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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
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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.
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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.
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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
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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
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(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
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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
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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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