K. CONSTITUTIVE EQUATIONS FOR NON-NEWTONIAN FLUIDS

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Basic Principles



satisfy these assumptions.27 The result will be a constitutive relationship that is mathematically acceptable, but the usefulness of this relationship depends on whether real fluids exist

that satisfy the physical assumptions! In the continuum mechanics framework, the only way

to tell is to compare predictions from the model with experimental observations, and this

is a long, difficult, and, often, nearly impossible task.

In fact, the field of experimental rheology is concerned with the initial steps in this

process. Measurements of stress components in shear-flow rheometers provide a basis for a

first-level test of any model. There are a number of different geometric configurations that

are used for such measurements: concentric cylinders that are rotated relative to one another;

a pair of parallel disks that are rotated relative to one another; and a similar geometry in

which one of the disks is replaced with a shallow-angle cone.28 The advantage of these

shear rheometers is that the flow can be approximated as the linear shear flow sketched

in Fig. 2–9 (assuming that the fluid remains homogeneous) for both Newtonian and nonNewtonian fluids (we will discuss this in the next chapter). It is only the stresses resulting

from this flow that are different. Because the form of the flow is known a priori, we do

not need to solve a fluid mechanics problem to make predictions of the measurable stresses

from a proposed constitutive model. If a proposed model cannot predict the measurable

behavior in shear flow, it is certainly not a generally acceptable model (though, of course,

it might still find use for other types of flow if the behavior in these flows was better).

However, even if a continuum-based model performs perfectly in predicting shear-flow

behavior, there is no guarantee that it will perform adequately in other flows. This must be

tested. However, at this point the situation becomes extremely difficult because there is no

other flow configuration in which the non-Newtonian behavior does not significantly alter

the flow. Hence it is necessary to both measure and predict not only the stress components

as before, but also the details of the flow. Not only are the experiments demanding, but

it is difficult in many instances to even solve the equations of motion corresponding to a

proposed constitutive model (either numerically or analytically) for comparison with the

experiments. In a worst-case scenario, it is possible that a solution does not even exist

because it is not known that the model represents the behavior of any realizable fluid.

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanicsbased theory and the desire to obtain results that apply to at least some real fluids, there has

been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for

example, to seek a constitutive equation that allows quadratic as well as linear dependence

on strain rate, but to retain the other assumptions) or to make assumptions of such generality

that they must apply to some real materials (for example, we might suppose that stress is

a functional over past times of the strain rate, but without specifying any particular form).

The former approach tends to produce very specific and reasonable-appearing constitutive

models that, unfortunately, do not appear to correspond to any real fluids. The best-known

example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function

of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and

dependence on E only at the same point and at the same moment in time, it can be shown

(see, e.g., Leigh29 ) that the most general form allowed for the constitutive model is

T = (− p + α)I + βE + δ(E · E).



(2–97)



Here, α, β, and δ are material coefficients that can depend on the thermodynamic state, as

well as the invariants of E, namely tr E, det E, and (tr2 E − tr E2 ). The Stokesian model

appears to be a perfectly obvious generalization of the Newtonian fluid model. However, no

real fluid has been found for which the model with δ = 0 is an adequate approximation. We

should, perhaps, not be surprised by this result as the examples in the all seem to suggest

that the assumptions of “isotropy,” plus instantaneous and linear dependence on E, all seem

to break down at the same time in real, complex fluids.

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The other end of the spectrum, namely invoking extremely general assumptions, produces models of excessive generality that typically cannot provide detailed predictions of

material behavior. The best-known example is the so-called “simple” fluid developed in the

1950s and 1960s by Collman, Truesdal, Noll, and others.30 Although some useful results

have been retained from this general theory, in the form of the “nth-order fluid approximation,” for description of the first weak departures from the Newtonian fluid limit (i.e.,

the limit of small Weissenberg number, W i

1), it cannot generally provide any detailed

predictions under more general flow conditions. We may conclude that the continuum

mechanics approach, which worked fine at the level of a linear (Newtonian) material, is

basically unsatisfactory for effective or productive treatment of constitutive nonlinearities.

The necessary continuum-level assumptions are largely arbitrary, and their relationship with

the material behavior of a complex fluid at a macromolecular or particulate level is unclear.

In the rest of this section, we consider the second basic approach, which is “molecular” or

“microscale” modeling.

The starting point for molecular/microscale modeling is a mathematical description of

the material at the scale of the macromolecules or particles. This is followed, in principle, by rigorous statistical mechanical calculations to obtain (1) an equation relating the

microstructural state of the fluid at a material point to the macroscale (or continuum) stress

at that point; and (2) an equation or equations that describe the evolution of the microstructural state of the material from some initial equilibrium-state-due to flow. The limitations

of this approach are quite different from those associated with the continuum mechanics

approach. The most obvious is that the microscale description of the fluid may be too simple

to provide a direct match with any specific example of fluids in this class. One example is

the most common models of polymer molecules in a solution, which envision them as a

freely jointed bead-spring chain, in which the hydrodynamic interactions with the solvent

are approximated by (point) centers of friction (the beads) separated by springs (or rods)

that do not interact with the solvent, with flexible joints at the position of the beads. An

even simpler model is the dumbbell with just two beads and a spring, intended to represent

the end-to-end vector of the polymer chain.31 Although certainly different in detail from an

actual polymer molecule, both of these models can potentially capture the tendency of the

polymer chain (or chain segments) to orient and stretch in a flow. This may be sufficient

for purposes of estimating the polymer contribution to the stress, without requiring a more

detailed description of the polymer chain. However, it is clear that this simple model, with

its simplistic descriptions of polymer chains, will not correspond precisely to any specific

polymer-solvent system, and will not model all aspects of polymer dynamics.

The underlying assumption in the “molecular” modeling approach is that the primary,

generic features of nonlinearity, finite relaxation time, and flow-induced anisotropy can

be captured by means of simple mechanical models, and this is sufficient to provide a

first -order approximation to the non-Newtonian rheology. One important point is that the

model system, even a solution of bead-spring “molecules,” is a “real” viscoelastic liquid

in the sense that it is physically realizable (at least in principle). This is a fundamental

improvement over the pure continuum mechanics approach, which may produce models

that do not correspond to any physically realizable material. If the microscale description

is overly simplified, the resulting constitutive models are at least a simplified version of

the “exact” model. Furthermore, the discussion of the preceding section leads us to expect

that the non-Newtonian behavior of complex fluids is likely to be rather generic and thus

able to be modeled without a large number of details. At the very least, the molecular

modeling approach should provide useful clues on the expected form of constitutive models

for complex fluids (the exact models, whatever these may be, cannot be simpler or of

a fundamentally different form). Furthermore, the molecularly derived models provide

specific values of the material constants, and useful insight into what micromechanical

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features they depend on. Although it may be necessary to “fit” constants by comparisons with

experimental data when the simplified models are applied to real systems, the theoretical

formulae can then often be used successfully to extrapolate from measured values for one

fluid to predicted values for a similar system (e.g., if the solvent viscosity is changed, or the

molecular weight of the polymer, etc.).

Molecular models typically consist of a pair of equations: One relates the stress to the

microscale structure, and the second describes the evolution of the microscale structure by

means of the competing effects of flow and Brownian diffusion (or some other mechanism

for relaxation of the system back toward an equilibrium state), i.e.,

(1)



␶ = ␶ (structure),



(2)



structure evolution in flow.



(2–98)



It is perhaps useful to briefly discuss the details of such a model for the specific and

relatively simple example of the dilute solution or suspension of rigid, rodlike (i.e., axisymmetric) macromolecules or particles immersed in a Newtonian solvent-suspending fluid,

which was introduced in the preceding section. We consider a material point that is subjected to a general linear flow, i.e., a flow that may be expressed in the form.

u = Γ · x,



(2–99)



where Γ is the velocity gradient tensor. It is convenient in what follows to express Γ as the

sum of the strain-rate and vorticity tensors:

u = (E + Ω) · x.



(2–100)



As stated in the preceding section, we assume that the macromolecules/particles remain

uniformly dispersed, so that it is only their rotational motion in the flow (2–100) that is

relevant to the microstructural derivation of a constitutive model. It was shown, almost 80

years ago,21 that this rotational motion can be expressed in terms of the time dependence

of a unit vector p that is aligned along the principal axis of the rod as

˙

p = Ω · p + G E · p − p(p · E · p) ,



(2–101)



where G is a shape factor that has the following values:

G = 0 for a sphere,

G = 1 for a long rod of infinite length-to-diameter ratio,

G ≡ [(r 2 − 1)/(r 2 + 1)] for an axisymmetric ellipsoid with axis ratio r.

This is known as Jeffery’s solution.21 It may be noted that the last term in (2–101) is

added to ensure that |p| = 1. When Brownian motion is present, the orientation of a particle

must be given statistically by means of an orientation probability distribution function

N (p, t),

which satisfies a Fokker–Planck equation in the orientation space occupied by p, i.e.,

DN

˙

(2–102)

+ ∇P · N p − D∇P N = 0.

Dt

Here ∇P is used to denote that the gradient operates with respect to the orientation space of

p,32 and D is the diffusion coefficient for rotational Brownian motion.

We use the convected derivative D/Dt to remind us that this equation is to be applied to

the ensemble of particles/macromolecules belonging to a fixed material point. The second

˙

term represents the effect of the flow and contains p [Eq. (2–101)], whereas the last term

represents the rotational diffusion process. Finally, we require an expression relating the

bulk (i.e., continuum) stress to the orientation distribution. For a dilute suspension, we

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can do this by calculating the contribution to the stress for a single isolated particle with

an arbitrary orientation p and averaging over all possible orientations weighted with the

orientation probability, N(p, t), for particles at the specified material point. The result33 is

σ = − pI + 2μE + 2μφ 2A pppp : E + 2B ( pp · E + E pp ) + CE + F pp D ,

(2–103)

where μ is the solvent viscosity, φ is the volume fraction of particles, A, B, C, and F are

known material coefficients that depend on particle shape, and D is again the orientational

diffusivity. The angle brackets denote an average over possible orientations, i.e.,

pp =



all

orientations



ppN (p, t)dp.



(2–104)



Hence we see that the stress depends on only the second and the fourth moments of the

orientation distribution, pp and pppp . However, in general, it is necessary to know

the complete orientation distribution function N(p, t) to calculate exact results for these

moments by means of (2–104).

We can see that Eqs. (2–101) (2–104) are sufficient to calculate the continuum-level

stress σ given the strain-rate and vorticity tensors E and Ω. As such, this is a complete

constitutive model for the dilute solution/suspension. The rheological properties predicted

for steady and time-dependent linear flows of the type (2–99), with Γ = Γ(t), have been

studied quite thoroughly (see, e.g., Larson34 ). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the

solution/suspension is assumed to be dilute, the volume fraction φ is very small, φ

1.

Nevertheless, the qualitative nature of the particle contribution to the stress is found to be

quite similar to that measured (at larger concentrations) for many polymeric liquids and

other complex fluids. For example, the apparent viscosity in a simple shear flow is found

to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities

are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far

as we are aware, however, the full model has not been used for flow predictions in a fluid

mechanics context. This is because the model is too complex, even for this simplest of

viscoelastic fluids. The primary problem is that calculation of the stress requires solution

of the full two-dimensional (2D) convection–diffusion equation, (2–102), at each point in

the flow domain where we want to know the stress.

The fact that the stress depends only on moments of N(p,t) and not on N(p,t) itself

would seem to provide an opportunity to simplify the model. Instead of calculating N(p,t)

and using that to calculate the moments, it would be much easier if we could calculate

the moments directly. The procedure to derive differential equations for pp or pppp is

actually quite simple. We simply multiply all terms in Eq. (2–102) by pp (or pppp) and

integrate over all possible orientations. The result of this procedure for pp is a DE of the

form

D pp

= g pp , pppp ,

Dt



(2–105)



whereas multiplying by pppp and integrating gives

D pppp

= h pppp , pppppp .

Dt



(2–106)



Unfortunately, the equation for pp involves the higher moment pppp , whereas that for

pppp involves pppppp . This is an example of a classical “closure” problem of statistical

physics. The only way to utilize (2–105) and (2–106) in conjunction with (2–103) to form a

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Basic Principles



simpler constitutive model is to introduce a so-called closure approximation. This is either

an equation relating the fourth moment directly to the second, or the sixth to the fourth, etc.,

depending on the level at which we wish to truncate the hierarchy of moment equations.

The simplest choice for truncation is the quadratic form

pppp = pp pp .



(2–107)



If we substitute this into (2–105) and (2–103) we obtain a specific, closed constitutive model that involves only E and the so-called configuration (or moment) tensor, pp .

Although this would seem to “solve the problem” of a relatively simple, constitutive model

for the specific material, it must be emphasized that the closure assumption, (2–107), is only

one arbitrary choice from a huge set of possible alternatives. There is, in fact, no guarantee

that there is any complex fluid for which (2–107) is an adequate approximation. Indeed, the

necessity to invoke an ad hoc mathematical approximation may produce a model that has

distinctly different properties from those of the full unapproximated model. It may even

exhibit pathologies that render it more or less useless. Again, the only way to verify the

usefulness of the resulting model is by comparison with predictions from the exact model,

but this is generally difficult and tedious, and there are many possible choices in lieu of

(2–107). It is perhaps not surprising that there are extensive efforts in current research to

develop rational or deductive paths for generating accurate closure approximations (see, e.g.,

Cintra and Tucker,35 Chaubal and Leal36 ). Recent research has also attempted to develop

alternative approaches to approximating the system (2–101)-(2–104). (Chaubal et al.,35

Ottinger36 ).

In spite of the complexity of the complete constitutive model, (2–101)–(2–104), and the

difficulties of deriving approximations to it, it does appear that the microscale/molecular

modeling approach is capable of generating constitutive theories for complex fluids. Thus it

is important to recognize some additional limitations of this approach so that the reader may

appreciate why it has not yet been completely successful in providing constitutive models

for the complete class of complex fluids. One intrinsic problem is that the resulting models

are generally too complex to use with present-day computing power to provide the engineer

a tool for solving realistic flow problems of the type that might be encountered in processing

applications. Effective means of approximating the full model to a simpler form have not

yet been developed.

A second issue is the derivation of the structural evolution equation. In the dilute suspension case, a complete and exact fluid dynamics description of the motion of individual

particles is possible, and the structural evolution equation can then be derived by a rigorous

statistical averaging procedure. However, in more general circumstances, this exact deductive procedure is not possible. In the case of nondilute suspensions of rigid, nonspherical,

Brownian particles (perhaps the next simplest of complex fluids), we cannot solve the multiparticle fluid dynamics problem to obtain a formula for the rotation of particles in a flow,

and, in addition, we also do not have an exact description for the multiparticle diffusion process that provides the relaxation mechanism for return to an equilibrium state. Hence we are

forced to invoke models to approximate these processes, and this introduces major uncertainties into the structural evolution equation. For polymeric liquids, the situation is even

more difficult. In this case, we do not even start with an exact description of a single-polymer

chain, but instead must introduce a model even for a single chain. At higher concentrations

in many polymer solutions, the “fluid” consists of an extremely complex system of interacting polymer molecules, which may be highly intertwined. These systems are known as

“entangled.” New models must be invoked to describe both the flow-induced orientation

and stretching of chains and the complex diffusion process by which the system can return

to an equilibrium state. Although major progress has been made (the contributions of the

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L. Boundary Conditions at Solid Walls and Fluid Interfaces



French physicist, P. G. DeGennes, were responsible, in part for his achieving the Nobel

prize in physics in 1991), there are still many open issues in modeling the microdynamics

of entangled polymers, and we are only now approaching constitutive models that can be

used for fluid mechanics predictions.37

It is likely, in the interim, while we await models from the molecular modeling perspective for the more “difficult” complex fluids, that the most success in predicting fluid

mechanics results for non-Newtonian fluids will come from a hybrid approach – combining

some elements of both continuum mechanics and molecular modeling to produce relatively

simple empirical models. There is a great deal of current research focused on all aspects

of constitutive model development; on numerical analysis of flow solutions based on these

models; and on experimental studies of many flows. There are a number of books and references available, but this is a complicated field that really requires a textbook/class of its

own. At this point, it is time to return from our little sojourn into the land of complex fluids

and come back to the principle subject of Newtonian fluids.

L. BOUNDARY CONDITIONS AT SOLID WALLS AND FLUID INTERFACES



We are concerned in this book with the motion and transfer of heat in incompressible,

Newtonian fluids. For this case, the equations of motion, continuity, and thermal energy,

ρ



∂u

+ u · ∇u = ρg − ∇ p + μ∇ 2 u + ∇μ · (∇u + ∇uT ),

∂t

∇ · u = 0,



ρC p



∂θ

+ u · ∇θ

∂t



=−



θ

ρ



∂ρ

∂θ



p



Dp

+ 2μ(E : E) + ∇ · (k∇θ ),

Dt



(2–108)

(2–109)

(2–110)



provide a complete set from which to determine the velocity u, pressure p, and temperature

θ within any homogeneous fluid regime. By homogeneous, we mean any region in which

material properties such as density, viscosity, heat capacity, or thermal conductivity vary

only “slowly” on length scales that are proportional to the size of the flow domain itself.

There are, inevitably, regions within the fluid domain where fluid properties vary on much

shorter length scales. These include the immediate vicinity of either a phase boundary (i.e.,

solid–liquid, liquid–gas, etc.) or a liquid–liquid interface between two homogeneous fluids

that are immiscible. In a purely molecular theory, no special treatment would be required

for encompassing these regions. However, at the continuum mechanics level of resolution,

these transition zones appear as surfaces. Equations (2–108)–(2–110) apply right up to

these bounding surfaces or interfaces. Material properties that are seen as varying rapidly,

but continuously, across the transition zones in a molecular theory are approximated in

the continuum mechanics description as suffering a discontinuous jump at the surface or

interface from the value in one bulk phase to the other. To determine solutions of (2–108)–

(2–110) for the velocity or temperature distributions in the “homogeneous” or bulk fluid

domains, boundary conditions must be specified for these variables or their derivatives at

these bounding surfaces or interfaces.

The obvious question is this: What conditions should be imposed? Without a molecular

or microscopic theory for guidance, there is no deductive route to answer this question. The

application of boundary conditions then occupies a position in continuum mechanics that

is analogous to the “derivation” of constitutive equations in the sense that only a limited

number of these conditions can be obtained from fundamental principles. The rest represent

an educated “guess” based to a large extent on indirect comparisons with experimental data.

In recent years, insights developed from molecular dynamics simulations of relatively simple

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molecular theories have also played a useful role. Our focus in this section is on boundary

conditions at either a solid boundary or at a fluid–fluid interface.

However, let us begin from a slightly more general perspective. There are really two

types of boundary conditions encountered in theoretical analyses of fluid flow or heat

transfer problems when (2–108)–(2–110) are used. In particular, when we are interested

in the temperature or velocity fields in the vicinity of an object of finite size in a much

larger fluid domain, it is often a convenient and reasonable approximation to assume that

the fluid domain is unbounded. This is particularly useful if the form of the temperature or

velocity field far from the object of interest is known in advance. In this case, the form of

the temperature or velocity field far from any boundaries is prescribed in lieu of boundary

conditions at an actual wall or surface. An example is the translation of a heated sphere

through a cooler, quiescent fluid that is held in a large container. Now, if the sphere is

much smaller than the container, and if it is not close to any of the container boundaries,

the velocity and temperature perturbations caused by the sphere will be relatively localized

in the vicinity of the sphere and be independent of where the sphere is located in relation

to the distant container walls. In this case, instead of solving for the temperature and

velocity fields in the complete fluid domain, with boundary conditions applied at the sphere

surface and container walls, an adequate approximation will be to treat the fluid domain

around the sphere as though it were unbounded (i.e., infinite in extent) and then require

that the temperature and velocity fields take the “ambient” form far from the sphere that

would exist in the container in the absence of any disturbance from the sphere. Although

the approximation of applying far-field boundary conditions “at infinity,” in lieu of actual

boundary conditions at some distant boundary, may at first seem questionable, the farfield conditions themselves are generally assumed to be known. Indeed, the unbounded

fluid approximation is useful only if we know the undisturbed form of the temperature or

velocity fields in advance.

The other class of boundary conditions is those applied at bounding surfaces, i.e., either

at solid surfaces or at an interface if there are two (or more) distinct “homogeneous” fluids in

the flow domain. We denote these surfaces with the generic symbol S. The transition between

two bulk materials occurs over a finite but thin region. In the continuum description, we

approximate this as a surface of discontinuity in material properties. An immediate question

that may arise is what surface we should choose within the finite, but thin, surface or interface

region for the purpose of applying the macroscale or continuum boundary conditions. For

that matter, we may equally ask whether it makes any difference. From a purely geometrical

point of view, there is no difference what choice we make for S as long as it remains within

the interfacial/surface zone. This whole region is vanishingly thin, in any case, compared

with the continuum scale of resolution L. Nevertheless, it has historically proven to be

extremely convenient to adopt a specific convention. To explain this convention, let us

initially limit ourselves to an interface separating two pure bulk fluids A and B. As we move

across the interfacial region there is a rapid, but smooth variation of the density from ρ A to

ρ B . However, from the continuum viewpoint, we model this as though there were simply

the two homogeneous fluids of density ρ A and ρ B right up to the surface S, across which

the density jumps discontinuously from ρ A to ρ B . The position of S is chosen so that the

total mass is the same in either description of the system, i.e.,

L

−L



ρ(z)dz = Lρ A + Lρ B ,



(2–111)



where z is the coordinate direction normal to S, the interface is (locally) at z = 0, and

the range −L to L covers a large but finite region such that ρ − ρ A → 0 as z → L and

ρ − ρ B → 0 as z → −L. If a different choice were made for S, there would be either more

or less mass in the idealized continuum description than in the real system, and it would be

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