms of which characteristic functionssthe (capillary) pressure differences between the phases, and their relative permeabilitiesscan be defined. Once known, these constitutive functions, characteristic of any particular rock, can be used in Darcy-scale model reservoir equations. These also have been put into CFD form and are routinely used for reservoir simulation.42 However, what might be termed anomalies in this simple treatment are known to arise with complex fluids, particularly if one fluid is being displaced from the porous medium by another fluid with different rheological properties. For rheologically complex fluids the details of the pore-scale flows becomes important: the defining REoS are nonlinear and evolutionary in time. The stream lines are therefore not independent of pressure gradient. If Darcy-scale variables are used, the effective (or apparent) Darcy viscosity is flow rate

dependent and, importantly, the viscosity function (which can be made dimensionless with the zero-shear-rate bulk viscosity) is not simply related to the REoS for the fluid, nor is it independent of rock type.43 For the case of long-chain flexible polymer solutions, it was noticed that at sufficiently high fluxes (Darcy velocities) increases in apparent Darcy viscosity arose when the corresponding simple shear bulk viscosity continued to decrease: this was attributed to high Trouton (transient extensional) bulk viscosity effects whenever fluid had to be accelerated and stretched to pass through throats in the porous medium. For the case of WLMF, differences between the ratedependent Darcy and bulk shear viscosities have been observed and explained qualitatively in terms of thixotropy and shear viscoelasticity. Network models using various geometric elements (flow channels) to represent pores and throats (nodes and connectors) have been developed to represent particular porous media. Single-phase-flow calculations have been carried out for generalized Newtonian fluids, treating each flow channel as a “unit” for which a nonlinear relation between pressure drop and flux can be prescribed (on a standard continuum mechanical basis). The hope has been that quantitative agreement with experimental data would allow reliable simulation using network models. An alternative approach has been to represent the boundaries of the pore space as faithfully as possible from tomographical data in some discretized or mathematical form and then to use an internal lattice on which to carry out LB simulations. These methods can be employed in two-phase and possibly three-phase immiscible sytems, and for some nonlinear fluid rheologies. Mixing processes can in principle be followed at the pore scale by Monte Carlo (random walk) methods superposed on streamline advection. When complex fluids are involved, it is becoming clear that polymeric, surfactant, or microparticle systems are best modeled in terms of their separate constituents. Detailed pore-scale simulation can be carried out by Brownian dynamics methods. In all these cases, the aim is to derive Darcy-scale relations by what can be termed homogenization or up-scaling based on volume averages over the pore-scale system. It is too early to say which approach will be the most appropriate for simulation. An underlying issue is whether the objective of simulation is qualitative (seeking to “understand” the observed phenomenas the approach of a physicist) or quantitative (seeking to predict the outcome of experiments on specified systems and hence avoiding the necessity for a large number of trials to achieve an “optimum design”sthe approach of an engineer). This article is aimed at the latter. 6. Discussion It should be clear from the previous sections that in my view the answer to the question posed in the title is “no”, not for complex flows such are met in engineering contexts. Improvements in engineering design methods for processes involving complex fluidssthe norm in much of chemical engineerings will continue to demand strong collaboration between engineering scientists specializing in the foundations of transport phenomena, including rheology, and process engineers concerned with industrial objectives. Each process (current or proposed) will have particular features that are critical to its perceived behavior and which must be adequately represented in any useful model of that process. Equally there will probably be many fine details that can be neglected for engineering purposes. Some parts of the flow field need only be modeled crudely and empirically, though

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the fine detail of other parts may be crucial to the process, and for which detailed simulations, on several length and time scales, may be necessary not only to “understand” the relevant part of the process but also to design, monitor, and control it. There is nothing new about these statements, for they are really no more than a restatement of the “unit operations” approach. The theorist has always placed great faith in engineering simulations, if only to derive trends in behavior as input and control parameters are varied. What is being suggested here is that the choice of model, with its associated (limited) number of parameters, is as important as the numerical scheme used in the simulations, if any form of selection or optimization is to be feasible within a realistic time scale. The model need not be a fully general one, applicable to a complete range of materials and processes, but should be carefully tailored to do no more than capture the most important phenomena governing a particular technological objective.11 The parameters defining such a model need not all be basic characteristics to be found in standard texts, or dependent on fine experimental techniques chosen to elucidate fundamental behavior, but could be quantities arising at an overall level in a simple experiment covering one or more of the most important phenomena arising in the process. The use of dimensionless formulations will continue to yield advantages by reducing the number of necessary parameters by up to three (or more if chemical reaction and/or electromagnetic effects are included) and by showing which possible phenomena can be neglected when certain dimensionless groups are very small (or very large). Put differently, this provides a way of suggesting and justifying helpful approximations. In developing relevant and useful models for complex situations, it is now clear that separate modeling and simulation for an increasing sequence of length and time scales, from the molecular, O(10-9 m,10-15 s), to those of the largest engineering projects, O(103 m,1011 s), may be the simplest way to provide useful overall predictions. Traditional thermodynamics has effectively considered just the two extremes. Engineers of the future will have to become adept at multiple up-scaling. Literature Cited (1) Truesdell, C.; Noll, W. The Non-linear Field Theories of Mechanics; Encyclopedia of Physics Vol. III/3; Springer: Berlin, 1965. (The relevant chapter is A, where Section 3 provides a important distinction between a purely continuum theory, statistical mechanical theories, and structural theories. Of particular interest is the sentence, “Continuum physics stands in no contradiction with structural theories, since the equations expressing its general principles may be identified with equations of exactly the same form in sufficiently general statistical mechanics.” The rest of the text is not relevant to my thesis.) (2) Truesdell, C. The Elements of Continuum Mechanics; Springer: Berlin, 1966. (This is a more readable version of the Truesdell-Noll approach; of interest is the phrase “transition from discrete point elements to a continuous manifold allows of microstructure”.) (3) Sedov, L. I. Foundations of the Non-linear Mechanics of Continua; Pergamon: London, 1966. (The relevant section is the Introduction.) (4) Oldroyd, J. G. On the formulation of rheological equations of state. Proc. R. Soc. London, Ser. A 1950, 200, 523. (5) Drew, D.; Passman, S. L. Theory of Multicomponent Fluids; Springer: New York, 1999. (6) Barenblatt, G. I.; Entov, V. M.; Ryzhik, V. M. Theory of Fluid Flows through Natural Rocks; Kluwer: Dordrecht, 1990. (See mainly Chapters 1 and 5 for two-phase theory; the notion of “interpenetrating” is implied by the first paragraph of p 237.) (7) Rheology; Eirich, F. R., Ed.; Academic Press: New York, 1967; Vol. 4. (See particularly Chapter 9 on terminology and its importance.) (8) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: Oxford, 1999. (This provides a good description of the Larsen/McLeish pom-pom model.)

(9) Doi, M.; Edwards, S. F. Theory of Polymer Dynamics; Oxford University Press: Oxford, 1986. (10) Theoretical Rheology; Hutton, J. F., Pearson, J. R. A., Walters, K., Eds.; Applied Science: London, 1975. (Section 2 contains some relevant material on thermomechanics, a complication that is rarely considered in REoS.) (11) Pearson, J. R. A. Wider Horizons for Fluid Mechanics. J. Fluid Mech. 1981, 106, 229. (The issue of how to choose models for industrial processes was raised in this paper, indicating that the then current state of knowledge lay between (a) and (b); the main aim was to suggest that traditional fluid mechanics needed considerable extensions to satisfy practical needs. It is interesting to note that the nature of papers in the Journal of Fluid Mechanics has developed in the required direction.) (12) Bird, R. B.; Weist, J. M. Constitutive Equations for Polymeric Liquids. Annu. ReV. Fluid Mech. 1995, 27, 169. (13) Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids; Wiley-Interscience: New York, 1987; Vol. 2. (14) Lodge, A. S. Elastic Liquids; Academic Press: London, 1964. (15) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids; Wiley-Interscience: New York, 1987; Vol. 1. (Table 7.3-2 is particularly relevant; the nonlinear variant of Giesekus is equally versatile, as shown in Table 7.3-4.) (16) Denn, M. M. Issues in Viscoelastic Fluid Mechanics. Annu. ReV. Fluid Mech. 1990, 22, 13. (This review addresses directly the question posed in the title above; Denn’s concluding remarks are those of an expert in the field.) (17) Agassant, J. F.; et al. (corresponding author, Mackley, M.) The Matching of Experimental Polymer Processing Flows to Viscoelastic Numerical Simulation. Int. Polym. Process. 2002, 17, 1. (This is a short report of an international consortium that has collaborated to try to reach what I have called a knowledge level (c); the need for multimodal models is made clear in their Section 2.) (18) Keunings, R. Finite Element Methods for Integral Viscoelastic Fluids. Rheol. ReV. 2003, 1, 167. (The article by A. N. Beris on p 37 is also relevant.) (19) Suen, J. K. C.; Joo, Y. L.; Armstrong, R. C. Molecular Orientation Effects in Viscoelasticity. Annu. ReV. Fluid Mech. 2002, 34, 417. (This introduces the important technique of hybrid scale simulation.) (20) McKelvey, J. M. Polymer Processing; Wiley: New York, 1962. (21) Pearson, J. R. A. Mechanical Principles of Polymer Melt Processing; Pergamon: Oxford, 1966. (22) Tadmor, Z.; Klein, I. Engineering Principles of Plasticating Extrusion; Kreiger: Huntingdon, NY; 1978. (23) Pearson J. R. A. Mechanics of Polymer Processing; Elsevier Applied Science: London, 1985. (24) Denn, M. M. Extrusion Instabilities and Wall Slip. Annu. ReV. Fluid Mech. 2001, 33, 265. (See also Chapter 9 in ref 23.) (25) Silva, L.; Gruau, C.; Agassant, J.-F.; Coupez, T.; Mauffrey, J. Advanced Finite Element 3D Injection Molding. Int. Polym. Process. 2005, 20, 3. (This is one of the few papers published on this subject that tackles a complex commercial design problem by detailed computation. Even so, a simplified temperature-dependent purely viscous modelswhich happens to be adequatesis used for the chosen polymer. Most other computational contributions to what is the leading world journal are satisfied by an ability to match phenomenological effects. What cannot be known is whether large machine manufacturers and processors have made much more progress and not published their results.) (26) Zoetelief, W. F.; Peters, G. M.; Meijer, H. E. H. Numerical Simulation of the Multi-Component Injection Molding Process. Int. Polym. Process. 1997, 12, 216. (27) Zhang, Z.; Lafleur, P. G.; Bertrand, F. Effect of Aerodynamics on Film Blowing Process. Int. Polym. Process. 2006, 21, 527. (28) Economides, M. J.; Nolte, K. G. ReserVoir Stimulation, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1989. (29) Anderson, V. J.; Pearson, J. R. A.; Boek, E. S. The Rheology of Worm-like Micellar Fluids. Rheol. ReV. 2006, 4, 217. (This provides over 100 references to recent literature; special mention must be made of ref 79 by Cates, which I found an excellent introduction.) (30) Chen, S.; Doolen, G. D. Lattice Boltzmann Method for Fluid Flows. Annu. ReV. Fluid Mech. 1998, 30, 329. (See reference to viscoelastic fluids in the Conclusions.) (31) Shook, C. A.; Roco, M. C. Slurry Flow; Butterworth-Heinemann: Boston, 1991. (32) Acrivos, A. The Rheology of Concentrated Suspensions of Noncolloidal Particles. Particulate Two-phase Flow; Roco, M. C., Ed.; Butterworth-Heinemann: Boston, 1993; Chapter 5. (Many of the other chapters are relevant.) (33) Stickel, J. J.; Powell, R. L. Fluid Rheology of Dense Suspensions. Annu. ReV. Fluid Mech. 2005, 37, 129.

Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6139 (34) Kunii, D.; Levenspiel, O. Fluidization Engineering; ButterworthHeinemann: Boston, 1991. (35) Brady, J. Stokesian Dynamics Simulation of Particulate Flows. Particulate Two-phase Flow; Roco, M. C., Ed.; Butterworth-Heinemann: Boston, 1993; Chapter 26. (36) Pearson, J. R. A. On Suspension Transport in a Fracture: Framework for a Global Model. J. Non-Newtonian Fluid Mech. 1994, 54, 503. (37) Marsily, G. de. QuantitatiVe Hydrogeology; Academic Press: New York, 1986. (Chapters 2 and 4 provide an introduction.) (38) Bear, J.; Bachmat, Y. Introduction to Modelling of Transport Phenomena in Porous Media; Kluwer: Dordrecht, 1990. (39) Bear, J. Modelling and Applications of Transport Phenomena in Porous Media; Bear, J., Buchlin, J.-M., Eds.; Kluwer: Dordrecht, 1991; Chapter 1. (Chapter 6 is also worth reading.) (40) Adler, P. M. Porous Media; Butterworth-Heinemann: Boston, 1992. (This gives a brief description of the homogenization method in section 4.5.1. On the whole the book is not for beginners.)

(41) Saffman, P. G. A Theory of Dispersion in a Porous Medium. J. Fluid Mech. 1959, 6, 321. (42) Gerritsen, M. G.; Durlofsky, L. J. Modeling Fluid Flows in Oil Reservoirs. Annu. ReV. Fluid Mech. 2005, 37, 211. (43) Pearson, J. R. A.; Tardy, P. M. J. Models for Flow of nonNewtonian and Complex Fluids through Porous Media. J. Non-Newtonian Fluid Mech. 2002, 102, 447. (44) Jung, H. W.; Hyun, J. C. Instabilities in Extensional Deformation Polymer Processing. Rheol. ReV. 2006, 4, 131.

ReceiVed for reView February 5, 2007 ReVised manuscript receiVed April 16, 2007 Accepted April 18, 2007 IE0702072