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inverse-scattering transform (IST), admits multisoliton
solutions, has an infinite number of conserved
quantities, and possesses many other interesting
properties. Some of these are discussed below.
The Inverse-Scattering Transform
The IST method allows one to linearize a large class
of nonlinear evolution equations and can be con-
sidered as a nonlinear version of the Fourier trans-
form. An essential prerequisite of IST method is the
association of the nonlinear evolution equation with
a pair of linear problems (Lax pair), a linear
eigenvalue problem, and a second associated linear
problem, such that the given equation results as a
compatibility condition between them. A key
research breakthrough on NLS systems appeared in
1972, in the papers of Zakharov and Shabat (1972,
1973), who first analyzed the scalar NLS equation
in the form
iq
t
¼ q
xx
2 q
jj
2
q ½27
( correspond to the focusing/defocusing case,
respectively) and found the associated Lax pair
v
x
¼
ikq
q
ik
v ½28
v
t
¼
2ik
2
ijqj
2
2kq iq
x
2kq
iq
x
2ik
2
ijqj
2
v ½29
where v(x, t) is a two-component vector. The
compatibility of [28] and [29] yields eqn [27],
assuming that the eigenvalue parameter k is
constant in time (so that [27] isoftensaidtobe
isospectral).
The solution of the initial-value problem of a
nonlinear evolution equation by IST proceeds in
three steps, as follows:
1. the forward problem – the transformation of the
initial data from the original ‘‘physical’’ variables
to the transformed ‘‘scattering’’ variables;
2. time dependence – the evolution of the trans-
formed data accordi ng to simple, explicitly
solvable evolution equations; an d
3. the inverse problem – the recovery of the evolved
solution in the original variables from the
evolved solution in the transformed variables.
The implementation of steps 1–3 described above is
more concretely carried out as follows. The initial
(Cauchy) datum q(x, 0) for eqn [27] is mapped into
scattering data S(k, 0) (comprising, in general, discrete
eigenvalues and associated normalization constants,
and reflection coefficients) by means of eqn [28].The
data S(k, 0) are evolved via eqn [29] to get S (k, t)atan
arbitrary time t > 0. Finally, by employing the
methods of inverse scattering, eqn [28] allows one to
reconstruct the evolved solution q(x, t)fromS(k, t ).
One can easily note the ‘‘formal’’ resemblance to
the well-known method of Fourier transform for
linear differential equations.
There is considerable literature on the subject and
the interested reader is encouraged to consult, for
instance, some of the following references: Ablowitz
and Segur (1981), Caloger o and Degasperis (1982),
Novikov et al.
(1984), Ablowitz and Clarkson
(1991), Ablowitz et al. (2004).
Linear Stability Analysis
Consider a special solution of eqn [27] in the
focusing (þsign) case: q = a exp(2ia
2
t). If this
solution is perturbed as
qðx; tÞ¼ae
2ia
2
t
ð1 þ ðx; tÞÞ
where jj1, it is found that satisfies the
condition
i
t
¼
xx
þ 2a
2
ð þ
Þ
On the periodic spatial domain 0 < x < L, has the
Fourier expansion
ðx; tÞ¼
X
1
1
^
n
ðtÞe
i
n
x
where
n
¼
2n
L
½30
Assuming a solution of the form
^
n
^
n
¼
e
i
n
t
one finds that
n
satisfies
2
n
¼
2
n
2
n
4a
2
½31
It then turns out that when aL= < n the system is
unstable. Note that there are only a finite number of
unstable modes (i.e., for fixed a, L, sufficiently high
mode numbers n will not satisfy the above inequal-
ity). In the context of water waves, this corresponds
to the famous experimental and theoretical result by
Benjamin and Feir that the Stoke’s water wave is
unstable. Later, Benney and Roskes (1969) showed
that all periodic wave solutions of the generalized
nonlocal NLS equation resulting from water waves
in (2 þ 1)-dimensions are unstable. Also, in (2 þ 1)-
dimensions soliton solutions are unstable to weak
transverse modulations.
Nonlinear Schro¨ dinger Equations 557
Wave Collapse
The equation
i
t
þ þj j
2
¼ 0; x ðx ; yÞ2R
2
½32
has the following conserved quantities:
P ¼
Z
jj
2
dx
M ¼
Z
r dx
H ¼
Z
r
jj
2
1
2
jj
4
dx
that is, mass (power), momentum, and energy
(Hamiltonian) are conserved. Remarkably, Talanov
(1965) showed that eqn [32] satisfies the following
equation:
@
2
V
@t
2
¼ 8H ½33
where
V ¼
Z
ðx
2
þ y
2
Þj j
2
dx dy
Equation [33] is also known as the ‘‘virial’’ theorem.
Hence, it follows that
V ¼ 4Ht
2
þ c
1
t þ c
2
and if H < 0 initially, then a singularity in eqn [32]
results since V must be positive. Actually, one can
further show (see, e.g., C Sulem and P L Sulem
(1999), and references therein) that there exists a
time t
such that
Z
r
jj
2
dx
becomes infinite as t ! t
, which in turn implies
that also becomes infinite as t ! t
(blowup in
finite time).
Note also that for the more general equation
i
t
þ
d
þ
jj
2
¼ 0; x 2 R
d
where
d
is the d-dimensional Laplacian, one has
the following types of solutions:
Supercritical (d > 2): the solution blows up.
Critical (d = 2): blowup can occur or global
solution can exist.
Subcritical (d < 2): global solutions exist.
Vector NLS Systems
In many applications vector NLS (VNLS) systems are
the key governing equations. Physically, the VNLS
arise under conditions similar to those described by
NLS with the additional proviso that there are
multiple wave trains moving nearly with the same
group velocities (Roskes 1976). Importantly, VNLS
also models systems where the field has more than
one component. For example, in optical fibers and
waveguides, the propagating electric field has two
components transverse to the direction of propaga-
tion. The nondimensional system
iq
ð1Þ
z
¼ q
ð1Þ
xx
þ 2 jq
ð1Þ
j
2
þjq
ð2Þ
j
2
q
ð1Þ
½34a
iq
ð2Þ
z
¼ q
ð2Þ
xx
þ 2 jq
ð1Þ
j
2
þjq
ð2Þ
j
2
q
ð2Þ
½34b
is an asymptotic model which governs the propaga-
tion of the electric field in a waveguide, where z is
the normalized distance along the waveguide and x
a transversal sp atial coordinate. It was first exam-
ined by Manakov (1974) (see also Anastassiou et al.
(1999) and Solja
cic
´
et al. (2003)). Subsequently, this
system was derived as a key model for light-wave
propagation in optical fibers. More precisely, in
optical fibers with constant birefringence
(i.e., constant phase and group velocities as a
function of distance) Menyuk (1987) has shown
that the two polarization components of the
electromagnetic field E = (u, v)
T
which are orthogo-
nal to the direction of propagation, z, along the fiber
asymptotically satisfy the following nondimensional
equations (assuming anomalous dispersion):
iðu
z
þ u
t
Þþ
1
2
u
tt
þðjuj
2
þ jvj
2
Þu ¼ 0 ½35a
iðv
z
v
t
Þþ
1
2
v
tt
þðjuj
2
þjvj
2
Þu ¼ 0 ½35b
where represents the group velocity ‘ ‘mismatch’’
between the u, v components of the electromagnetic
field, is a constant that depends on the polarization
properties of the fiber, z the distance along the fiber, and
t a retarded temporal frame. In deriving eqn [35],itis
assumed that the electromagnetic field is slowly varying
(as in the scalar problem); certain nonlinear (four-wave
mixing) terms are neglected in the derivation of eqn
[35], because the light wave is rapidly varying due to
large, but constant, linear birefringence. In this context,
birefringence means that the phase and group velocities
of the electromagnetic wave in each polarization
component are different. In a communications environ-
ment, due to the distances involved (hundreds to
thousands of kilometers), the polarization properties
evolve rapidly and randomly as the light wave evolves
along the propagation distance, z. Not only does the
birefringence evolve, but it does so randomly, and on a
scale much faster than the distances required for
558 Nonlinear Schro¨ dinger Equations
communication transmission (birefringence polariza-
tion changes on a scale of 10–100 m). In this case, the
relevant nonlinear equation is eqn [35] above, but with
= 0and = 1. Indeed, this is the integrable VNLS
equation first derived by Manakov (1974).
It should be remarked that the VNLS equation
[34] and its generalization to an arbitrary number of
components,
iq
t
¼ q
xx
2 q
kk
2
q ½36
where q is an N-component vector and
kk
is the
Euclidean norm, are integrable by the IST. One has
to suitably extend the analysis discussed earlier in
this article (cf. e.g., Ablowitz et al. (2004)).
Discrete NLS Systems
Both the NLS and the VNLS equations discussed
above admit integrable discretizations which,
besides being used as the basis for constru cting
numerical schemes for the continuous counterparts,
also have physical applications as discrete systems.
A natural discretization of NLS [27] is the
following:
i
d
dt
q
n
¼
1
h
2
q
nþ1
2q
n
þ q
n1
ðÞ
q
n
jj
2
q
nþ1
þ q
n1
ðÞ½37
which is referred to as the integrable discrete NLS
(IDNLS). It is an O(h
2
) finite-difference approxima-
tion of [27] which is integrable via the IST and has
soliton solutions on the infinite lattice (Ablowitz and
Ladik 1975, 1976). Note that if the nonlinear term in
[37] is changed to 2 q
n
jj
2
q
n
, the equation, which is
often called the discrete NLS (DNLS) equation, is
apparently no longer integrable. It should be
remarked that the (apparently nonintegrable) DNLS
equation arises in many important physical contexts.
Correspondingly, one can consider the discretiza-
tion of VNLS given by the following system:
i
d
dt
q
n
¼
1
h
2
q
nþ1
2q
n
þ q
n1
kq
n
k
2
q
nþ1
þ q
n1
½38
where q
n
is an N-component vector. Equation [38]
for q
n
= q(nh) in the limit h ! 0, nh = x gives VN LS
[36]. The discrete vector NLS system [38] is also
integrable (Ablowitz et al. 1999, Tsuchida et al.
1999). The interested reader can find further details
in Ablowitz et al. (2004).
See also: Boundary-Value Problems for Integrable
Equations; Dynamical Systems in Mathematical Physics:
An Illustration from Water Waves; Evolution Equations:
Linear and Nonlinear; Ginzburg–Landau Equation;
Integrable Systems and Discrete Geometry; Integrable
Systems: Overview; Partial Differential Equations: Some
Examples; Riemann–Hilbert Methods in Integrable
Systems; Schro
¨
dinger Operators.
Further Reading
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communication systems with strong dispersion management.
Optics Letters 23: 1668–1670.
Ablowitz MJ, Biondini G, and Blair S (1997) Multi-dimensional
propagation in non-resonant
(2)
materials. Physics Letters A
236: 520–524.
Ablowitz MJ, Biondini G, Chakravarty S, Jenkins RB, an d
Sauer JR (1996) Four-wave mixing in wavelength-division
multiplexed soliton systems: damping and amp lification.
Optics Letters 21: 1646.
Ablowitz MJ and Clarkson PA, Nonlinear Waves, Solitons and
Symmetries, London Mathematical Society Lecture Notes Series.
Cambridge: Cambridge University Press (to be published).
Ablowitz MJ and Clarkson PA (1991) Solitons Nonlinear
Evolution Equations and Inverse Scattering. London Mathe-
matical Society Lecture Notes Series, vol. 149. Cambridge:
Cambridge University Press.
Ablowitz MJ, Hirooka T, and Biondini G (2001) Quasi-linear
optical pulses in strongly dispersion managed transmission
systems. Optics Letters 26: 459–461.
Ablowitz MJ and Ladik JF (1975) Nonlinear differential-difference
equations. Journal of Mathematical Physics 16: 598–603.
Ablowitz MJ and Ladik JF (1976) Nonlinear differential-
difference equations and Fourier analysis. Journal of Mathe-
matical Physics 17: 1011–1018.
Ablowitz MJ, Ohta Y, and Trubatch AD (1999) On discretiza-
tions of the vector nonlinear Schro¨ dinger equation. Physics
Letters A 253: 253–287.
Ablowitz MJ, Prinari B, and Trubatch AD (2004) Continuous and
Discrete Nonlinear Schro¨ dinger Systems, London Mathema-
tical Society Lecture Notes Series, vol. 302. Cambridge:
Cambridge University Press.
Ablowitz MJ and Segur H (1981) Solitons and the inverse scattering
transform. Soceity for Industrial and Applied Mathematics 4.
Anastassiou C, Segev M, Steiglitz K, Giordmaine JA, Mitchell M
et al. (1999) Energy-exchange interactions between colliding
vector solitons. Physical Review Letters 83: 2332.
Benney DJ and Newell AC (1967) The propagation of nonlinear
wave envelopes. Journal of Mathematical Physics 46: 133–139.
Benney DJ and Roskes GJ (1969) Wave instabilities. Studies in
Applied Mathematics 48: 377–385.
Calogero F and Degasperis A (1982) Spectral Transform and
Solitons I. Amsterdam: North-Holland.
Chiao RY, Garmire E, and Townes CH (1964) Self-trapping of
optical beams. Physical Review Letters 15: 479–482.
Davey A and Stewartson K (1974) On three-dimensional packets
of surface waves. Proceedings of the Royal Society of London
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Gabitov I and Turitsyn S (1996) Averaged pulse dynamics in a
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Ginzburg VL (1956) On the macroscopic theory of superconductivity.
Soviet Physics – JETP 2: 589 (russian: Journal of Experimental
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Ginzburg VL and Pitaevskii LP (1958) On the theory of super-
fluidity. Soviet Physics – JETP 7: 858–861 (russian: Journal of
Experimental and Theoretical Physics. USSR 34: 1240–1245).
Nonlinear Schro¨ dinger Equations 559
Gordon JP and Haus HA (1986) Random walk of coherently amplified
solitons in optical fiber transmission. Optics Letters 11: 665.
Gross EP (1961) Structure of quantized vortex. Nuovo Cimento
20: 454.
Hasegawa A (ed.) (2000) Massive WDM and TDM Soliton
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Hasegawa A and Kodama Y (1995) Solitons in Optical Commu-
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Hasegawa A and Tappert F (1973a) Transmission of stationary
nonlinear optical pulses in dispersive dielectric fibers I.
Anomalous dispersion. Applied Physics Letters 23: 142.
Hasegawa A and Tappert F (1973b) Transmission of stationary
nonlinear optical pulses in dispersive dielectric fibers II.
Normal dispersion. Applied Physics Letters 23: 171.
Kalinikos BA, Kovshikov NG, and Patton CE (1997) Decay-free
microwave envelope soliton pulse trains in yittrium iron
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Kelley PL (1965) Self-focusing of optical beams. Physical Review
Letters 15: 1005–1008.
Kodama Y and Hasegawa A (1992) Generation of asymptotically
stable optical solitons and suppression of the Gordon–Haus
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self-focusing of electromagnetic waves. Soviet Physics – JETP
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Non-Newtonian Fluids
C Guillope´ , Universite
´
Paris XII – Val de Marne,
Cre
´
teil, France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The flow of a fluid, liquid or gas, is described by
three conservation laws, the conserved physical
quantities being the mass, the linear momentum,
and the energy, and by constitutive equations. The
constitutive equations are specific to each fluid, and
link deformations to stresses.
A fluid is said to be Newtonian if it satisfies the
simplest constitutive equation, which gives the stress
tensor as a linear function of the rate of
deformation tensor D = (1=2)(ru þru
T
), namely
¼ð tr D pÞ I þ 2D ½1
where u is the fluid velocity, p is the hydrostatic
pressure (p 0), and and are the Lame´ viscosity
coefficients of the fluid, satisfying 0and þ
2=3 0. The superscript T designates the transpose
operation, the abbreviation ‘‘tr’’ the trace operator
of a tensor, and I the unit tensor. Water and glycerin
are examples of Newtonian liquids.
560 Non-Newtonian Fluids
Non-Newtonian fluids are fluids for which the
behavior is not described by eqn [1]. Silicone oils,
polymers (melted or in solution), egg yolks, and
blood are examples of non-Newtonian liquids.
Other examples include liquid crystals, rubbers,
suspensions, paints, etc.
In the following we shall first describe flows
which show Newtonian or non-Newtonian
behaviors. Then we shall describe the requirements
a constitutive equation needs to satisfy to be
considered, introducing the notions of continuum
mechanics we need. After giving the most commonly
used constitutive equations, we will give a few ideas
about the mathematical study of the set of equa-
tions, and their numerical study, in the particular
case of viscoelastic fluids.
Numerous kinds of materials are already known
to exist, and more might exist in the future. This
report, however, will be limited to the most
commonly materials used nowadays, which are
polymers, liquid crystals and polymeric liquids
crystals, and paints. Moreover, we shall only
consider isothermal flows, even though temperature
might be an important parameter in experiments
or in industry, because in particular most theoretical
or numerical studies concern isothermal problems.
Non-Newtonian fluids will always be liquids, and
we shall use the terms liquid or fluid indifferently.
Non-Newtonian Behaviors
We describe a few experiments to show how
differently both types of fluids, Newtonian or non-
Newtonian, might react in some experimental
situations. We also give some mechanical explana-
tion when possible.
Shear Thinning or Shear Thickening
In a Poiseuille experiment, wher e a fluid flows in
a tube under the action of a pressure drop, the
volumetric flow rate of a Newtonian fluid is
inversely proportional to the constant fluid viscosity.
Under the same pressure-drop condition, a polymer
melt flows much faster out of the tube, which means
that there is a decreasing apparent viscosity with
increasing shear rate: this is referred to as shear
thinning effect. Other fluids might exhibit the
opposite behavior and flow out of the tube more
slowly: this is called the shear thickening effect.
Rod Climbing
When a rotating rod is inserted in a beaker filled with
a Newtonian fluid, it is observed that the liquid near
the rotating rod is pushed outwards by centrifugal
force and that a dip on the surface of the liquid near
the rod results. On the contrary, if we make the same
experiment with a polymer, the fluid climbs along the
rod. Moreover, for comparable rotation speed, the
difference in behaviors might be quantitatively con-
siderable. This is explained by totally different
pressure repartitions in both fluids, Newtonian or
non-Newtonian: in particular, the pressure in the
polymer along the rod is much larger than that along
the beaker, so that this pressure difference fights the
centrifugal force; this is in contrast with the situation
in a Newtonian fluid.
Extrudate Swell
If a fluid is forced to flow from a large reservoir out
of a circular tube of small diameter, the swell at the
exit is much larger for a polymer solution than for a
Newtonian fluid. A polymer flowing out of a die
might also show a delayed die well, which means
that the swell is not at the exit but on the jet at a
certain distance of the exit. The explanation of this
phenomenon is not unique: it is due partly to
memory effects (the fluid remembers its former
shape, the one in the reservoir), partly to the release
of normal stresses, to interfacial forces, compressi-
bility, viscous heating, and the complicated flow
near the die exit.
Difference in Normal Stresses
In a shearing flow of a Newtonian fluid, the two
normal stress differences are both zero, whereas for
a polymer the first normal stress difference might be
very large, the second one being nearly zero. These
differences in stresses in shearing flow might be a
partial answer to the extrudate swell and to rod
climbing experienced by polymers.
Presence of a Yield Stress
Some materials, when subjected to shear stress,
flow only after a critical value is attained. Such
fluids are referred to as Bingham fluids: some
cements, slurries, paints, and biological fluids
might exhibit such a behavior. It is actually a
well-known property of paints: if put in large
quantities on a vertical wall, the paint will flow,
whereas if put as a very t hin film on the s ame wall,
the paint will not flow, but stay in place, and dry to
form a nice colored covering.
Preferred Orientation of the Particles of Fluid
Fluids with properties as above, Newtonian or
non-Newtonian, are isotropic in nature, even though
they are constituted of atoms, or of long chains of
material. They are the same everywhere, optically,
Non-Newtonian Fluids 561
magnetically, or electrically. Some fluids, liquid
crystals, or polymeric liquid crystals in particular,
have remarkable properties of nonanisotropy, being
able to orient themselves, on average, along a
particular direction: this is the nematic phase, which
is used in many devices (screens for clocks, hand
calculators, and cell phones), because the average
orientation may be changed by applying an electric
field. Other phases of liquid crystals include smectic
A, C, and C
phases, where one sees a preferred
orientation (tilted for C phases) of the fluid, and also
a layer-like structure. As an example, let us mention
discotic nematic liquid crystals, which are precursors
for carbon-based materials, such as fibers, compo-
sites, and films, which possess excellent mechanical
and thermal properties. Sails for race sailing boats are
made of Kevlar, which is one of these new materials
with remarkable properties.
Modeling
The flowing fluid will be described by its (Euler-
ian) velocity at time t and position x,sayu(x, t),
for x belonging to the domain of the flow and
the time t to R
þ
, by its mass density (x, t), its
pressure p(x, t)(p > 0 defined up to an additive
constant), and its stress (x, t)–whichisa
symmetric tensor.
The partial differential equations describing the
flow are satisfied in the domain of the flow and read
as follows:
@
@t
þ divðuÞ¼0
@u
@t
þðu rÞu
¼ div þ f
½2
where f denotes some external forces applied to the
fluid. These equations describe the conservation of
mass and the conservation of linear momentum. To
close the system, we need a constitutive equation for
the stress as well as initial conditions and
boundary conditions.
Moreover, most non-Newtonian fluids are practi-
cally incompressible in most regions of the flow, so
that we shall only consider this case: the first
equation in [2] is replaced by condition div u = 0in
the domain of the flow.
Notions of Continuum Mechanics
At time t, a body S occupies a region
t
of the
Euclidean space E
3
, called the configuration at time t,
of the body. Points p of S are called material points
or particles of fluids. The configuration
t
is assumed to be regular in the following sense:
t
is closed, its interior is connected and dense
everywhere, its boundary is piecewise regular, C
0
at
least.
A mapping :
0
!
t
is a deformation if is a
bijection from
0
onto
t
and is a C
1
–diffeomorph-
ism from the inte rior of
0
onto the interior of
t
,
with positive Jacobian.
The motion of a body S is given by a set of
deformations (t, t
0
):
t
!
t
0
, satisfying
ðt; tÞ¼Id; ðt
00
; tÞ¼ðt
00
; t
0
Þðt
0
; tÞ
The trajectory of the material point which is in X at
t
0
is the set
ðt; t
0
ÞðXÞ; t t
0
fg
A body is said to be rigid if the deformation (t, t
0
)
is an isometry for all times t and t
0
. A material point
p is said to be attached to the rigid body S if the
body p [ S is rigid.
The motion of a fluid might be described in terms
of the Lagrangian coordinates X 2
0
of each
particle of fluid:
0
is called the reference config-
uration and is the fixed configuration occupied by
the body of fluid at the time of reference, say t
0
. The
motion of the fluid might also by described in terms
of the Eulerian coordinates x = (X, t), which
represent the position of a particle at time t which
has position X at t
0
. The Lagrangian and Eulerian
coordinates of the same particle of fluid are linked
by the differential equation
_
ðX; tÞ¼uððX; tÞ; tÞ; for t t
0
ðX; t
0
Þ¼X
For defining the constitutive equations, we shall
use a few tensors that we define now. The defor-
mation gradient is defined by F(X, t) = @
(X, t)=@X, and the right Cauchy–Green tensor by
C = F
T
F (also called Cauchy strain). To define
relative tensors, we denote by =
t
(x, s)the
position at time s t of the material point, which
is at x at time t. T he relative tensors are defined in
the following way:
the relative deformation gradient F
t
(s) = r
t
(x, s),
the relative right Cauchy–Green tensor C
t
(s) = F
T
t
(s)
F
t
(s), and
the relative Finger tensor C
t
(s)
1
.
Note that the rate of deformat ion tensor is obtai ned
as the time derivative of the relative Cauchy strain
tensor:
D ¼
1
2
@C
t
ðsÞ
@s
j
s¼t
562 Non-Newtonian Fluids
Principle of Objectivity and Frame Invariance
A f rame of reference is defined in the spacetime
E
3
R attached to the observer by giving a
chronology and a system of reference. The chron-
ology is a timescale, which will be assumed to be
the same f or all observers. The system of reference
is a set of at least four points attached to a rigid
body (this is the o bserver), which are not
coplanar.
The constitutive equation needs to satisfy the
principle of frame invariance and of frame indiffer-
ence (or objectivity), which means that the equation
does not depend on rigid motions of the observer. In
the mathematical framework, it means that the
equation has to be invariant under a change of
orthonormal frame of reference x
= Q(t)x, where
Q(t) is an orthogonal tensor: the transformed
equation has to have the same expression, and also
to be frame indifferent. We define a scalar quantity
’, a vector field u, or a tensor field , as being frame
indifferent if, under the change of variables
x
= Q(t)x, they satisfy the relations ’(x, t) =
’
(x, t), u(x, t) = Q(t)
T
u
(x
, t), and (x, t) = Q(t)
T
(x
, t)Q(t), respectively.
The velocity gradient ru is not frame indifferent,
but its symmetric part is. The vorticity, which is the
antisymmetric part W = (ru ru
T
)=2 of the velo-
city gradient, satisfies the equation
_
W = Q
T
W
Q
Q
T
_
Q, where the dot denotes the convective deriva-
tive d=dt = @=@t þ (u r).
Note that the convective derivative of a
scalar function ’ is frame indiffe rent, which
means that
@’
@t
þðu rÞ’ ¼
@’
@t
þðu
r
Þ’
but the convective derivative of a vector or a tensor
is not frame indifferent.
It can be easily checked that the derivative
D
0
Dt
¼
d
dt
þ W W ½3
of a (frame-indifferent) tensor is frame indifferent,
which means that
D
0
Dt
¼ Q
T
D
0
Dt
Q
To obtain another frame-indifferent derivative of a
tensor , we need to start with the expression [3],to
which we may add other terms containing frame-
indifferent quantities, for example, combinations of
and D. A derivative which is often considered is
the Oldroyd derivative, as introduced by Oldroyd in
1958:
D
a
Dt
¼
d
dt
þ W W aðD þ DÞ½4
where a is a real parameter, chosen in the interval
[1, 1]. (This restriction on a is necessary for
viscometric reasons, and obtained when simple
flows, such as Couette or Poiseuille flows, are
studied.)
The case a = 1 corresponds to the upper convected
derivative, and the case a = 1 to the lower
convected derivative. The case a = 0 refers to the
corotational or Jaumann derivative. Derivatives
corresponding to cases a = 1, 0, or 1 might
actually be obtained by derivating in a frame
fixed locally to the body of fluid, and which rotates
and/or deforms with the body. Moreover, we shall
see that the derivatives corresponding to a = 1or1
have very simple integral expressions.
Constitutive Equations
The constitutive equation of a non-Newtonian fluid
is a nonlinear relationship between the stress tensor
and objective variables depending on the flow, such
as the pressure, the rate of deformation, frame-
indifferent derivatives of such quantities, etc.
Analogously to the constitutive equation for an
incompressible Newtonian fluid, we may also write
the stress tensor in the form = pI þ . The extra
stress tensor could be either a function of objective
variables, which characterize the flow, or defined by
a differential equation or by an integral equation.
The point here is to model the fact that the fluid
might have some elasticity or some memory, or
might experience, for example, yield stress or
orientational properties.
Shear dependent viscosity fluids A very simple
generalization of the incompressible Newtonian
fluid consists in making the visc osity dependent on
the rate of deformation tensor, = (D). This
generalization has been introduced by O A Ladyz-
henskaya in 1970 and, if the function is chosen
properly, this model reproduces the behavior of
existing fluids, at least in certain parts of their flow.
For power-law fluids, the viscosi ty depends on the
second invariant I
D
= (1=2)tr D
2
of the symmetric
tensor D (the first invariant tr D is zero because of
incompressibility), and reads as
ðDÞ¼
0
þ mI
n1
D
½5
where
0
0, m > 0, and n 0. If n = 1, we recover
the Newtonian case, whereas for n < 1 this equation
describes a shear thinning fluid, and for n > 1 a shear
thickening fluid. The power law is not valid for I
D
Non-Newtonian Fluids 563
close to 0, so that the Carreau–Yasuda law is
preferred:
1
0
1
¼ 1 þðI
D
Þ
2
ðn1Þ=ð2Þ
½6
where
0
is the zero-shear rate viscosity,
1
is the
infinite-shear rate viscosity, a time constant, n a
dimensionless power-law index, n 0, and >0a
parameter (generally equal to 1 for a monomolecu-
lar polymer).
Oldroyd models and related models Oldroyd mod-
els are differential models built with one of the
Oldroyd derivatives, and are very commonly used
for polymer solutions or melts. The stress tensor is
given as a solution of a differential equation in the
following way:
þ
1
D
a
Dt
þ gð;DÞ¼2 D þ
2
D
a
D
Dt
½7
where
1
> 0 is a relaxation time,
2
is a retardation
time, 0
2
<
1
, and g(, D) is a tensor-valued
function, constrained to certain restrictions due to
objectivity, and which is at least quadratic.
The Johnson–Segalman model has g = 0, and 1
a 1. Other models of differential type often
suppose the parameter a to be 1, because it has
been noticed that with a close to 1 the model is able
to reproduce some experimental behavior, whereas
for a = 1orcloseto1, the model does not work
at all. Among the models with a = 1, the following
ones are fairly popular: the model of Phan-Thien and
Tanner has g(, D) = tr ,where is a constant;
this model can be generalized by defining g(, D) =
2
þ , and being functions of the trace of
and of its determinant; the model of Giesekus is the
particular case where is a constant and = 0. The
Oldroyd eight-constant model is given by
gð;DÞ¼
0
ðtr ÞD þ
1
trðDÞI
þ
2
D
2
þ
2
trðD
2
ÞI
where
0
,
1
,
2
, and
2
are constants.
In [7] , the limit case
2
= 0 corresponds to
Maxwell’s type models, where there is no New-
tonian viscosity, while the case
2
> 0 corresponds
to the Jeffreys’ type models. The cases where a = 1
and g = 0, are often considered in mathematical or
numerical studies: this is the upper convected
Maxwell (UCM) model for
2
= 0, and the Oldroyd
B model for
2
> 0.
The parameters
1
,
2
, and might also depend
on I
D
: such a model where the uppe r convected
derivative (a = 1) is chosen is referred to as the
White–Metzner model, and reads as follows:
þ
I
D
1
Dt
¼ 2
I
D þ
1
D þ
I
D
1
D
Dt
where
1
is also the Newtonian viscosity.
Integral equations Other constitutive equations for
viscoelastic fluids include integral equations. Actu-
ally, some differential equations have integral
counterparts: this is the case for the differential
equations associated with the upper or lower
convected frame-indifferent derivatives. For the
upper convected derivative (a = 1), the extra stress
is given by the integral expression
ð x; tÞ¼2
2
1
Dðx; tÞþ2
1
2
2
1
Z
t
1
e
ðtsÞ=
1
ðr
X
xÞDðX; sÞðr
X
xÞ
T
ds
where X is the position, at time s, of the point which
is at x at time t. A similar expression might be
obtained for the lower convected derivative.
A very common integral equation is the K–BKZ
equation (introduced independently by Kaye and
Bernstein, Kearsley, and Zapas in 1962–63). In a
simplified form, the extra-stress tensor is given as
the integral of a combination of the relative Cauchy
strain tensor C
t
and its inverse:
ðx; tÞ¼2
Z
t
1
Gðt sÞ
@WðI
1
; I
2
Þ
@I
1
C
1
t
ðsÞ
@WðI
1
; I
2
Þ
@I
2
C
t
ðsÞ
ds
where I
1
= tr C
1
t
(s) and I
2
= tr C
t
(s). The function G
is a given kernel, and W a given scalar potential.
The upper convect ed Maxwell model is obtained
from the K–BKZ model by setting W(I
1
, I
2
) = I
1
and
G(s) = (
1
2
=2) e
1
s
.
Models issued from kinetic theories or micro–macro
models Polymeric fluids could also be modeled by
coupling a macroscopic viewpoint – the one of
continuum mechanics, as described above – and
a microscopic viewpoint. A polymer is, in general,
made of long chains of molecules. Rather than trying to
represent the polymer behavior by a sophisticated
constitutive equation, one describes the mean behavior
of the molecules by using their microscopic description.
To take an example, we consider a dilute solution
of polymer, where each chain of polymer is modeled
as a collection of dumbbells, each of them consisting
of two beads connected by a spring. The configura-
tion of the spring, namely its length and orientation,
is described by a random vector field Q 2 R
3
. The
dumbbells are convected and stretched by the flow.
564 Non-Newtonian Fluids
The probability (x, Q, t)dQ of finding a dumbbell
with a configuration Q at (x, t) is governed by a
Fokker–Planck equation:
d
dt
þ div
Q
ððruÞQ Þ
¼
2
div
Q
ððr
Q
WÞ Þþ
2kT
Q
where is the friction coefficient of the dumbbell
beads, T the temperature, and k the Planck constant,
and W the spring potential. The extra stress is given
by the constitutive equation
¼
Z
ðr
Q
WQÞ ðx; Q; tÞdQ
The simplest potential is the linear one (also called
Hookean potential) W(Q) = HjQj
2
,wherejQj is
the length of Q,andH the elasticity constant.
In fact, in the case of the Hookean potential, this
set of equations is equivalent to the Oldroyd B
model. Another potential corresponds to finitely
extendable nonlinear elastic (FENE) chain of
dumbbells,
WðQÞ¼
HQ
2
0
2
log 1
jQj
2
Q
2
0
!
for jQjQ
0
, and gives the FENE model, for which
there is no macroscopic constitutive equation known.
We have only made here a short incursion in these
micro–macro models: research is in progress, both
analytical and numerical (O
¨
ttinger 1996, Suen et al.
2002, Keunings 2004).
Liquid crystals and polymeric liquid crystals As an
example, we present the constitutive equations for a
uniaxial nematic liquid crystal.
In the theory of Leslie and Ericksen, established in
the 1960s and the 1970s, the stress tensor is given as
a function of the orientation unit vector n, through
the Oseen–Frank elastic energy,
2Wðn; rnÞ¼
1
ðdivnÞ
2
þ
2
ðn curl nÞ
2
þ
3
jn curl nj
2
where
1
> 0,
2
> 0, and
3
> 0 are the three basic
modes (splay, twist, and bend, respectively). The extra
stress tensor is precisely given by the relation
¼ðrnÞ
T
@W
@rn
þ
1
ðn DnÞn n
þ
2
N n þ
3
n N
þ
4
D þ
5
Dn n þ
6
n
where N =
_
n Wn is the corotational derivative of
the director, and
i
, i = 1, ..., 6, the six Leslie
viscosity coefficients.
The director satisfies a differential equation
derived from continuum mechanics,
1
€
n ¼ G þ g þ div
where
1
is the moment of inertia per unit volume,
G the external director body force (torque per unit
volume), the director stress tensor, and g the
intrinsic director body force. Precisely,
g ¼ n ðrnÞ
@W
@n
1
N
2
Dn
¼ n þ
@W
@rn
where is a Lagrange multiplier vector, and =
2
=
1
is the reactive parameter, with
1
=
3
2
the rotational viscosity, and
2
=
6
5
=
3
þ
2
the irrotational torque coefficient.
Polymeric liquid crystals might have other variables
entering in the modeling, such as order parameters,
order tensors, etc.
Because of the complexity of modeling , most
studies concern either very simple flows, such as
Couette or Poiseuille flows, or steady flows, or
flows for which the coefficients satisfy specific
relationships.
Reports about earlier studies, theoretical as well
as numerical, can be found in Coron et al. (1991),
and references therein. The study of polymeric liquid
crystals, or of the smectic phase of liquid crystals is
at its very early stage and one could look into it in
specialized journals, such as the Journal of Non-
Newtonian Fluid Mechanics,orsee Liquid Crystals.
Yield stress fluids Bingham materials have the
property of flowing only when the stress magnitude
is greater than a critical value, and being a solid
otherwise. Precisely, in the simplest and the most
widely used model, the Bingham model, the extra
stress tensor is given by the relations
¼ 2D þ
D
I
D
if I
D
6¼0
jj
if I
D
¼0
½8
where
> 0 is the yield limit. The Bingham model
is generalized in taking the viscosity to be a
function of the shear stress: is given by the
relation
¼ 1 þ 2
I
D
1=2
Non-Newtonian Fluids 565
for the Casson law, and by the power law [5] for the
Herschel–Bulkley model.
The mathematical study was started by Duvaut
and Lions (1976), and regained interest recently
(Malek and Rajagopal 2005), especially in relation
with other recent studies in polymeric liquids.
Theoretical and Numerical Problems
for Viscoelastic Flows
The mathematical study of viscoelastic fluid flows
amounts to studyin g systems of partial differential
equations, which all include either the incompres-
sible Euler equation or the incompressible Navier–
Stokes equation as particular cases. In particular, it
means that the results obtained from such a study
are similar to the ones obtained for Euler or Navier–
Stokes equations, and, because of the complexity of
the system, the results are expected to be qualita-
tively as good, actually more often less good, than
for these equations. For example, the existence of
weak three-dime nsional solutions to the Navier–
Stokes system is known, while for non-Newtonian
flows, this result will be true only in very specific
cases. Moreoever, when a result is not known for
the Navier–Stokes problem, such as the uniqueness
of solution for all data in a three-dimensional
problem, there is no hope something similar could
be proved for non-Newtonian fluid flows.
As an example, we consider the case of Johnson–
Segalman fluids, which are described by constitutive
equation [7] with g = 0. Recall that the limit case
2
= 0 corresponds to the purely elastic case, and
2
=
1
to the purely Newtonian case. Equation [7]
is coupled with the equations of motion:
du
dt
þrp ¼ div þ f
div u ¼ 0
½9
Equations [7] and [9] have to be solved in the
domain of the flow, which might be the whole
space R
3
(or R or R
2
in case of symmetries), or a
domain , bounded or not, in R
n
, n = 1, 2, or 3.
These equati ons are supplemented by appropriate
boundary conditions and ini tial conditions for the
velocity u and the extra stress (no boundary
condition on is needed if the homogeneous
nonslip boundary condition u = 0 is chosen).
We first make explicit the Newtonian contribu-
tion to the stress by setting =
s
þ
p
and
s
= 2
s
D. The differential equation for
p
is then
p
þ
1
D
a
p
Dt
¼ 2
p
D
where
p
= (1
2
=
1
) is the so-called polymeric
viscosity,
s
= (
2
=
1
) the so-called Newtonian
viscosity (or solvent viscosity).
We then use nondim ensional variables, so as to
make explicit the characteristic parameters, which
the flow depends on. The non-Newtonian fluid
considered in this model will always be homoge-
neous: its density is a constant independent of x
and t. The dimensional variables are now asterisked.
We define quantities which are characteristic of the
flow: a length L, a velocity magnitude U, a stress
magnitude T, a force magnitude F, and a pressure P.
We operate the change of variables and functions
x = x
=L, u = u
=U, t = Ut
=L, and also introduce the
nondimensional functions
¼
T
; p ¼
p
P
; f ¼
f
F
After choosing the parameters T, P, and F in
an appropriate way, namely T = P = U=L, and
F = U=L
2
, we obtain the following system
Re
du
dt
þrp ¼ð1 !Þu þ div þ f
div u ¼ 0
We
D
a
Dt
þ ¼ 2!D
½10
Here the three nondimensional parameters which
the flow depends on are the usual Reynolds number
Re =
0
UL= and two other numbers: the Weissen-
berg number We = U=L measures the elasticity per
unit time (sometimes also called the Deborah
number), and the parameter ! =
p
= is the ratio of
elastic viscosity to total viscosity (! = 0 corresponds
to the Newtonian case, while ! = 1 corresponds to
the purely elastic case).
System [10] couples a transport equation (the
equation for the stress ), and either a Navier–
Stokes type equation when !<1, or a Euler type
equation when ! = 1 (for the velocity u). This
system is not hyperbolic, parabol ic, or elliptic.
Maxwell’s type models (! = 1) display two striking
phenomena. First, the Cauchy problem (with initial
data) can present Hadamard instabilities, that is,
instabilities to short waves. It means, in particular, that
the Cauchy problem is not well posed in any good class
but analytic. Moreover, the partial differential system
for Maxwell’s type steady flows may experience a
change of type, analogous to the situation in gas
dynamics, if the ‘ ‘Mach number’’ Re We is larger than 1.
Jeffreys’ type models (!<1), because of the
presence of a Newtonian viscosity, do not exhibit
such phenomenon, but their study does not enter in
566 Non-Newtonian Fluids