J. COMPLEX FLUIDS – ORIGINS OF NON-NEWTONIAN BEHAVIOR

P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



J. Complex Fluids – Origins of Non-Newtonian Behavior



in terms of a concentration, say c, that is assumed to be independent of position. Then

specification of the microstructural state requires only the orientation distribution function,

N (p; x, t).

The probability of finding a particle/macromolecule with an orientation within a solid angle

dp of some orientation p is

N (p; x, t)dp.

In general, this probability will be a function of both position x and time t. However,

at equilibrium, all orientations are equally probable. Hence the equilibrium orientation

distribution, normalized so that the total probability of orientation in some direction is

unity, is just

1

N (p; x, t) =

.

4π

In general, we seek a description of the solution/suspension from a macroscopiccontinuum perspective. Hence we generalize our notion of a material point to be a volumetric

region that is small enough compared with the (macroscopic) length scales of a flow domain

to be considered as a point, but large enough to contain a statistically meaningful number

of particles/macromolecules. Any macroscopic property of the suspension/solution at a

material point will then depend on the orientation distribution at that point and time. In

the equilibrium state, the orientation probability is uniform, as already noted, and hence

the material at equilibrium is statistically isotropic, in spite of the fact that individual particles/macromolecules are rodlike in shape.

Now, let us think qualitatively about what happens to the material when it is subjected

to a flow. For simplicity, let us imagine that the flow is a simple shear flow

u = γ y ix ,

˙



(2–94)



with a shear rate γ . Simple shear flow is a good approximation to the flows generated in

˙

many rheometers – namely, the instruments designed to measure the flow properties of

liquids – and thus it is an appropriate choice for the present discussion.

When an individual particle/macromolecule is subjected to a flow of the type (2–94),

it will tend to rotate. In general, the rotational motion of a rod in a shear flow is complex,

though a complete theory for rodlike ellipsoids had been developed already in the early

1900s by Jefferey (1922). 21 For present purposes, let us simply consider a rod that lies in

the plane of the flow (i.e., the x–y plane). This rod will rotate. However, the rate of rotation is

not uniform. If the rod is oriented in the y direction (across the flow), it is intuitively obvious

that the hydrodynamic torque on it will be much larger than if the same rod were oriented

with its axis in the x direction (parallel to the flow). Hence it will rotate more rapidly in the

cross-flow orientation than when it is aligned with the flow. Even though the rod will continue

to tumble over and over, it will spend relatively longer in orientations that are near alignment

with the flow and less time in orientations that are across the flow. Thus the action of the flow

is to produce a nonuniform orientation distribution, with larger N values in the flow-aligned

direction and smaller values away from this direction. Of course, Brownian motion is still

present – i.e., “rotational diffusion” – and this continues to try to maintain the orientation

distribution in a uniform or completely random state. The orientation distribution that results

can be viewed as a consequence of the competition between the flow-induced tendency

toward a nonuniform orientation distribution and the tendency of Brownian diffusion to

maintain a uniform state. The strength of the flow effect will clearly be proportional to γ ,

˙

whereas the diffusion process is characterized by a so-called rotational diffusivity D. Both

of these quantities have units of inverse time. D, in particular, provides a measure of the

53



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



Basic Principles

Applied tangential

force per unit area, F/A

U



y



d

x



Figure 2–9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate,

and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement

of the fluid viscosity (see, e.g., Fig. 3–5). In the present discussion, we approximately represent the flow in

these devices as the flow between two plane boundaries as described in the text and sketched in this figure.



time scale required for a system with a nonuniform initial orientation distribution to return

to the uniform equilibrium state by means of Brownian motion in the absence of flow. The

magnitude of flow-induced deviations from the uniform equilibrium state then depends on

the dimensionless ratio

Wi ≡



γ

˙

.

D



(2–95)



This parameter is known in the rheological literature as the Weissenberg number (also

sometimes – mistakenly – called the Deborah number).22

As noted previously, we expect the macroscopic properties to depend on the orientation

distribution. Because the latter will, in general, change in the presence of flow, it follows

that properties, such as the viscosity, as measured in a standard shear rheometer, will have

different values at different shear rates. In such a rheometer, the fluid is between two plane

solid boundaries (as sketched in Fig. 2–9). The shear flow is induced by the movement of

one of the boundaries (in its own plane) relative to the other. The shear rate is determined by

the differential velocity (U) divided by the gap width d. The force F required for maintaining

the velocity U is then measured. For a Newtonian fluid, the ratio

(F/A)

(U/d)

is equal to the viscosity μ (here the ratio F/A is equal to τx y and the ratio U/d is equal to

E x y ). The viscosity, being a material constant according to the model, is independent of the

shear rate (U/d). Hence the ratio (F/A)(U/d)(U/d) will be independent of U if the fluid is

truly Newtonian. For the suspension/solution that we have been considering, experimental

measurements show that the ratio (F/A)/(U/d) actually decreases with increase of the

shear rate, U/d, at least when U/d is large enough so that the magnitude of the Weissenberg

number

(U/d)

Wi =

D

is of the order of unity or larger. As the shear rate increases, the flow-induced tendency for

alignment of the particles/macromolecules in the flow direction is enhanced. However, all

else being equal, it seems intuitively obvious that the work (or force) required to produce a

certain shear-rate will be decreased as the degree of flow alignment is increased, and thus

the apparent viscosity, (F/A)/(U/d) will decrease with increase of the shear rate U/d.

This is illustrated pictorially in Fig. 2–10.

54



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



J. Complex Fluids – Origins of Non-Newtonian Behavior



Newtonian fluid



log



τxy



( )

Exy



= log



(F/A)



non-Newtonian

(shear-thinning)

fluid



(U/d)



log Wi = log[(U/d ) /D]



Figure 2–10. A qualitative sketch of the dependence of the ratio of the shear stress to the shear rate in a

simple shear flow. The upper curve represents the result that would be obtained if the fluid were Newtonian.

The lower curve illustrates the shear-thinning behavior that is described in the text for the solution of rodlike

macromolecules (or the suspension of rigid-rod particles) when the shear rate is large enough to produce

partial alignment of the macromolecules (the particles) in the flow direction.



A fluid that exhibits an apparent viscosity that decreases with increasing shear rate is

often called a “shear thinning” fluid. Shear thinning is one easily measurable signal that

a fluid cannot be described as a Newtonian fluid. It should be cautioned, however, that

shear-rate-dependent apparent viscosity is but one way that non-Newtonian fluids may

differ from a Newtonian fluid. Not all non-Newtonian fluids exhibit this particular property,

though certainly every fluid that does shear thin is non-Newtonian. The reader may have

noted that the word “apparent” has been used in describing the “viscosity” measured in

our solution/suspension (or in any non-Newtonian fluid). This is because the measurement

previously described, although clearly giving a direct measure of the material viscosity μ

when the fluid is Newtonian, can only be said to measure the same ratio of macroscopic

quantities, (F/A) and (U/d), when the fluid is not Newtonian. Although it is sometimes

proposed, on a purely empirical basis, that we might describe the non-Newtonian behavior

as reflecting a shear-rate-dependent viscosity, it is not true for any real non-Newtonian or

complex fluid that the Newtonian constitutive equation, (2–81), can be replaced with

T = − pI + 2μ(γ )E,

˙



(2–96)



as would be true if the ratio (F/A)/(U/d) were measuring a true material function, μ(γ ). In

˙

truth, constitutive equations required to describe the behavior of complex or non-Newtonian

fluids are much more complex than the so-called “generalized Newtonian fluid” form

(2–96), even for the relatively simple system made of rigid Brownian macromolecules in a

Newtonian fluid that we have been discussing. Hence the measured quantity (F/A)/(U d)

is not a true viscosity except when the fluid is Newtonian. It is nevertheless convenient to

refer to it in terms of a shear-rate-dependent viscosity, and we add “apparent” to emphasize

that it is not a true material constant. We shall briefly discuss constitutive models for nonNewtonian complex fluids in the next section. First, we should complete the discussion of

our specific example.

The preceding description of the expected behavior of our solution/suspension of rodlike

macromolecules/particles led to the realization that this is a fluid whose behavior is not that

of a Newtonian fluid. However, there is more that we can learn about such fluids by taking

the discussion somewhat further.

The first, and most obvious but critical point is that the solution/suspension is nonNewtonian in behavior because its statistical structure at the macromolecular/particulate

level is modified when it is subjected to flow. This may be contrasted with typical small

molecule liquids whose constitutive behavior is generally well approximated as Newtonian.

55



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



Basic Principles



If we think about it, even the small molecule liquid, when subjected to flow, will experience

forces at the molecular level that will tend to produce a nonisotropic statistical structure.

In the simplest case, at least conceptually, this may be simply a tendency to produce a

nonuniform orientation distribution for molecules that are not strictly spherical in shape.

As in the case of our suspension/solution, the random motions corresponding to orientational

diffusion will oppose this tendency. The difference between Newtonian fluids and complex

fluids is the enormous change in the magnitude of the relevant diffusivities. In the case

of our macromolecular solution or colloidal suspension, the characteristic time scales for

relaxation from a nonequilibrium to an equilibrium configuration may be as large as seconds

or even tens of seconds, depending on the solvent viscosity and the molecular weight or

size of the rodlike macromolecules/particles. For a small-molecule liquid, on the other

hand, the same time scale could be fractions of a microsecond or smaller. Hence, whereas

a shear rate of order 1–10 s−1 leads to Weissenberg numbers of order 1 or larger for the

suspension/macromolecular solution, the magnitude of shear rates normally accessible in

technological or naturally occurring flows are almost never large enough to produce anything

other than extremely small values of Wi for a small-molecule liquid. Thus the microstructural

state in the latter case tends to be completely denominated under all normal flow conditions

by random, thermal motion, and there is no perceptible effect of the flow on the statistical

structure of the fluid at the molecular level. Thus the fluid properties remain the same under

flow conditions as they are at equilibrium (i.e., in the absence of flow) and the fluid can be

approximated as Newtonian provided only that it is isotropic at equilibrium.23

An important conclusion from this discussion is that it is not appropriate to describe a

particular fluid as Newtonian (or not) without coupling this statement to a characterization

of the flow conditions (i.e., in the case of the simple shear flow previously discussed to the

range of shear rates). Clearly it should be possible under especially severe flow conditions of

high shear rate for a small-molecule liquid to exhibit non-Newtonian behavior. An example

is some types of motor oil that appear as Newtonian under conventional conditions for

shear rheometry (maximum shear rates in common commercial rheometers are generally

less than 100 s−1 ), but are subjected to extremely large shear rates in reciprocating piston

engines and are then known to exhibit shear thinning unless they have additional (usually

polymeric) additives included. Even a fluid like water is expected to exhibit non-Newtonian

characteristics if subjected to extremely high shear or rapidly varying flow conditions. On the

other hand, a much more common situation is that polymeric liquids or other complex fluids

may be approximated as Newtonian fluids if they only experience flows with shear rates

such that W i

1. Hence, to emphasize again, no fluid should be thought of as categorically

Newtonian or non-Newtonian. The behavior depends on expected flow conditions.

Finally, let us return to review the various assumptions that were made in the derivation

of the Newtonian constitutive model. These were, first, that the stress depends linearly

on the strain rate E. This is not true in our example of the solution/suspension of rods.

Evidently, if we are given a fixed material configuration, the stress may depend linearly

on E, as in small-molecule liquids. In our case, however, the configuration also depends

on E (i.e., on γ in shear flow), and thus the stress must depend on E in some nonlinear

˙

fashion. This is the source of shear thinning that was discussed earlier. The second condition

required for the Newtonian fluid model to apply is that the stress is assumed to respond

instantaneously to the current value of the strain rate at the particular material point of

interest. This assumption is also clearly violated by the solution/suspension of rods. In

general, following a change in shear rate, the orientation distribution must also change,

and thus too the stress. However, the orientation distribution responds to a change in flow

conditions in only a finite increment of time. This is most easily seen if the shear rate

is suddenly put to zero. In a Newtonian fluid, the stress would instantaneously return to

its equilibrium value. In the solution/suspension, however, this occurs over only the finite

56



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



J. Complex Fluids – Origins of Non-Newtonian Behavior



time scale D −1 that is required for the orientation distribution to return to its equilibrium

state. As a consequence, the statistical structure (orientation distribution) and thus the stress

at a material point at any instant of time must depend on the past history of flow of that

material point. On the other hand, it is apparent that the configuration (and stress) at an

instant of time will depend most strongly on the recent flow history and less strongly on

the flow at earlier times. In the language of continuum mechanics, and “rheology”, it is

said that a material such as ours exhibits a “memory” for past shear-rate history, which

“fades” on a time scale proportional to D−1 . The suspension always returns to the same

configuration state of equilibrium and in that sense resembles an elastic solid. Fluids with

this feature are called viscoelastic liquids. We note that the elasticity in this case comes from

the entropic tendency of the orientation distribution to return to an isotropic equilibrium

state. The individual particles/macromolecules are modeled as solid–rigid objects, hardly

the direct source of any elasticity. Finally, in the Newtonian fluid model, it is assumed that the

material is completely isotropic. In our example case, this is true at equilibrium. However,

as soon as flow is added to the picture, the structure becomes anisotropic as the orientation

distribution becomes nonuniform. The only one of the original assumptions that actually

applies for complex/non-Newtonian fluids is the statement that the stress at a material point

depends on only the material configuration at that material point. However, because material

points move with finite velocity, the dependence of the microstructural state on past history

does imply a dependence on the flow history along the fluid pathline that is followed by

the material point. It may also be noted that the size of the region that comprises a material

point will generally be larger for a complex fluid than for a small-molecule liquid. Hence

the conditions for applicability of the continuum approximation will generally be somewhat

more restrictive.

We have focused our discussion on the specific example of a dilute suspension of Brownian rodlike particles or macromolecules. However, as indicated at the beginning of this

section, there are many examples of complex or non-Newtonian materials. Others that come

immediately to mind include emulsions or blends of two immiscible fluids (generally with

one dispersed as bubbles or drops within the other), polymeric liquids (either solutions or

melts), and colloidal suspensions of solid spherical particles (which might be immersed in

either a Newtonian or non-Newtonian liquid). It will not be difficult for the reader to think of

other examples. The common characteristic is that the material contains or is made up of relatively large molecules, or particles, that, if perturbed from their equilibrium configurational

state, require a relatively long time scale (of the order of milliseconds or longer) to return to

the equilibrium state. This means that flows with moderate-velocity gradients can produce

a significant departure from the equilibrium state. Because macroscopic properties depend

on the configurational state of the material, all such fluids will exhibit non-Newtonian (i.e.,

flow-dependent) properties. In fact, though the various types of complex fluids are quite different at a detailed, microscopic level, there is a common conceptual framework that applies

to all. This also suggests a certain commonality to the macroscopic material behavior, and

quite likely, also some similarity in the form of constitutive models that could be used to

describe this behavior (though, of course, one should expect that any material coefficients

would be strongly dependent on the specific material.).

Let us consider the three additional examples just mentioned. First, we need to identify

the features, in each case, that define the microstructural state. In the case of the emulsion

or blend, the most important microscale feature that can be influenced by the flow is the

orientation and shape of the disperse-phase bubbles or drops (the mean drop size and dropsize distribution will also generally be important and can be influenced by flow-induced drop

breakup and coalescence events, but we will ignore this extra complication for purposes of

our current discussion). At equilibrium, the drops will be spherical and the microstructure

isotropic. For polymeric liquids, it is the statistical configuration of the polymer molecules

57



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



Basic Principles



(both individually and collectively) that can be influenced by flow. For example, in dilute

solutions of flexible polymer molecules, the flow can induce a transformation from the

random coil configuration, which exists at equilibrium, to a stretched and oriented threadlike

conformation. More generally, flow will tend to produce an aligned and stretched statistical

state, beginning from an isotropic state at equilibrium. Finally, for a suspension of solid

spherical particles, it is obvious that individual particles cannot change shape. However, in

the presence of a flow, hydrodynamic interactions between particles can change the nearestneighbor probability density distribution from an isotropic distribution at equilibrium that is

established by random thermal motion (diffusion), to a distribution with a higher probability

in some directions than others.

In each case, the macroscopic properties will be changed by these flow-induced changes

in the material microstructure. For all three materials, there is a “relaxation” mechanism that

tends to drive the system back toward an isotropic, equilibrium state; the drops are spherical

at equilibrium and are uniformly dispersed; the polymer molecules are characterized by

a random configuration of chains and/or chain segments; and the colloidal particles are

uniformly dispersed. The mechanism by which the system relaxes toward equilibrium obviously is different in each case. The drops achieve a spherical shape at equilibrium because

of the action of interfacial or surface tension (as we shall discuss in more detail later in this

chapter), and any tendency of flow to produce a nonisotropic distribution of neighboring

drops (n-drop distribution function) is overcome by Brownian diffusion (on a time scale that

increases with drop size). Polymer molecules undergo random fluctuations in configuration

because of random thermal motions (Brownian) associated with the solvent and/or other

polymer chains, and this leads to an isotropic equilibrium configuration of “undeformed”

chains. Finally, the suspension of colloidal, spherical particles is subject to translational

Brownian diffusion that drives the system back toward a uniformly dispersed state. The

time scales for these “relaxation” processes are finite in each case: inversely proportional to

the surface tension for emulsions, μa/σ , where μ is the suspending fluid viscosity, a is the

undeformed drop radius, and σ is the interfacial tension; and proportional to an appropriate

diffusivity D−1 in the other two cases. The degree of departure from equilibrium in each

case is then determined by a Weissenberg number

W i = τrelxn · |∇u|

where |∇u| is a measure of the magnitude of velocity gradients in the flow (i.e., it is the

shear rate for a simple shear flow).

Hence we see that there is a great deal of the overall behavior that is common to all four

of the specific examples that we have considered. Each is isotropic in the equilibrium state,

maintained by a relaxation process (or more than one in some cases) that is characterized by

a finite relaxation time, τrelxn . In the presence of flow each becomes anisotropic in structure,

with the degree of departure from equilibrium determined by a balance between the flowinduced deformation and the relaxation process(es) with the relative importance of each

characterized by the ratio of time scales

W i = τrelxn /(|∇u|)−1

that is known as the Weissenberg number. Each has a preferred equilibrium state – and so

will be expected to exhibit viscoelastic behavior. Further, each will exhibit a so-called “fading memory” for past flow conditions and a nonlinear dependence of the macroscopic stress

on |∇u|. Although the various complex fluids are quite different in terms of their microscopic makeup, it should be clear that there is a great deal in common insofar as expected

macroscopic flow behavior is concerned. It should also be clear that many fluids of common experience (indeed all fluids under sufficiently extreme flow conditions) will exhibit

viscoelastic, non-Newtonian rheology, and this is true even if the microscale particles or

58



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



K. Constitutive Equations for Non-Newtonian Fluids



macromolecules are rigid. The “elasticity” in such cases is essentially “entropic” in origin –

a measure of the “deterministic” tendency to revert to a completely random equilibrium

structure.

In the next section, we briefly discuss some specific constitutive models and their

derivation.

K. CONSTITUTIVE EQUATIONS FOR NON-NEWTONIAN FLUIDS



In the preceding section, we have seen that there are many fluids that cannot be approximated

as Newtonian under normal flow conditions. An obvious question is whether successful

generalizations of the Newtonian fluid model exist that can be used to solve flow and

transport problems for this class of materials?

If we interpret this question as asking whether models exist for the general class of

complex/non-Newtonian fluids that are known to provide accurate descriptions of material behavior under general flow conditions, the current answer is that such models do

not exist. Currently successful theories are either restricted to very specific, simple flows,

especially generalizations of simple shear flow, for which rheological data can be used to

develop empirical models, or to very dilute solutions or suspensions for which the microscale

dynamics is dominated by the motion deformation of single, isolated macromolecules or

particles/drops.24

It should be emphasized that many constitutive models have been proposed especially

for polymeric solutions and melts, and there is a great deal of current research that is aimed at

both new models25 and numerical analysis of fluid motions by use of the existing models.26

The problem is that few have been carefully compared with the behavior of real fluids

outside the highly simplistic flows of conventional rheometers, and then mainly under flow

conditions in which the perturbations in material structure are weak. Thus there is currently

no model that is known to provide quantitatively accurate or even qualitatively reliable

descriptions of real complex fluids for a wide spectrum of flows.

Assuming this to be an accurate assessment of the current state of affairs, the reader

may wonder at the existence of the present section of this book. Indeed, it is unique (with

respect to the rest of the book) in the sense that we are unable to follow the general goal

of introducing specific methods and ways of thinking that will allow us the opportunity to

address new or unresolved problems of fluid motion. Nevertheless, the class of complex

fluids is so important, and the lack of sections on “non-Newtonian” fluid mechanics appears

as such a glaring omission from this textbook, intended as it is for scientists and engineers

who will likely encounter complex fluids throughout their careers, that I believe that it is

worthwhile to provide some additional discussion of past and current attempts to develop

the necessary constitutive theories. The reader who is strictly concerned with Newtonian

fluids may skip this section with no effect on understanding later sections of this book.

There are, in fact, two quite distinct approaches to the generation or derivation of new

constitutive models. The first is the purely continuum mechanics approach that was so successful in obtaining the Newtonian fluid approximation. This approach has a long history,

extending over at least the past 50–100 years in a form that resembles its current use and is

responsible for the vast majority of constitutive models that have been proposed up to the

present time. The steps are familiar. First, and most critical, is that a guess is made about

the general nature of the proposed relationship: What independent dynamical variables are

required (i.e., only the rate of strain in the case of the Newtonian fluid model) and what

assumptions are to be imposed on the relationship between these variables and the macroscopic stress (e.g., linear, instantaneous, local, and isotropic in the case of a Newtonian

fluid)? Given this input, the mathematical constraints of coordinate invariance and material

objectivity are then used to determine admissible forms for the functional relationships that

59



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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.

60



10:7