# Understanding viscoelastic flow instabilities: Oldroyd-B and
beyond111Published in the Oldroyd-100 special issue of the Journal of non-
Newtonian Fluid Mechanics

Hugo A. Castillo Sánchez 222The authors are listed alphabetically and
contributed in their areas of expertise to this review, whose preparation was
coordinated by V. Shankar and G. Subramanian. Department of Applied
Mathematics and Statistics, Institute of Mathematics and Computational
Sciences, University of Sao Paulo, Brazil<EMAIL_ADDRESS>Mihailo R.
Jovanović  Ming Hsieh Department of Electrical and Computer Engineering,
University of Southern California, Los Angeles, California 90089, USA
<EMAIL_ADDRESS>Satish Kumar  Department of Chemical Engineering and Materials
Science, University of Minnesota, Minneapolis, MN 55455, USA<EMAIL_ADDRESS>Alexander Morozov  SUPA, School of Physics and Astronomy, The University of
Edinburgh, Edinburgh EH9 3FD, UK<EMAIL_ADDRESS>V. Shankar
Department of Chemical Engineering, Indian Institute of Technology, Kanpur
208016, India<EMAIL_ADDRESS>Ganesh Subramanian  Engineering Mechanics
Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore
560064, India<EMAIL_ADDRESS>Helen J. Wilson  Mathematics Department,
University College London, Gower Street, London WC1E 6BT, UK
<EMAIL_ADDRESS>
###### Abstract

The Oldroyd-B model has been used extensively to predict a host of
instabilities in shearing flows of viscoelastic fluids, often realized
experimentally using polymer solutions. The present review, written on the
occasion of the birth centenary of James Oldroyd, provides an overview of
instabilities found across major classes of shearing flows. These comprise (i)
the canonical rectilinear shearing flows including plane Couette, plane and
pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines
such as those in the Taylor-Couette, cone-and-plate and parallel-plate
geometries; (iii) non-viscometric shearing flows with an underlying
extensional flow topology such as the flow in a cross-slot device; and (iv)
multilayer shearing flows. While the underlying focus in all these cases is on
results obtained using the Oldroyd-B model, we also discuss their relation to
the actual instability, and as to how the shortcomings of the Oldroyd-B model
may be overcome by the use of more realistic constitutive models. All the
three commonly used tools of stability analysis, viz., modal linear stability,
nonmodal stability, and weakly nonlinear stability analyses are discussed,
with supporting evidence from experiments and numerical simulations as
appropriate. Despite only accounting for a shear-rate-independent viscosity
and first normal stress coefficient, the Oldroyd-B model is able to
qualitatively predict the majority of instabilities in the aforementioned
shearing flows. The review also highlights, where appropriate, open questions
in the area of viscoelastic stability.

###### keywords:

Oldroyd-B fluid; purely elastic instability; elastic turbulence; elasto-
inertial turbulence; nonmodal stability; nonlinear stability.

##  1 Introduction

Compelling differences between Newtonian and viscoelastic flow phenomena in
the same geometry have been well highlighted in textbooks [1], and this
contrast also applies to instabilities occurring in the same base-flow
configuration. Viscoelastic flows are prone to novel instabilities that arise
due to elasticity alone, or due to a combination of elastic and inertial
effects, and such instabilities evidently have no Newtonian counterparts.
Initial interest in the understanding of these instabilities was driven by the
need to prevent their occurrence during polymer processing operations [2, 3],
and thereby circumvent the associated restrictions on processing rates; fluid
inertia is usually negligible in these processes, and the focus is therefore
on purely elastic instabilities. For instance, during extrusion of highly
viscous entangled polymer melts, the extrudate often exhibits a spiral or wavy
distortion, a phenomenon referred to as ‘melt fracture’ [4], and thought to
occur via a hydrodynamic instability; although, physicochemical effects such
as wall slip likely play a role [5]. A second, more recent, motivation for
studying viscoelastic flow instabilities arose from their discovery in the
standard rheometric geometries (e.g. the Couette, cone-and-plate and parallel-
plate set ups); for instance, see [6, 7]. The occurrence of purely elastic
instabilities, in dilute polymer solutions subject to simple curvilinear
shearing flows in rheometric devices, hampered their use for purposes of
rheological characterization; an understanding of these instabilities is
clearly essential for defining the operating range of these devices. The
elastic instabilities above have their origins in normal stress differences
present in viscoelastic shear flows. For the curvilinear shearing flows in
particular, the first normal stress difference leads to a hoop stress, on
account of streamline curvature, which drives an instability.

Viscoelastic flow instabilities can also be beneficial depending on the
scenario. For instance, with regard to the rheometric example above,
increasing the shear rate causes the initial elastic instability to eventually
saturate in a complex disorderly nonlinear flow state termed ‘elastic
turbulence’ (ET) [8, 9, 10]. There have been reports of similar ET-like states
in rectilinear flows of dilute polymer solutions through micro-channels,
especially when the flow is perturbed by finite-amplitude obstacles at the
channel inlet [11, 12, 13, 14]. The inaccessibility of Newtonian turbulence on
microfluidic scales (owing to the modest Reynolds numbers) implies that the ET
state, in either of the two cases above, can instead be exploited towards
increasing mixing efficiencies, as has indeed been demonstrated in earlier
efforts [9].

While the above instances probed the low Reynolds number ($Re$) regime where
fluid inertial effects are negligible, there have also been many reports of
‘early turbulence’ in pipe flow of polymer solutions at higher Re, albeit
still lower than the oft-quoted Newtonian threshold of $Re\sim 2000$ [15, 16,
17, 18, 19, 20, 21]; these early studies were, however, not systematically
corroborated in the subsequent literature. In contrast, the recent experiments
of Hof and coworkers [22, 23], involving pipe flow of polymer solutions at
concentrations below or close to the overlap value, have unambiguously
demonstrated transition from the laminar state at $Re\sim 1000$, thereby
confirming the aforementioned observations. The ensuing flow was found to be
neither laminar, nor to bear a resemblance to Newtonian turbulence, and was
therefore christened ‘elasto-inertial turbulence’ (EIT) to emphasize the
importance of both fluid elasticity and inertia, in contrast to the ET states
discussed above. This EIT state was further shown to be linked [24] to the
asymptotic maximum drag reduction (MDR) regime, a universal state that arises
with the progressive addition of polymers to turbulent Newtonian pipe flow
[25]. This link is an important one. The phenomenon of turbulent drag
reduction [26] is undoubtedly one of the most spectacular manifestations of
viscoelasticity, and while there exists a vast body literature in this regard
[27, 28, 29], the prevailing viewpoint regards the aforementioned MDR regime
as a drag-reduced state accessible only from the Newtonian turbulent state;
even relatively recent dynamical-systems-based interpretations have attempted
to understand MDR in terms of the existence of essentially Newtonian coherent
structures, modified by elasticity [30, 31]. As a consequence, advances in
viscoelastic stability and drag reduction have occurred largely independently
with very little cross-pollination of theoretical viewpoints. However, as we
discuss later in this review, the aforementioned pipe flow experiments,
together with recent theoretical work [32], show that the MDR regime, at least
for moderate $Re$, can be viewed as a ‘drag-enhanced’ state arising from an
elastoinertial instability of the laminar state. The above efforts for pipe
flow, and other recent efforts that include theoretical [33], computational
[34] and experimental [7] investigations of other rectilinear and curvilinear
shearing flow configurations, reflect an increasing interest in the origin and
dynamics of elastoinertial instabilities.

The present review article, written on the occasion of the birth centenary of
James Oldroyd who proposed the now-eponymous constitutive equation [35],
attempts to provide a state-of-the-art summary of the understanding of flow
instabilities using the Oldroyd-B model. That said, however, where
appropriate, we also discuss the use of more refined constitutive models, with
physics outside of the Oldroyd-B framework, that are often crucial to
obtaining better agreement with experimental observations. Indeed, from a
fundamental standpoint, an accurate prediction of viscoelastic flow
instabilities may also be regarded a rigorous test that aids in the eventual
development of physically sensible constitutive equations [7, 36]. While there
have been many earlier review articles on the subject of viscoelastic flow
instabilities [2, 3, 6, 7], the present review focuses on the developments
over the last two decades. Further, in contrast to some of the review articles
above, which have focused almost exclusively on the hoop-stress-driven elastic
instabilities that arise in the curvilinear rheometric geometries, this review
covers instabilities in both the canonical rectilinear and curvilinear
shearing flows, with the latter encompassing both viscometric and non-
viscometric flow configurations. In fact, the experimental observations
mentioned in the preceding paragraph, pertaining to the moderate-to-high $Re$
regimes in flow through pipes and channels, have spawned renewed interest in
viscoelastic instabilities that occur in rectilinear shearing flows, and the
present review lays a greater emphasis on these more recent findings. The
review by Renardy and Thomases [37] in this special issue presents a different
perspective, by focusing on open mathematical challenges related to the
Oldroyd-B model, and the write-up of Hinch and Harlen [38] provides a
conceptual account of Oldroyd’s formulation of upper- and lower-convected
derivatives. While the review by Shaqfeh and Khomami [39], also a part of this
special issue, addresses instabilities in curvilinear viscoelastic flows based
on the Oldroyd-B model, as already mentioned, we also place an emphasis (in
Sec. 4 below) on how the use of genuinely nonlinear constitutive equations
that account for shear-thinning effects plays an important role in more
accurate predictions of experimental observations. Another recent multi-author
review article [40], based on the virtual workshop on viscoelastic flow
instabilities and elastic turbulence organized by the Princeton Center for
Theoretical Sciences, also provides a state-of-the-art summary of the various
challenges in this broad area.

It is worth recalling that Oldroyd, in his seminal 1950 paper [35], proposed a
constitutive model for viscoelastic flows purely from a continuum viewpoint,
by requiring the model to satisfy the principle of material frame
indifference. As discussed by Hinch and Harlen [38], Oldroyd argued that the
frame indifference implied that the time derivative of the stress tensor be
computed in a reference frame that undergoes local translation, rotation, and
deformation with the material. The usual material derivative in fluid
mechanics denotes the instantaneous rate of change of a fluid property (e.g.
velocity, temperature) in a reference frame that translates with a given fluid
element, and is thereby appropriate only for a point microstructure. In
contrast, the upper convected derivative introduced by Oldroyd denotes the
rate of change in a reference frame that, in addition, deforms (affinely) with
the fluid motion, thereby accounting (implicitly) for an underlying orientable
microstructure.

Interestingly, the Oldroyd-B constitutive equation can also be derived from a
coarse-grained, mesoscopic model wherein a dilute solution of polymer
molecules is modeled as a suspension of non-interacting Hookean ‘dumbbells’ in
a Newtonian solvent, each polymer molecule being idealized as a dumbbell
comprising an infinitely extensible Hookean spring connecting two beads which
account for the drag experienced by the polymer molecule [1, 41]. Any
additional physical effects entering this mesoscopic picture, for instance, a
nonlinear spring force, a configuration-dependent bead drag, or hydrodynamic
interaction between the beads, invariably lead to a closure problem, and
derivation of the equation governing the polymeric stress, to be used in the
continuum mechanical formulation, requires appropriate pre-averaging
approximations. The stress tensor in the Oldroyd-B constitutive equation is
conveniently divided into two parts that correspond to the polymeric and
Newtonian solvent contributions. Thus, the stress is written as
$\bm{\tau}=\bm{\tau}_{p}+\bm{\tau}_{s}$, where the solvent contribution
$\bm{\tau}_{s}=\eta_{s}(\nabla\bm{v}+\nabla\bm{v}^{T})$, $\eta_{s}$ being the
solvent viscosity, and the polymeric contribution $\bm{\tau}_{p}$ is given by
the ‘upper-convected Maxwell’ (UCM) constitutive equation:

$\lambda\left(\frac{\partial\bm{\tau}_{p}}{\partial
t}+\bm{v}\cdot\nabla\bm{\tau}_{p}-\left(\nabla\bm{v}\right)^{T}\cdot\bm{\tau}_{p}-\bm{\tau}_{p}\cdot\left(\nabla\bm{v}\right)\right)+\bm{\tau}_{p}=\eta_{p}\left(\nabla\bm{v}+\nabla\bm{v}^{T}\right)\,.$
(1)

The terms within parentheses on the left-side of the above equation represent
the upper-convected derivative of $\bm{\tau}_{p}$ described above, $\lambda$
is the relaxation time of the dumbbell, being related to the longest
relaxation time of the polymer molecule, and $\eta_{p}$ is the polymeric
contribution to the steady-shear viscosity. The nonlinear coupling between the
velocity and the stress in the upper-convected derivative suggests that, even
in the absence of fluid inertia, one may anticipate bifurcations from a given
base flow, leading to qualitatively new instabilities - such bifurcations are
indeed the basis of the purely elastic instabilities mentioned earlier. In the
limit of zero solvent viscosity, $\eta_{s}=0$, the Oldroyd-B model reduces to
the UCM model.

For steady simple shear flow, the Oldroyd-B model predicts a shear-rate-
independent viscosity and first normal stress coefficient
$\Psi_{1}=N_{1}/\dot{\gamma^{2}}$ (where $N_{1}$ is the first normal stress
difference and $\dot{\gamma}$ is the shear rate), and yields a zero second-
normal stress difference ($N_{2}=0$). The shear-rate-independence is due to
the infinite extensibility of the Hookean dumbbells in the mesoscopic picture
underlying the Oldroyd-B model. The Hookean response is valid only for small
deformations from the equilibrium conformation of a polymer molecule.
Consequently, the Oldroyd-B model is not applicable for strongly shear-
thinning systems such as polymer melts or water-based dilute polymer
solutions. The Oldroyd-B model also cannot describe phenomena ascribable
exclusively to an $N_{2}$ which, in the present context, include spanwise
instabilities in a rectilinear shearing flow [42, 43]. While the model
correctly predicts a thickening behavior in extensional flows, it also
predicts an unbounded growth of the extensional viscosity beyond a threshold
extension rate, this growth again arising from the infinite extensibility of
the Hookean dumbbells. Thus, the Oldroyd-B model is expected to be more
relevant to shear-dominated flows.

One way to rectify the aforementioned shortcomings of the Oldroyd-B model is
to go beyond the Hookean dumbbell assumption, by incorporating a nonlinear
spring with the spring force diverging at maximum extension; the most commonly
used form of the force law leads to the so-called finitely extensible
nonlinear elastic (‘FENE’) springs [44]. The nonlinearity of FENE springs does
not allow for an analytical solution of the Smoluchowski equation in the
underlying kinetic theory framework [45]. Consequently, preaveraging
approximations are needed to obtain a closed-form relation between the stress
and the strain rate, and the constitutive equation thus obtained is referred
to as the ‘FENE-P’ model (‘P’ being Peterlin, who proposed this
approximation). The FENE-P model is characterized by an additional
dimensionless parameter, $L$, which is the ratio of the fully stretched length
to the equilibrium coil dimension; $L\rightarrow\infty$ recovers the Oldroyd-B
model. The nonlinear stiffening of the spring, and the resulting decrease in
the relaxation time implies that the FENE-P model predicts shear thinning of
both the viscosity and the first normal stress coefficient at high shear
rates. The inclusion of a nonlinear spring force also removes the divergence
of the extensional viscosity, causing it to saturate at a large but finite
value, in accord with experimental observations [41]. A closely related
nonlinear constitutive equation that also incorporates a finitely extensibile
spring is the FENE-CR model (proposed by Chilcot and Rallison [46]), which
predicts a constant shear viscosity, while allowing a for a shear-rate
dependence of the first normal stress coefficient, and is thereby especially
suited for the so-called Boger fluids; see footnote 5 in Sec. 4.2.

A second way of addressing the deficiencies of the Oldroyd-B model, one
appropriate for concentrated polymer solutions, is to recognize the anisotropy
of a given dumbbell’s environment, this anisotropy arising from the the
stretched dumbbells in its neighborhood; this may be incorporated via an
anisotropic tensorial correction to the relaxation term. Giesekus [47]
postulated that the tensor characterizing the anisotropy is proportional to
the (deviatoric) stress itself, leading to the Giesekus constitutive equation.
The proportionality constant, $\alpha$, measures the amplitude of anisotropy,
with $\alpha=1$ denoting maximum anisotropy, and $\alpha=0$ denoting the
original isotropic relaxation in the UCM model [41]. For any non-zero
$\alpha$, the Giesekus model includes an additional term quadratic in the
stress tensor which becomes important in the nonlinear regime. Similar to the
FENE-P model, the Giesekus model predicts shear thinning of both the viscosity
and first normal stress coefficient, and does not exhibit any singularity in
the extensional viscosity.

Even for the simplest shearing flows driven by the motion of rigid boundaries,
and that are characterized by a single length ($H$) and velocity ($V$) scale,
the stability of an Oldroyd-B fluid is governed by three dimensionless
parameters: the Reynolds number $Re=HV\rho/\eta$, the Weissenberg number
$Wi=\lambda V/H$ which is the product of the polymer relaxation time and a
typical shear rate, and the ratio of solvent to solution viscosity
$\beta=\eta_{s}/(\eta_{s}+\eta_{p})$; here, $\rho$ and
$\eta=\eta_{p}+\eta_{s}$ are, respectively, the density and total viscosity of
the polymer solution. For purely elastic instabilities, $Wi$ and $\beta$ are
the relevant parameters; the elasticity number $E=Wi/Re=\lambda\eta/(\rho
H^{2})$, that is independent of the flow, being the ratio of the polymer
relaxation and the momentum diffusion timescales, may also be used (instead of
either $Re$ or $Wi$) when describing elastoinertial instabilities. We note in
passing that the capillary number, denoting the ratio of viscous to surface
tension forces, will become relevant for shearing flows with a free surface.
For viscoelastic flows in particular, the ‘elastocapillary’ number, which is
the ratio of Weissenberg and capillary numbers, and measures the relative
importance of elastic and capillary forces, may be used [48]. In flow
configurations involving multiple length scales and/or in the presence of an
unsteady shearing, the Deborah number ($De$), defined as the ratio of the
relaxation time to a characteristic flow time $T$, is also used; here, $T$ can
be either the residence time of a fluid element in the region of interest or
the time period of an oscillatory shear flow. The Deborah and Weissenberg
numbers are often interchangeably used for steady shearing flows. For the
curvilinear viscometric flows examined in Sec. 4, $De$ and $Wi$ will be seen
to be related to each other by the aspect ratio of the particular flow
configuration [49].

In light of the above, the subject of transition in viscoelastic shearing
flows, unlike their Newtonian counterparts, is not a ‘single problem’. For a
given viscoelastic shearing flow, one could be in an inertially dominant
regime with weak elasticity ($Wi\ll 1,Re\gg 1$), a strongly elastic near-
Stokesian regime ($Wi\sim O(1)$, $Re\ll 1$), or an elasto-inertial regime with
inertia and elasticity being of comparable importance ($Wi,Re>1,E\sim O(1)$);
the nature of transition would depend sensitively on the particular asymptotic
regime. In addition, the solvent viscosity ratio $\beta$, which is a proxy for
polymer concentration, allowing one to span the regimes from ultra-dilute
polymer solutions ($\beta\to 1$) to polymer melts ($\beta\rightarrow 0$), is
also expected to influence the transition. Thus, transition from the steady
laminar state, to states with nontrivial spatiotemporal dynamics, can occur
via multiple pathways in $Re$-$Wi$-$\beta$ space; we return to the aspect of
multiple transition scenarios, in the context of rectilinear flows, in Sec.
2.3.

The first step in analyzing the stability of a laminar flow is to consider its
response to infinitesimal disturbances, which allows for the linearization of
the governing equations about the laminar base state. Within this linear
stability framework, there are two different approaches. The classical
approach is modal stability, and involves expressing the perturbation fields
in the normal mode form with an exponential dependence in time, in turn
leading to an eigenvalue problem for the growth rate as a function of the
wavenumber and other relevant dimensionless parameters. According to the
convention usually followed, a change in sign of the imaginary part of the
eigenvalue (the growth rate), from negative to positive, corresponds to the
onset of instability. For sheared base states in particular, the non-normality
of the differential operator governing linear stability implies that the
aforementioned modal stability analysis only pertains to the asymptotic
behavior for long times; when unstable, this long-time evolution is dominated
by exponentially growing modes [50]. Even in the absence of such unstable
modes, however, small amplitude disturbances can grow algebraically for
shorter times. The machinery for a detailed analysis of this so-called
transient growth is now well developed, and has been extensively applied in
the Newtonian context [51].

Going beyond linear stability, finite amplitude disturbances are often
considered within the framework of an amplitude expansion, an approach that
originated in the efforts of Stuart [52] and Watson [53] in the Newtonian
context. In fact, the nonmodal and nonlinear stability approaches have
acquired prominence owing to the failure of the classical modal approach to
explain transitions in any of the canonical Newtonian shearing flows (plane
Couette, plane Poiseuille and pipe flows). All the three approaches above are
covered in this review within the context of the Oldroyd-B model and its
refinements. It is important to note that, in recent times, direct numerical
simulations of viscoelastic flows complement the aforementioned approaches,
providing detailed structural information in the nonlinear regime [54, 55, 56,
57, 58, 59, 60]. The computational expense, however, implies that the
parameter space explored by such simulations is necessarily restricted, a
limitation that is amplified by the high-dimensional nature of the parameter
space characterizing viscoelastic shearing flows.

The remainder of this review is organized as follows. We begin with
instabilities in simple rectilinear flows in Sec. 2. After a brief summary of
the principal features of the Newtonian spectrum in Sec. 2.1, we discuss the
nature of the viscoelastic spectrum in the inertialess limit in Sec. 2.2.
Herein, we point out that rectilinear shearing flows are generally linearly
stable in the $Wi$-$\beta$ space; although, plane Poiseuille flow has recently
emerged as an exception in this regard, becoming unstable for $Wi\gg 1$ and
$\beta\to 1$. In Sec. 2.3, the elasto-inertial spectrum for the canonical
rectilinear shearing flows is discussed, and it is shown that while plane
Couette flow is always stable in the $Re$–$Wi$–$\beta$ space, both plane- and
pipe-Poiseuille flows are unstable in significant domains of this space. The
nature of instabilities in these flows is discussed briefly, and based on our
current knowledge, we provide an overview of various possible transition
scenarios in the $Re$-$Wi$-$\beta$ space. In Sec. 3, we discuss instabilities
in two-layer flows of viscoelastic fluids wherein a jump in the first normal
stress difference leads to a novel instability absent for Newtonian two-layer
flows. This section also includes a brief summary of instabilities in shear-
banded flows; while shear banding itself is outside the purview of the
Oldroyd-B model, owing to the absence of nonmonotonicity in the constitutive
curve, the instabilities in the banded state can nevertheless be usefully
interpreted using the Oldroyd-B model.

Instabilities in curvilinear viscometric shearing flows are surveyed in Sec.
4. Here, we first discuss (Sec. 4.1) the role of elasticity on the Newtonian
(centrifugal) instability in the Taylor-Couette geometry, before moving on to
a discussion of the purely elastic instability in the same geometry (Sec.
4.2). The effects of finite-gap widths (relative to the radius of the inner
cylinder) and nonaxisymmetric disturbances are summarized in Sec. 4.3. The
role of inertia in leading to additional instabilities, and the nature of the
resulting dominant mode in the $Wi-Re$ plane, is discussed in Sec. 4.4.
Elastic instabilities in other curvilinear viscometric flows, including the
those in the cone-and-plate and parallel-plate geometries are discussed in
Sec. 4.5. The issues that bedevil the comparison of experimental observations
and theoretical predictions of purely elastic instabilities are then discussed
in Sec. 4.6. The role of rheological features beyond the scope of the
Oldroyd-B model (such as a nonzero second normal-stress difference) is
analyzed in Sec. 4.8; for the rectilinear shearing flows surveyed in Sec. 2,
the relatively nascent state of understanding, of instabilities that drive
transition, has meant that the consequences of refinements to the Oldroyd-B
model are only beginning to be explored. We end Sec. 4 with a discussion of
the Pakdel-McKinley criterion for instabilities in shearing flows with curved
streamlines, and in particular, its use in understanding the role of shear
thinning in limiting the elastically unstable domain in the relevant parameter
plane (Sec. 4.9). Section 5 examines instabilities in non-viscometric settings
such as the cross-slot geometry (Sec. 5.1), contraction-expansion flows (Sec.
5.2), and flow past a circular cylinder (Sec. 5.3). While the approach used in
the sections above is the classical modal one, Sec. 6 provides an account of
recent efforts, within the non-modal transient growth framework, applied to
viscoelastic flows. The role of finite-amplitude disturbances, within a
nonlinear stability framework, is discussed in Sec. 7, the emphasis being on
the purely elastic scenario. Finally, in Sec. 8 we end with a brief discussion
on some of the outstanding issues in this field.

##  2 Rectilinear shearing flows: Results from modal analyses in the
$Re$-$Wi$-$\beta$ space

In this section, we examine the stability of canonical rectilinear shearing
flows, comprising plane Couette, plane Poiseuille and pipe Poiseuille flows,
from the modal perspective. Further results from the non-modal viewpoint are
presented later in Sec. 6. A prerequisite to understanding the non-trivial
structure of the full elastoinertial spectrum, and associated instabilities,
is an understanding of the spectra arising from inertial and elastic forces
acting separately. Thus, we begin with a discussion of the Newtonian spectrum
below, and follow it up with a discussion of the inertialess elastic spectrum
associated with an Oldroyd-B fluid. This discussion also highlights the
recent, and unexpected, discovery of a purely elastic instability in plane
Poiseuille flow.

Figure 1: The eigenspectrum for Newtonian plane Poiseuille flow at
$Re=10^{4}$, $k=1$ illustrating the A, P, and S branches characteristic of the
Newtonian spectrum. One of the modes belonging to the A branch is unstable for
the chosen parameters. The spectrum is obtained for two different values of
$N$ (the number of collocation points in the spectral collocation method), and
the convergence of the eigenvalues with $N$ demonstrates that the eigenvalues
are physically genuine.

###  2.1 The Newtonian Spectrum

It is useful to first recall features of the $Re$-dependent Newtonian
eigenspectrum for the canonical rectilinear shearing flows [51]. By way of
illustration, consider plane Poiseuille flow in the $x$-direction with
velocity profile $U(y)\propto[1-(y/H)^{2}]$; here, $2H$ is the separation
between the walls with the spanwise base-state vorticity pointing along the
$z$-direction. In the linear stability analysis, and within the modal
framework, the perturbed velocity field is assumed to be of the form
$[U(y)+v_{x}^{\prime}(x,y,z,t),v_{y}^{\prime}(x,y,z,t),v_{z}^{\prime}(x,y,z,t)]$,
where the primes denote the perturbation components which are taken to be
Fourier modes $v_{x,y,z}^{\prime}(x,y,z,t)=\tilde{v}_{x,y,z}(y)\exp[ikx+ilz-
ikct]$; here, $k$ and $l$ are the wavenumbers in the $x$ and $z$-directions,
$\tilde{v}_{x,y,z}(y)$ are the shapes (eigenfunctions) of the perturbations in
the $y$ direction, and the complex wavespeed $c=c_{r}+ic_{i}$ is the (unknown)
eigenvalue. If $c_{i}>0$, the flow is temporally unstable, and if $c_{i}<0$,
the flow is asymptotically stable in that perturbations decay away
exponentially for sufficiently long times. On account of Squire’s theorem,
which remains valid for both Newtonian [50] and Oldroyd-B [34] fluids, it is
sufficient to restrict attention to two-dimensional perturbations ($l=0$).
Note, however, that the theorem is applicable only within the normal-mode
ansatz, and it is possible to have a larger nonmodal growth of three-
dimensional perturbations, as will indeed be seen in Sec. 6.

Figure 1 shows the Newtonian spectrum at $Re=10^{4}$ and $k=1$, which has a
characteristic ‘Y-shape’ structure in the $c_{r}$-$c_{i}$ plane. It consists
of (i) the A branch, corresponding to ‘wall modes’ with phase speeds
approaching zero with decreasing decay rates, (ii) the P branch corresponding
to ‘center modes’ with phase speeds approaching the base-state maximum, again
with decreasing decay rates, and (iii) the vertical S branch corresponding to
modes having a common phase speed intermediate between the wall velocity and
the base-state maximum, and with decay rates extending to infinity. The
aforementioned Y-shaped structure only emerges above a threshold $Re$. Below
this threshold, whose value is dependent on the particular base-state profile,
the Newtonian spectrum comprises only of the S-modes. For sufficiently large
$Re$, the Y-locus itself becomes invariant, with the density of modes along
each of the three (invariant) branches increasing with increasing $Re$. As
evident from Figure 1, the first mode belonging to the A branch is unstable
for the chosen parameters, and corresponds to the well known Tollmien-
Schlichting (TS) instability. The Newtonian pipe flow spectrum is
qualitatively similar to that of plane Poiseuille flow, with the
characteristic A, P, and S branches, albeit being stable at all $Re$ [61]. In
addition, Newtonian plane Couette flow has also been found to be linearly
stable. However, unlike plane and pipe Poiseuille flow, on account of an exact
antisymmetry about the centerline, plane Couette flow does not possess a P
branch; instead, both arms of the Y correspond to A branches with the
corresponding eigenfunctions exhibiting a mirror symmetry about the
centerline.

###  2.2 The purely elastic spectrum and the elastic centermode instability
($Re=0,Wi\neq 0$)

In the absence of inertia, the governing equations in the Newtonian case
reduce to the Stokes equations. For Stokes flows driven by the motion of rigid
boundaries, the quasi-steady nature of the governing equations and boundary
conditions implies there can be no associated spectrum. With reference to the
preceding subsection, the $S$-modes in the finite-$Re$ Newtonian spectrum
recede down to negative infinity in the Stokes limit. In contrast, the stress
relaxation term in the Oldroyd-B equation provides for an intrinsic time
scale, and as discussed below, gives rise to a non-trivial spectrum even in
the absence of inertia. We discuss below the nature of this inertialess
spectrum whose structure is a function of $Wi$ and $\beta$. Unstable modes in
this spectrum correspond to purely elastic instabilities.

The simplest flow is, of course, plane Couette flow. The elastic plane Couette
eigenspectrum was first examined in the UCM limit by Gorodtsov & Leonov [62],
who showed, analytically, that there is a continuous spectrum (abbreviated as
‘CS’ henceforth) along with two discrete modes, all of which are stable. We
refer to the two stable discrete modes as the zero-Reynolds number Gorodtsov-
Leonov (‘ZRGL’) modes. The elastic continuous spectrum is a generic presence,
and owes its origin to the spatially local evolution of the polymeric stress
(in accordance with the simple fluid paradigm, which stipulates a local
relation between stress and deformation in a fluid [41]); the CS
eigenfunctions decay exponentially on the scale of the polymer relaxation
time. The above picture was generalised to the Oldroyd-B fluid by Wilson,
Renardy & Renardy in 1999 [63]. While the flow continues to remain stable, the
spectrum becomes considerably more complicated, with new discrete modes
appearing for any non-zero $\beta$, thereby pointing to the singular nature of
the UCM elastic spectrum. The continuous spectrum associated with the UCM
fluid is qualitatively unchanged, as are the two ZRGL modes. But, there is an
additional stable continuous spectrum which moves in from $c_{i}=-\infty$ as
$\beta$ increases from zero. Further, unlike the UCM-continuous spectrum, this
new so-called viscous continuous spectrum is associated with a branch cut, and
discrete eigenvalues can emerge from, or disappear into, the viscous
continuous spectrum with varying $\beta$. The number of discrete modes
increases with decreasing $\beta$, with there existing an infinite sequence of
discrete modes in the limit $\beta\rightarrow 0$; for moderate $\beta$, all of
these discrete modes are all more stable than the viscous continuous spectrum
modes.

In the aforementioned effort, the authors also analyzed the spectrum of plane
Poiseuille flow. In the UCM limit, the authors showed that the equivalent of
the Gorodtsov–Leonov spectrum has six discrete modes (instead of the two for
plane Couette flow above); numerical computations showed that the discrete
modes continued to remain stable. The addition of a solvent viscosity, leading
to the Oldroyd-B model, again resulted in a spectrum similar to plane Couette
flow; thus, a second viscous continuous spectrum arose for any non-zero
$\beta$, along with a large family of stable discrete modes that disappear
into this spectrum with increasing $\beta$.

Figure 2:  The inertialess plane-Poiseuille spectrum of an Oldroyd-B fluid
shows five discrete modes for $\beta=0.997$, $k=0.75$, $Wi=2500$, with three
in the main figure and the remaining two visible in inset A; $N$ is the number
of Chebyshev polynomials in the spectral expansion. Inset A shows an enlarged
view of the region near the unstable center mode (‘unstable CM’). Inset B
shows the variation of $c_{r}$ and $c_{i}$ with $Wi$. Figure reproduced with
permission from Khalid et al. [64].

The preceding two paragraphs had, until very recently, represented our
understanding of the elastic stability characteristics of rectilinear shearing
flows. Thus, although never proven, such shearing flow configurations have
nonetheless been thought to be linearly stable (this is the scenario even for
Newtonian pipe flow, although in this case observations clearly point to
nonlinear mechanisms). As a consequence, purely elastic linear instabilities
are synonymous with curvilinear flow configurations [6], with the analog of
such instabilities in rectilinear flows thought to have a nonlinear character
(see section 7.1); in either case, streamline curvature is regarded as a
necessary prerequisite for instability [65]. However, recent work by Khalid,
Shankar, and Subramanian [66] has demonstrated that inertialess plane
Poiseuille flow of an Oldroyd-B fluid is, in fact, linearly unstable at
sufficiently high $Wi$ (of $O(1000)$), and for $\beta>0.99$. Figure 2 shows
the structure of the elastic spectrum at such high $Wi$’s, and Fig. 3 shows
neutral curves, which are in the form of the unstable tongues in the $Wi-k$
plane; the instability appears to arise due to a critical-layer
mechanism333The term ‘critical-layer’ refers to the location where the base-
flow velocity equals the phase speed of the eigenmode. The instability arises
due to stretched polymers being rotated away from flow-alignment by the
perturbation shear, as they are swept past by the base-state parabolic flow.
The differential rate of convection becomes small near the critical layer,
owing to the phase speed of the eigenmode approaching the base-flow velocity.
As a result, the time available for the perturbation-shear-induced rotation
(of the stretched polymers) increases, and the resulting accumulation of
perturbation elastic shear stress drives a reinforcing flow, leading to
exponential growth. Close to neutrality, this mechanism leads to stress
eigenfunctions that exhibit singular features in the neighborhood of the
critical layer. [66], in contrast to the hoop-stress-based mechanism that is
operative in curvilinear shearing flows. The work of Buza, Page, and Kerswell
[67], using the FENE-P model, has confirmed that the aforementioned
instability continues to exist with the incorporation of finite extensibility.
The said authors have also carried out a weakly nonlinear stability analysis
to show that the instability in the creeping-flow limit is subcritical,
pointing to a potentially larger unstable region in the $Wi-\beta$ plane. At
present, it is not yet clear whether this instability is directly relevant to
recent experimental observations from the Paulo Arratia [12, 13] and Victor
Steinberg groups [14] which clearly indicate an ET-like state for $Re\ll 1$
even in rectilinear shearing flows, albeit for smaller $\beta\sim$
$0.5$–$0.7$.

Figure 3:  Neutral curves in the $Wi$–$k$ plane for different $\beta$’s in the
creeping-flow limit; inset A shows the collapse for $\beta\rightarrow 1$ when
plotted as $Wi\,k(1-\beta)$ vs $k$, and the results obtained from the reduced
equations in that limit. Inset B shows the variation of the critical
Weissenberg number $Wi_{c}$ and wavenumber $k_{c}$ with $(1-\beta)$. Figure
reproduced with permission from Khalid et al. [64].

###  2.3 The elasto-inertial spectrum ($Re,Wi\neq 0)$: the wall- and center-
mode instabilities instabilities at finite $Re$

The work of Gorodtsov and Leonov[62], referred to in the section above in the
context of the inertialess elastic spectrum, also analyzed plane Couette flow
of a UCM fluid for small but finite $Re$. In addition to the aforementioned
pair of stable ZRGL modes, the authors found a new class of modes,
corresponding to damped shear waves in a viscoelastic fluid with phase speeds
of $O(\sqrt{G/\rho})$, $G=\eta/\lambda$ being the shear modulus. We refer to
this family of modes as the high-frequency-Gorodtsov-Leonov (‘HFGL’) modes
since, in units of the base-state velocity scale, the phase speed (frequency)
of the HFGL modes is $O(WiRe)^{-\frac{1}{2}}$, and therefore these modes
recede to infinity (parallel to the $c_{r}$-axis) in the inertialess limit.
Although Gorodtsov and Leonov [62] predicted an instability due to the HFGL
modes in the limit $kWi\gg 1$, this was later shown to be incorrect [68]; the
HFGL modes remain damped for any finite $Wi$, with $c_{i}\rightarrow-1/2kWi$
for $Re\rightarrow 0$. Note that the original elastic continuous spectrum
continues to be present at finite $Re$, with the CS-modes having phase speeds
in the base-state range of velocities, with decay rates of $c_{i}=-1/kWi$
(this corresponds to the dimensional decay rate equaling the inverse
relaxation time, as mentioned in section 2.2). Thus, the elastoinertial
spectrum of plane Couette flow of a UCM fluid has been shown [69, 68] to
consist of the finite-$Re$ continuation of the ZRGL modes, the elastic
continuous spectrum, and the HFGL modes. Although a rigorous proof does not
exist, plane Couette flow does appear to be stable in the $Re$–$Wi$ plane for
$\beta=0$; the conclusion remains unchanged on consideration of an Oldroyd-B
fluid. [70, 68, 71, 72]. The stability of viscoelastic plane Couette flow
therefore mirrors that of its Newtonian counterpart [50], although there
exists a rigorous proof in the latter case [73]. In summary, there appears no
evidence of a linear instability of plane Couette flow in the
$Re$–$Wi$–$\beta$ space.

(a) Plane Couette flow

(b) Plane Poiseuille flow

Figure 4: Eigenspectra for (a) plane Couette and (b) plane Poiseuille flows of
a UCM fluid ($\beta=0$) for $E=0.001,k=1.0,Re=10^{4}$ with $N=$ $500$ and
$600$. The distinctive features are labelled within the spectra. Reproduced
with permission from Ref [72]. Figure 5: Eigenspectrum for plane Poiseuille
flow of a UCM fluid showing three unstable modes for $E=0.005,k=1.3,Re=8000$.
Only the converged eigenvalues are shown, and thus the CS balloons are absent
in this filtered spectrum. Reproduced with permission from Ref. [72].

In contrast, plane Poiseuille flow of a Newtonian fluid ($Wi=0$) becomes
susceptible to the TS instability [50] at $Re_{c}\approx 5772$. As already
shown in section 2.1, the unstable TS eigenvalue belongs to the A-branch, and
is therefore a wall mode. A continuation of this instability is expected for
small $Wi$ regardless of $\beta$, including for the case of a UCM fluid. The
key question is whether there are new unstable modes in plane Poiseuille flow
of a UCM fluid that have an essentially elastic origin, and are therefore
absent in the Newtonian limit. This question was first addressed by Porteus
and Denn [74], who found three unstable modes for sufficiently high
$Re\,(>2000)$, only one of which was a continuation of the TS mode; the other
two unstable modes are absent in the Newtonian limit. The authors showed that,
for the TS mode, increasing elasticity in the range $0<E<10^{-2}$ resulted in
a decrease in $Re_{c}$ from its Newtonian value to $Re_{c}\sim 2000$.
Elasticity was also shown to have a destabilizing effect on one of the other
two unstable modes, albeit with limited data. On the other hand, plane
Poiseuille flow of a UCM fluid was found to be stable at low $Re$ [75, 70].
Sureshkumar and Beris [76] found two different unstable families, of which one
was a continuation of the Newtonian TS mode. The critical Reynolds number
$Re_{c}$ showed a nonmonotonic behavior, showing an initial decrease for very
small $E$’s and an eventual decrease at higher $E$. Both the modes analyzed in
Ref.[76], and the initial decrease found in $Re_{c}$ with $E$, are consistent
with the earlier results of Porteus and Denn [74].

Figure 6: Neutral stability curves in the $Re$–$k$ plane corresponding to the
different wall modes for $E=0.005$. The modes labeled A1(b), A2, A3, and A4
are antisymmetric, while S1, S2 and S3 are symmetric. Reproduced with
permission from Ref. [72].

The recent study of Chaudhary et al. [72] presented a more comprehensive
picture of the elasto-inertial spectrum of a UCM fluid, emphasizing features
common to both plane Couette and Poiseuille flows. As shown in Fig. 4, for
$Re\sim 1000$ and higher, the elastoinertial spectrum for both flows contains:
(i) a ballooned manifestation of the horizontal line ($c_{i}=-1/k\,Wi$;
$c_{r}\,\epsilon\,[-1,1]$ for plane Couette, and $c_{r}\,\epsilon\,[0,1]$ for
plane Poiseuille) corresponding to the elastic continuous spectrum, (ii) a
horizontal string of eigenvalues corresponding to the aforementioned HFGL
modes, and (iii) a roughly ‘hourglass’ shaped structure that extends above and
below the HFGL line; note that the length of the HFGL sequence obtained is a
function of the numerical resolution of the spectral method, and is smaller
for plane Poiseuille flow due to the lower $N$. Despite both spectra
conforming to a common template, all modes remain stable for plane Couette
flow, as mentioned above, while some of the eigenvalues belonging to the
small-$c_{r}$ ‘arm’ of the hourglass become unstable at sufficiently high $Re$
and $E$, for plane Poiseuille flow; see Fig. 5.

Chaudhary et al. [72] further showed that plane Poiseuille flow of a UCM fluid
is susceptible to an apparently infinite hierarchy of elasto-inertial wall
mode instabilities. In contrast to the antisymmetric Newtonian TS mode, these
unstable elastoinertial modes can have either symmetry (symmetry, here, is
based on the variation of the streamwise velocity eigenfunction, about the
centerline, in the wall-normal direction). The multiple unstable tongues in
the $Re-k$ plane, for both the antisymmetric and symmetric wall-mode
instabilities, are shown in Fig. 6 for $E=3.5\times 10^{-3}$. The lowest
critical Reynolds number was found to be $Re_{c}\approx 1210.9$ for
$E=0.0066$; $Re_{c}$ was found to diverge in the limit $E\ll 1$, although the
scalings differed for the symmetric ($Re_{c}\propto E^{-1}$) and antisymmetric
($Re_{c}\propto E^{-\frac{5}{4}}$) modes. Both the unstable wall modes above,
that are part of the hour-glass structure, and the HFGL modes, are found to be
strongly stabilized on introduction of a solvent viscosity component (non-zero
$\beta$) [77, 64]. Thus, although relevant to the UCM limit, the wall-mode
instabilities are not relevant to the dilute solutions on which most
experiments have been performed.

Figure 7: Eigenspectrum for pipe flow of an Oldroyd-B fluid for $E=0.1$,
$\beta=0.8$, $Re=600$, and $k=3$, and $N=200$ (red filled circles). The inset
shows the unstable center mode. The modes labelled $2^{\prime}$, $3^{\prime}$
and $4^{\prime}$ are discrete center modes that emerge out of the CS as $E$ is
increased. The vertical locations of the CS lines and the pipe-flow Newtonian
spectrum (open blue circles) at the same $Re$ and $k$ are shown for reference.
Reproduced with permission from Ref. [78].

(a) Pipe-Poiseuille flow

(b) Plane-Poiseuille flow

Figure 8: Variation of the critical Reynolds number $Re_{c}$ with $E(1-\beta)$
for different $\beta$ for both pipe and plane-Poiseuille flows. Figures from
Ref. [78] and Ref. [64] reproduced with permission.

In contrast to the many studies that have focused on the stability of plane
Poiseuille flow of an Oldroyd-B fluid, rather surprisingly, there had not been
a single study, until recently (see [32, 78]), analyzing the stability of pipe
flow of an Oldroyd-B fluid. The only stability analysis in the literature by
Hansen [79, 17] had neglected the crucial convected nonlinearities in the
Oldroyd-B model. The lack of emphasis on pipe flow could perhaps be attributed
to the linear stability of Newtonian pipe flow for all $Re$ [61], in turn
leading to the assumption of viscoelastic pipe flow also being linearly stable
in $Re$–$Wi$–$\beta$ space; an assumption that has often found an explicit
mention in the literature [80, 81, 11, 82]. This is despite the absence of a
systematic exploration of the larger (three-dimensional) parameter space, and
inspite of the recent pipe flow experiments of Samanta et al. [22] showing the
existence of an perturbation-amplitude-independent threshold $Re$ for
transition from the laminar state in sufficiently elastic polyacrylamide
solutions, the amplitude independence being a clear signature of an underlying
linear instability.

It is worth pointing out here that two protocols were adopted by Samanta et
al.[22] in the experiments above: in the first protocol, the flow was forced
by radial fluid injection near the inlet, resulting in the oft-quoted
threshold $Re\approx 2000$ for the Newtonian case. The second protocol did not
involve any external forcing, corresponding therefore to a ‘natural’
transition, and occurred at $Re\approx 8000$ in the Newtonian limit. With
increase in the polyacrylamide concentration, the threshold $Re$ for the
natural transition decreased, while that for the forced transition is
increased, and for concentrations greater than $300$ppm, the threshold $Re$
became independent of the protocol. For the $500$ppm solution in particular,
the threshold $Re$ was found to be as low as $800$, and the transition was
bereft of signatures such as turbulent puffs that accompany the onset of
Newtonian turbulence. As mentioned in the Introduction, the flow state that
resulted after the non-hysteretic transition was referred to as elasto-
inertial turbulence, to distinguish it from both purely-elastic turbulence and
inertial Newtonian turbulence; this reduction in the transition $Re$ has been
corroborated by later experiments in pipes of much smaller radii [83, 84]. The
subsequent experimental study of Choueiri et al. [24] showed that, at a fixed
$Re<3600$, as the polymer concentration is increased, the frictional drag
decreased and approached the maximum-drag-reduction asymptote, in accordance
with the well-established paradigm of turbulent drag reduction. However, in a
significant departure from this scenario, further increase in polymer
concentration resulted in the drag reduction exceeding the MDR asymptote, with
the flow relaminarizing completely for a range of polymer concentrations. This
laminar state becomes unstable when polymer concentration is increased
further, eventually again approaching the MDR asymptote. As alluded to in Ref.
[32], the MDR regime could thus be viewed as a ‘drag-enhanced’ state directly
accessible via an instability of the laminar state, rather than as a drag-
reduced state accessible from Newtonian turbulence.

Motivated by the above experiments, and in a significant departure from the
Newtonian scenario (see end of section 2.1), the recent studies of Garg et al.
[32] and Chaudhary et al. [78] have shown that pipe Poiseuille flow of an
Oldroyd-B fluid is indeed linearly unstable to an axisymmetric center-mode,
consistent with the aforementioned experimental observations. An analogous 2D
center-mode instability is predicted for plane Poiseuille flow [32, 64].
Figure 7 highlights the existence of an unstable center-mode in the pipe
elastoinertial spectrum for $Re=600,E=0.1$ and $\beta=0.8$. The relevance of
exponentially growing axisymmetric/2D disturbances is consistent with the
signatures seen in recent444It is worth pointing out that these recent DNS
studies differ from the earlier ones on drag reduction [for instance, Refs.
85, 86] in not using an artificially high stress diffusivity that might have
led to the absence of spanwise EIT structures in those earlier efforts. DNS
studies [82, 87] of viscoelastic pipe and channel flows. In both cases, the
characteristic structures in the elastoinertial turbulent regime are found to
be spanwise oriented rolls, in contrast to the streamwise oriented spanwise
varying streaks and counter-rotating vortices which are known to underlie the
sub-critical Newtonian transition [88].

Unlike the wall-mode instabilities described above, the center-mode
instability is not restricted to small $\beta$ for either pipe or plane
Poiseuille flow. In fact, for both flows, the instability appears to require a
combination of fluid inertia, elasticity and solvent viscous effects. This may
be seen in the limit $Re\gg 1$ when the threshold Reynolds number
$Re_{c}\propto E^{-3/2}$ for both these flows, a scaling that can only be
obtained by balancing fluid inertia, elasticity and solvent viscous effects in
a thin layer near the pipe centerline/channel midplane [78]; see Figs 8(a) and
8(b). Thus, in sharp contrast to the known irrelevance of linear (modal)
stability theory vis-a-vis the Newtonian transition in the canonical shearing
flows, a common modal mechanism is predicted to underlie the transition to
EIT, in plane- and pipe-Poiseuille flows of an Oldroyd-B fluid, over a
significant domain of the $Re$-$Wi$-$\beta$ space, with supporting evidence
from both simulations [82, 87] and experiments [23].

While the center-mode instabilities for pipe- and plane-Poiseuille flows share
many similarities, there are crucial differences. The instability ceases to
exist for $\beta<0.5$ in channel flow [64], while continuing down to
$\beta\sim 10^{-3}$ for pipe flow [32]. Recent work by Wan et al. [89] has
found that the center-mode instability is present even in the UCM limit
($\beta=0$) for some isolated regions in the $Re$-$Wi$ space. The more
interesting limit is that corresponding to dilute solutions with $\beta\to 1$.
As shown in Fig. 8(a), for pipe flow, in the aforesaid limit, $Re_{c}\approx
63$ for $E\rightarrow\infty$, this being the minimum Reynolds number for
instability onset. In contrast, as shown in Fig. 8(b), $Re_{c}$ for plane
Poiseuille flow behaves in a similar manner only for $\beta<\beta_{c}\approx
0.990552$; for $\beta>\beta_{c}$, the center-mode instability continues to
down arbitrarily low $Re$, with $Re_{c}\propto E^{-1}$ (corresponding to a
threshold $Wi$) for $E\rightarrow\infty$; in the process, the elastoinertial
center-mode morphs into the purely elastic center-mode instability described
in section 2.2 [66]. While the predictions for $Re_{c}$ in Figs. 8(a) and 8(b)
correspond to an Oldroyd-B fluid, the use of a nonlinear constitutive
equation, such as the FENE-P model, is expected to lead to curves that remain
similar in the vicinity of the minimum Reynolds number, but that close up
beyond a second larger critical $Re$, owing to shear-thinning-induced
stabilization; this feature will again be seen, in the context of the
curvilinear instabilities, in Sec. 4.9.

Figure 9: Comparison of experimental flow structures with the eigenfunction
for the center mode. Top and Middle show streamwise velocity fluctuations
obtained from PIV measurements in a longitudinal cross-section. Top shows flow
structures at $Re=5$ and corresponds to an experiment with
$\beta=0.57,E=20.8,Wi=104$, whereas Middle corresponds to $Re=100$ with
$\beta=0.5,E=3.4,Wi=304$. Lower shows the most unstable mode in the linear
stability analysis; the solution plotted is intended for qualitative
comparison only and was computed for different flow parameters:
$Re=100,E=0.6,Wi=60,\beta=0.9,n=0$, and $k=1$; here, $n$ and $k$ are the
azimuthal and axial wave numbers, respectively. Flow direction is from right
to left. Reproduced with permission from Fig. 1b of Choueiri et al. [23]

Finally, getting down to actual numbers, linear stability theory [32, 78]
predicts a threshold $Re\approx 800$ for transition, similar to the
observations of Samanta et al. [22], albeit at much higher $E$’s. The
theoretical predictions are in better agreement with the pipe flow experiments
of Chandra et al. [83], an aspect that might have to do with the differing
methods used to determine the polymer relaxation times in the two efforts. The
recent pipe-flow experiments of Choueiri et al. [23], however, show excellent
agreement between their observations, and theoretical predictions [32, 78] for
the threshold $Re$, for $E\leq 0.1$. Further, the experiments demonstrate a
remarkable match (see Fig. 9) between the structures seen immediately after
transition and the linear center-mode eigenfunction, while also pointing to a
secondary transition to a wall mode. For $E>0.1$, the experiments reveal a
monotonic decrease of the threshold $Re$ with $E$, while the theoretical
predictions [32, 78] predict a sharp upturn in $Re_{c}$ (see Fig. 8(a)). The
experimental threshold appears to indicate a transition from an elastoinertial
instability to an elastic one, similar to plane Poiseuille flow, although the
elastic branch (corresponding to the higher values of $E$) must then
correspond to a subcritical (nonlinear) transition. For viscoelastic plane
Poiseuille flow, the predictions [64] are in good agreement with the limited
experimental data of Srinivas and Kumaran [90] for channels with a cross-
sectional aspect ratio of $10$:$1$.

Figure 10: Schematic representation of various transition scenarios in
viscoelastic pipe flow in the $Wi$–$Re$ plane.

###  2.4 Transition scenarios in rectilinear viscoelastic shearing flows

Figures 10 and 11 illustrate the various possible transition scenarios in the
$Wi$-$Re$ plane, for different fixed $\beta$, for pipe and plane Poiseuille
flows. In these schematic illustrations, we bring together ideas based on the
section above, on the centermode instability, and other hypotheses based on
earlier nonmodal and nonlinear analyses (see sections 6 and 7 respectively for
a detailed discussion); we also comment briefly on a recent independent line
of work by Graham and coworkers that proposes a new subcritical route to EIT
based on elastoinertial TS-wave analogs [57, 58, 91]. In the aforementioned
figures, the linearly unstable regions in the interior of the $Wi$-$Re$ plane
correspond to the domain of the elastoinertial centermode instability, and are
depicted using colored lines for different $\beta$. Regions adjacent to the
$Re$ and $Wi$ axes correspond to the onset of predominantly inertial and
elastic instabilities, respectively, with the former underlying the sub-
critical Newtonian transition. Recall from section 2.1 that Newtonian pipe
flow is believed to be linearly stable at all $Re$ in sharp contrast to the
observed transition at $Re\approx 2000$. Likewise, the presence of the
classical TS instability in Newtonian channel flow, at $Re\approx 5772$ (see
section 2.1), is now known to be irrelevant to the observed transition at
$Re\approx 1000$. Thus, the inertial Newtonian transition for either flow
configuration has a nonlinear subcritical character. Indeed, transition in
these flows is now understood to be a complex process triggered by the
emergence, via saddle-node bifurcations, of novel three-dimensional solutions
called ‘exact coherent states’ (ECS) [92, 93, 94], with a sufficiently large
number of such unstable solutions forming the scaffold of the turbulent
attractor in an appropriate phase space.

We begin with a brief discussion of the features common to the center-mode
instability in both pipe and channel flows, before going on to describe those
unique to channel flow in Figs. 11(a) and 11(b). It has been shown that
elasticity suppresses the 3D ECS solutions [30, 31, 95, 96, 97], making the
nonlinear Newtonian-ECS-based mechanism irrelevant for weakly elastic flows.
Although this suppression has been demonstrated specifically for plane
Poiseuille flow, the prediction should be valid for pipe flow as well, on
account of the similarity of the Newtonian ECSs across the different
rectilinear shearing flows [98, 92, 93]. The elasticity-induced suppression of
the ECS has been proposed to underlie delayed transition and eventual
disappearance of the Newtonian turbulent state in the flow of polymer
solutions. As a result, in the said figures, the Newtonian turbulent-like
state is confined to a region between the $Re$-axis and a curve that
corresponds to a critical $Re$-dependent $Wi$.

For smaller $\beta$, as shown in Fig 10, the aforementioned Newtonian
turbulent state likely gives way to a laminar one with increasing elasticity.
Indeed, the vertical arrow shown on the extreme right in Fig. 10 corresponds
to the experimental path of Choueiri et al. [24] who, starting from Newtonian
turbulence, first accessed an intermediate laminar state, and then the MDR
regime, with increasing $Wi$, as discussed above in Sec. 2.3. On the other
hand, for very dilute solutions, as shown in Fig 11(b), the intervening
laminar state gives way to overlapping Newtonian and elastoinertial turbulent
regions at higher $Re$. The vertical arrow shown in the figure, again on the
extreme right, now corresponds to a ‘reverse’ transition where the Newtonian
turbulent state exhibits an increasing degree of spatiotemporal intermittency
with increasing $Wi$, before giving way to EIT; this was observed to be the
pathway, at higher $Re$, in Ref. [24].

For sufficiently high elasticities, the linear center-mode instability,
discussed in section 2.3 above, becomes relevant. Although the extent of the
linearly unstable region depends sensitively on flow-type and $\beta$, the
unstable regions for both pipe and channel flows exhibit qualitative
similarities for $0.5<\beta<0.98$, with $Wi\propto Re^{1/3}$ along the lower
branch of the unstable region (this scaling corresponds to the $Re\propto
E^{-3/2}$ regime in Figs. 8(a) and 8(b)), and $Wi\propto Re$ along the upper
one (this corresponds to the near-vertical divergence of $Re_{c}$ in Figs 8(a)
and 8(b)). Note that $Wi\propto Re$, in corresponding to a constant $E$, also
represents an experimental path of increasing flow rate for a given flow
geometry and polymer solution. Thus, as shown in Figs 10, 11(a) and 11(b), for
both the plane and pipe Poiseuille geometries, the centermode eigenfunction is
likely to lead to supercritical nonlinear structures that, either directly, or
through secondary instabilities, might underlie the dynamics of the EIT state.
The centermode instability, for both pipe and channel flows, therefore
provides a continuous pathway from the laminar state to the EIT/MDR regime, a
prediction that now has been confirmed in experiments [23].

Beyond the aforementioned range of $\beta$, as mentioned above in section 2.3,
there exist significant differences between the pipe and plane Poiseuille
cases. Specifically, in the limit $\beta\rightarrow 1$, while the center-mode
instability appears to be restricted to $Re>63$ for pipe flow (Figs. 8(a) and
10), it morphs into a purely elastic instability for channel flow, continuing
to arbitrarily small $Re$ for $\beta>\beta_{c}\approx 0.990552$ (Fig. 8(b)).
Correspondingly, in Fig 11(b), the lower boundary of the linearly unstable
envelope (with $Wi\propto Re^{\frac{1}{3}}$) opens out into a plateau with
decreasing $Re$, approaching a threshold $Wi$ for $Re\rightarrow 0$. This
purely elastic instability might in turn lead to an ET state, and the implied
continuous (modal) pathway between the EIT and ET states is shown
schematically in Fig 11(b). Note that the blue curve in Figs 11(a) and 11(b)
corresponds to the neutral boundary for $\beta=\beta_{c}$ that demarcates,
within a linearized framework, the pure-EIT regime, and the one that exhibits
the EIT-ET connection.

(a) $\beta\leq\beta_{c}$

(b) $\beta>\beta_{c}$

Figure 11: Schematic representation of various transition scenarios in
viscoelastic channel flow in the $Wi$–$Re$ plane.

In regions of the $Re$-$Wi$-$\beta$ space where the centermode is linearly
stable, and the originally Newtonian ECS are stabilized by elasticity, novel
subcritical mechanisms are expected to dominate the transition process. For
plane Poiseuille flow, at moderate $Re$, recent work [57, 58, 91] has
identified a nonlinear mechanism based on elastoinertial wall modes closely
related to the stable Newtonian TS mode (although still disconnected from it
in phase space until a $Re$ of $10^{4}$). Such a pathway could be especially
relevant in a direct transition between the Newtonian and elastoinertial
turbulent states (as in Fig 11(b)), with the near-wall coherent structures in
the former state acting as possible seeds for the aforementioned TS-wave
analogs 555Given that recent experimental evidence points to EIT and MDR
states being one and the same, for low to moderate $Re$ values, it is worth
mentioning here that the 2D TS-wave-analogs recently proposed to underlie EIT
[57, 58, 91] stand in sharp contrast to an earlier interpretation that
regarded the MDR regime as corresponding to a hibernating state of turbulence
[86, 99] comprising 3D so-called edge-state solutions (lying on the basin
boundary between the laminar fixed point and the turbulent attractor in an
appropriate phase space). Such states already exist in Newtonian turbulence,
and their frequency of occurrence is thought to be progressively enhanced with
increasing polymer concentration (although, the hibernating periods have been
found to be strongly box-size dependent [29]). The relation between this
earlier edge-state-based hypothesis, and the more recent TS-analog-based
hypothesis, needs further investigation.. However, the fact that there is no
analog (linear or nonlinear) of the TS-mode in the Newtonian pipe-flow
spectrum, and that the centermode remains the least stable one even in the
weakly elastic regime [78], suggests that the TS-analog-based subcritical
mechanism may not be obviously applicable to pipe Poiseuille flow; more work
is clearly required in this regard.

The recent subcritical continuation of the unstable center mode, in
viscoelastic channel flow, to a nonlinear EIT structure [56] implies that
subcritical mechanisms based on the centermode might also be operative in
certain regions of $Re$-$Wi$-$\beta$ space, and thus the relevance of the
centermode might extend outside of the linearly unstable regions indicated;
see the dashed line in Fig 11(b). The very recent weakly nonlinear analyses of
Buza et al. [67] for channel flow and Wan et al. [89] for pipe flow further
confirm that the center-mode instability is likely subcritical in large parts
of the parameter space. Despite these developments, it is relevant to point
out that there still remain vast tracts of the viscoelastic parameter space
where the mechanism of transition is not understood. As an example, Khalid et
al. [64] have shown that for $\beta=0.97$, and for $0.02<E<0.5$, neither the
center mode nor the wall mode is the least stable. Instead, it is the singular
modes belonging to the continuous spectrum that are the least stable for these
$E$ values, and that might therefore dictate the nature of the
transition.While some work has been done on the role of continuous spectra in
pattern formulation, in Hamiltonian systems [100], more work is therefore
required to clarify continuous-spectra-dominated transition mechanisms in
viscoelastic shearing flows.

The above discussion of transition scenarios has been restricted to either new
elastoinertial modal pathways, or the elastic modification of essentially
Newtonian nonmodal nonlinear pathways (implicit in the examination of the
effects of finite $Wi$ on the Newtonian ECS’s, mentioned earlier). In the
opposite limit of $Re\ll 1$, pipe and plane Poiseuille flows, as indeed all
rectilinear shearing flows, are linearly stable for $Wi\sim O(1)$ [63, 78],
since a linear instability at such $Wi$’s requires a hoop-stress-based
mechanism (see section 4). The absence of a linear instability at moderate
$Wi$’s has led to the exploration of novel nonmodal pathways due to elasticity
alone [101, 102], or due to a non-trivial interplay of elasticity and inertia
[103, 104]. While such efforts are discussed in detail in Sec. 6 below, it is
worth summarizing a few salient points that appear in Figs 10, 11(a) and
11(b). The nonmodal pathways, in the inertialess limit in particular, point to
the importance of spanwise varying disturbances (much like the Newtonian case)
that are amplified by an elastic analog of the lift-up effect [105, 106], and
by an amount that increases with increasing $Wi$. Page and Zaki [107] have
examined an elastoinertial nonmodal pathwayfor streamwise varying (2D)
disturbances, termed the reverse-Orr mechanism, on account of the dominant
algebraic growth occurring during the phase where the disturbance aligns with
the ambient shear flow (in contrast to the Orr-mechanism-driven growth
dynamics in the Newtonian case [106]).

An alternate transition scenario, again relevant to the elasticity-dominant
limit, is that of a subcritical 2D nonlinear instability [108, 109] based on
the classical Stuart-Landau amplitude expansion, an approach originally
developed to describe the Newtonian transition [52, 53]. This approach,
described in more detail in Sec. 7 below, has been demonstrated only for
$Wi\sim O(1)$ and $\beta\rightarrow 0$. Both the elastic nonmodal and
nonlinear (modal) pathways above are shown in Figs 10, 11(a) and 11(b), in the
vicinity of the $Wi$-axis, and are believed to trigger transition to an ET
state (‘ET1’ in the figures). The existence of an additional linear
instability for $Wi\sim O(1000)$ and $\beta\rightarrow 1$[66], discussed in
section 2.2, implies a possible bifurcation to a distinct elastic turbulent
state. It is therefore possible to envisage (at least) two different ET states
(ET1 and ET2 in Fig. 11(b)), in inertialess plane Poiseuille flow, depending
on $Wi$. There, however, remains a wide intermediate range of $\beta$
($0<\beta<\beta_{c}$) for which the nature of the subcritical transition is
not fully understood.

##  3 Interfacial instabilities in multilayer and shear-banded flows

Next, we focus on interfacial instabilities which are a major concern in
applications (coating, coextrusion and others) that involve multi-layer
configurations, i.e. the flow of immiscible liquids as distinct layers in
contact with one another. The objective in the said applications is to obtain
composite materials with properties that are a desired combination of those of
the individual layers. In order for these layered composites to have the
desired properties, uniformity of the individual layers is crucial, in turn
implying that instabilities must be avoided during processing; the formation
of interfacial waves, for instance, can result in a significant deterioration
of the product. Interfacial waves in such configurations are often driven by a
stratification (i.e., a sufficiently rapid variation across the layers) of
either material (static or dynamic) or flow properties. For viscoelastic
liquids, there arises the specific scenario of a stratification in the elastic
characteristics.

Interfacial instabilities are well known to occur even in Newtonian fluids,
where stratification can be due to the (rapid) variation of density,
viscosity, and/or velocity across layers. A stratification in fluid density,
with the heavier fluid lying above the lighter one, leads to the well-known
Rayleigh-Taylor instability; for brevity, we will not discuss the role of
density differences here. A difference in the velocities of two co-flowing
fluid streams leads to the classic Kelvin-Helmholtz (‘shear layer’)
instability, which has an essentially inviscid origin. Azaiez and Homsy [110]
used the Oldroyd-B model (in addition to Giesekus and co-rotational Jeffereys
models) to show that elasticity has a stabilizing effect on the shear layer
instability, with an increasing $E$ reducing both the growth rates and the
unstable interval of wavenumbers. Even in the absence of a density and
velocity stratification, a jump in viscosity across an interface can lead to
an instability, as first demonstrated by Yih [111] for Newtonian fluids. Yih
analysed one of the simplest interfacial flows viz. wall-bounded two-layer
plane Couette flow, and found that viscosity stratification can cause a long
wave instability for any non-zero $Re$; see Ref. [112] for a discussion on the
underlying physical mechanism. The lack of a threshold $Re$ for the onset of
this instability should be contrasted with the Kelvin-Helmholtz instability
above. An analogue of the viscosity stratification instability is also seen,
for instance, in lubricated pipelining [113], where a viscous core fluid
(typically oil) is lubricated by a thin annulus of viscous fluid (water).

###  3.1 Predicting interfacial instabilities using the Oldroyd-B model

Moving beyond Newtonian fluids, there is the possibility of an elasticity
mismatch between fluids having identical shear viscosities and densities. The
first study of this scenario was carried out by Waters & Keeley [114] for a
two-layer plane Couette flow of Oldroyd-B fluids; however, the authors found
no instability due to an error in the interfacial boundary condition. Chen
[115] carried out a rather similar calculation for a core-annular coextrusion
flow of a pair of UCM fluids, corrected the error above, and discovered a new
instability. The predictions were experimentally verified by Bonhomme et al.
[116]. In the limit of long wavelengths, the instability arises due to a jump
in $N_{1}$ across the interface. However, the elastic stratification can be
stabilizing or destabilizing, depending on the ratio of the volumetric fluxes.
Hinch et al. [117] gave a simple physical explanation to show which fluid
arrangements would be stable or unstable to varicose (as studied by Chen) and
sinuous modes, and showed that for cases where both modes are unstable, the
sinuous modes were the more dangerous. Further analysis of interfacial
instabilities, of UCM fluids in Couette flow, was carried out by Renardy
[118]; this effort identified five modes in the short-wave limit, of which
only one is an interfacial mode. The interfacial mode was again shown to
become unstable in the said limit due a stratification in elasticity even when
the viscosities of the two layers are identical.

These results were extended, still using the Oldroyd-B model, to all
wavelengths in symmetric three-layer planar interfacial flows (essentially the
2D analogue of the axisymmetric coextrusion configuration) by Miller, Wilson &
Rallison [119, 120, 121], and to two-layer arrangements in plane Poiseuille
flow by Su & Khomami [122]. More recent works have extended the above efforts
in various directions: (i) to very high Weissenberg numbers [119, 121]; (ii)
to two-layer flows with moving boundaries (Couette-Poiseuille flow) [123];
(iii) by analyzing the effect of surfactants [124], and (iv) by exploring the
use of deformable solid boundaries as a way of suppression of interfacial
instabilities [125, 126].

###  3.2 Shear banding and instabilities in the banded state

As discussed in another paper in this special issue [38], Oldroyd introduced
the Oldroyd-A model [35] as well as the more famous Oldroyd-B, implying the
existence of everything between the two. In other words, there exist valid
constitutive equations with any of a one-parameter family of convected
derivatives (collectively dubbed the ‘Gordon-Schowalter’ (GS) derivative [41])
all of which are consistent with the principle of material frame indifference.
The aforementioned parameter appears as a slip parameter, $a$, in the GS
derivative. The Johnson-Segalman model replaces the upper-convected derivative
in the UCM model with the GS derivative [127]. With varying $a$, the convected
derivative in this model transitions from a lower convected derivative
(Oldroyd-A; $a=-1$) to an upper convected one (Oldroyd-B $a=1$) via the
corotational or Jaumann derivative $(a=0)$. The resulting response in
viscometric flows has a shear-thinning character for all $a$ values except
those corresponding to the Oldroyd-B and Oldroyd-A limits. Further, for some
values of $a$ (and $\beta$), the shear thinning is intense enough that the
shear stress exhibits a non-monotonic dependence on shear rate, as shown in
Fig. 12.

$\sigma$$\dot{\gamma}$$\sigma_{p}$$\dot{\gamma}_{1}$$\dot{\gamma}_{2}$$\dot{\gamma_{3}}$
Figure 12:  Non-monotonic flow curve, as exhibited by the Johnson–Segalman
model. If the applied stress is $\sigma=\sigma_{p}$, flow can coexist at two
shear-rates $\dot{\gamma}_{1}$ (low-viscosity band) and $\dot{\gamma}_{3}$
(high-viscosity band). The intermediate shear-rate $\dot{\gamma}_{2}$ is
thermodynamically unstable.

One of the most striking phenomena arising from the non-monotonicity of the
flow curve is shear banding, in which a simple shear flow of a complex fluid,
with a shear rate in the intermediate mechanically unstable portion,
spontaneously separates into high- and low-shear-rate bands [128, 129]. There
has been widespread interest in such banding flows since the phenomenon was
first reported in the early 1990sin the context of worm-like micellar
solutions [130, 131]. Surprisingly, shear-banded flows of worm-like micellar
solutions are themselves unstable [132, 133] and exhibit a variety of
instabilities ranging from purely elastic instabilities localised in the more
elastic band [134] to instabilities of the interface between the bands [135,
136, 137, 138]. Although the primary banding instability is beyond the scope
of the Oldroyd-B model, significant understanding of the secondary
instabilities of banded micellar systems can be gained from drawing analogies
with the interfacial instabilities discussed above, and the purely elastic
bulk instabilities discussed in detail in Sec. 4 below, both based on the
Oldroyd-B model.

Thus, in shear-banded flows, one can capture the subsequent instabilities in
the banded state by treating each band as a distinct Oldroyd-B fluid. For
instance, the interfacial instabilities seen in shear-banded Couette and plane
Poiseuille flow [139, 140, 36, 141] can be explained, at least in the long-
wave limit, by Chen’s mechanism discussed above [115], adapted to allow for
the fact that the two shear bands have mismatches of viscosity as well as of
$N_{1}$. It has also been shown that the Pakdel-McKinley criterion, discussed
below in Sec. 4.9, can be adapted to describe (bulk) instabilities in shear-
banded flows [141].

In a very recent paper, Castillo & Wilson [142] found an interfacial
instability while analyzing the stability of channel flow of a shear-banded
thixotropic-viscoelasto-plastic fluid, which exhibits shear banding; thus,
this instability shares some similarities with the instability of the shear
banded state discussed above. For the aforesaid configuration, $N_{1}$ varies
continuously across the interface, and there is only a jump in the viscosity;
thus, the instability may be regarded as the elastic version of the inertial
instability analyzed by Yih [111]. The authors were able to reproduce the main
points of the instability (seen in a highly shear-thinning fluid with many
constitutive complications) by using two Oldroyd-B fluids, with the the
interfacial value of $N_{1}$ matched, but having different shear viscosities.
The explanatory power of the Oldroyd-B model is again seen to extend well
beyond its expected realm of validity.

##  4 Instabilities in curvilinear shearing flows

The term ‘purely elastic’ instabilities has traditionally been used to refer
to the instabilities observed in flows of viscoelastic fluids, in geometries
with curved streamlines, including in particular viscometric configurations
such as the Taylor-Couette, cone-and-plate and parallel-plate geometries. This
class of instabilities is present even when inertial effects are not
significant. As already mentioned in the introduction, a precise prediction of
the domain of existence of these instabilities is of immense importance to
rheological characterization of polymeric liquids, since the inference of
rheological properties presupposes the existence of viscometric flows in the
aforementioned geometries; the occurrence of instabilities corrupts
rheological measurements, precluding characterization. Further, the
instabilities are of relevance to coating applications, and other polymer
processing scenarios where flow configurations akin to the said viscometric
flows occur. Excellent comprehensive reviews by the pioneers of this field,
earlier ones by Larson [3] and Shaqfeh [6], and the more recent one by Muller
[7], already exist in the literature; the goal of this section is to provide a
self-contained, but more up-to-date summary of this important and novel class
of instabilities. In fact, a discussion of these instabilities is all the more
pertinent to the present review, because their prediction is one of the
prominent success stories of the Oldroyd-B model.

###  4.1 Effect of viscoelasticity on the Newtonian Taylor-Couette instability

Purely azimuthal flow of a Newtonian fluid between concentric cylinders (the
Taylor-Couette configuration), becomes unstable due to (inertial) centrifugal
effects[143]. The instability is absent when only when both cylinders rotate
in the same direction with the angular velocity of the outer cylinder
exceeding that of the inner cylinder by the ratio $(R_{out}/R_{in})^{2}$,
$R_{out}$ and $R_{in}$ being the radii of the outer and inner cylinders,
respectively. This is consistent with the Rayleigh criterion for inviscid
instability that requires the base-state angular momentum to monotonically
decrease (in magnitude) with increasing radius [50]. The domain of existence
of both the primary linear instability and the various higher order
transitions has been well characterized in a parameter plane consisting of the
Reynolds numbers based on the radii and angular velocities of the inner and
outer cylinders [144]. Excluding the case of strong counter-rotation, the
unstable mode, at onset, is axisymmetric and stationary (i.e., with a zero
frequency). A similar centrifugal instability is also present in ‘Dean flow’
entailing pressure-driven flow through a curved channel, and originally
analyzed in the limit where the channel width is small compared to the radius
of curvature [145]. When a combination of cylinder rotation and a streamwise
pressure gradient drives the flow, the resulting centrifugal instability is
dubbed the ‘Taylor-Dean’ instability, and may be achieved experimentally by
inserting a meridional obstruction in the original Taylor-Couette
geometry[146].

Early efforts by Ginn and Denn [147] probed the role of weak viscoelasticity
on the centrifugal instability using a second-order fluid model. The analysis
showed that positive values of the first normal stress difference ($N_{1}$),
corresponding to a tension along the base-state azimuthal streamlines, had a
destabilizing effect. In contrast, a negative second normal stress difference
($N_{2}$), corresponding to a tension along the base-state axial vortex lines,
had a stabilizing effect. The latter effect could be interpreted as being due
to the resistance of the tensioned vortex lines to bending caused by axially
modulated perturbations. This is somewhat analogous to the work of Azaiez and
Homsy [110] mentioned in the subsection above, where the bending resistance of
tensioned streamlines acts to stabilize the viscoelastic shear layer. In the
narrow-gap limit of $\epsilon\ll 1$ ($\epsilon$ being the ratio of gap width
between the cylinders and the inner cylinder radius), the effect of an $N_{2}$
appears at a lower order in $\epsilon$ than $N_{1}$. Hence, although $N_{2}$
is usually negative and much smaller in magnitude than $N_{1}$ (typically
10-30% for polymer melts, and smaller for polymer solutions [148]), the
stabilizing or destabilizing action of viscoelasticity can nevertheless be
expected to depend sensitively on $N_{2}$ for $\epsilon\ll 1$.

The second-order model used in Ref. [147], being restricted to weak flows in
the quasi-steady limit, is only valid for $Wi\ll 1$ [149]; experiments,
particularly those examining instabilities, are most often performed outside
this regime, especially in the narrow-gap limit. Using the more realistic UCM
model, Walters and coworkers [150, 151, 152] again found that $Re_{c}$ for the
stationary Newtonian mode decreased with $E$ for small $E$, this decrease
being consistent with the destabilizing role of a positive $N_{1}$ mentioned
above. A new oscillatory ‘inertio-elastic’ mode was found to become more
unstable for higher $E$. Note that this new oscillatory unstable mode is not
connected to the Newtonian limit, and is therefore not captured by the second-
order fluid model. Beard et al. [152] found the $Re_{c}$ for this mode to also
decrease monotonically with $E$ in the range $0<E<1$, although the authors did
not extend their computations all the way down to the inertialess limit
($Re=0$ or $E=\infty$). The destabilizing effect of weak elasticity on the
Newtonian centrifugal instability also holds for the Dean and Taylor-Dean
configurations [153].

###  4.2 The purely elastic Taylor-Couette instability

While Giesekus reported evidence for the onset of a cellular instability in
Taylor-Couette flow of polymer solutions as early as 1966 [6] at Reynolds
numbers of $O(10^{-2})$, it is the pioneering theoretical-cum-experimental
efforts of Larson, Muller and Shaqfeh [154, 155, 156] that led to the
unequivocal establishment of an inertialess instability in viscoelastic
Taylor-Couette flow. In addition to carrying out a classical modal stability
analysis, using the Oldroyd-B model in the inertialess limit, the authors also
characterized the transition experimentally using a Boger fluid666The term
‘Boger fluid’ refers to a class of fluids prepared by dissolving small amounts
of high-molecular weight polymer in a very viscous solvent [157], which leads
to a high elasticity (owing to the long relaxation time) but negligible shear-
thinning (in the viscosity); further, on account of an intermediate-shear
plateau in $\Psi_{1}$ [158], these fluids have served as model systems
reasonably well described by the Oldroyd-B model over a range of shear rates
corresponding to the aforementioned plateau. Since the elasticity number $E$
scales directly with the solvent viscosity, and inversely with the square of
the flow length scale, elastic effects can be enhanced, and inertial effects
simultaneously suppressed, by use of Boger fluids in microscale flows; figure
13 shows the secondary (toroidal) recirculation patterns for both the
Newtonian and purely elastic cases [154]. The theoretical predictions,
obtained for axisymmetric disturbances in the thin gap limit, were in
qualitative agreement with experimental observations. Unlike the Newtonian
case, the unstable mode existed with rotation of either cylinder, and was
found to be oscillatory at onset, with the dominant measured frequency in good
agreement with theory. However, experiments showed the vertical length scale
of the cellular pattern at onset to correspond to an axial wavenumber smaller
than the theoretical prediction. Interestingly, the cellular pattern in the
experiments continued to evolve over times much longer than the nominal
polymer relaxation time, with the cell height eventually shrinking to half its
initial value. The authors attributed this discrepancy to the relatively flat
neutral curve, implying the excitation of unstable modes across a broad
spectrum of wavenumbers even in the immediate vicinity of the threshold, and
the resulting nonlinear interactions then contributing to the aforementioned
evolution (as discussed below, consideration of nonaxisymmetric disturbances
leads to better agreement). Further, the measured critical Weissenberg number
was typically found to be between $0.5$–$0.9$ times the predicted value.
Plausible reasons behind this discrepancy are discussed in Sec. 4.6 below.
Analogous elastic instabilities have also been predicted in the Dean flow
[159] and Taylor-Dean flow configurations [146], with the unstable mode in the
latter case changing from an oscillatory to a stationary one, as the pressure
gradient becomes dominant in relation to cylinder rotation.

Figure 13: Visualization of Taylor-Couette flow using reflective mica flakes.
Panel A shows the patterns for flow of a Newtonian fluid with Taylor number
$3.8\times 10^{3}$, while panel B pertains to viscoelastic Taylor-Couette flow
with $Ta=O(10^{-8})$ and $De=20$; here, $Ta=Re^{2}\epsilon$. Figure reproduced
with permission from Fig. 1 of Muller et al. [154]

Both the elastic Taylor-Couette and Dean instabilities owe their origin to
either the base-state or perturbation hoop stress fields that arise on account
of tension along the curved base-state streamlines. For the former flow,
Larson et al. [155] proposed a physical mechanism based on a dumbbell model
for a polymer molecule, consistent with the microscopic picture underlying the
Oldroyd-B equation used for the stability analysis. The toroidal circulation
associated with the axisymmetric unstable eigenmode leads to an extensional
flow in the meridional plane that stretches the dumbbell in the radial
direction. This stretched dumbbell is now acted upon by the base-state
azimuthal shear, which tilts it, leading to an increased separation between
the beads along the azimuth (see Fig. 12 of Ref. [155]). This drives a
perturbation normal stress in the $\theta\theta$ direction (the hoop stress),
which in turn produces a radial perturbation pressure gradient. The radial
flow driven by this pressure gradient is out of phase with the original
extensional flow, but for sufficiently high $Wi$, overwhelms the original
radial perturbation, leading to overstability, and a growing oscillatory
response. Note that, on account of the underlying hoop stress, the mechanism
above is relevant only for flows with curved streamlines, and and is absent,
at linear order, in the rectilinear shearing flows discussed in Sec. 2.
However, the mechanism can operate at a nonlinear order in rectilinear flows,
where the streamline curvature itself arises on account of the perturbation,
and this scenario is discussed in more detail in Sec. 7.

###  4.3 Finite-gap effects and nonaxisymmetric disturbances

The early theoretical efforts [154, 155] were restricted to small gap-widths
($\epsilon\ll 1$) and axisymmetric disturbances. In order for the effects of
curvature, in the Oldroyd-B model, to remain important for $\epsilon\ll 1$,
one requires $Wi\sim\epsilon^{-1/2}$, and expectedly, the linear analysis
[155] yields a threshold value of $Wi_{c}\,\epsilon^{\frac{1}{2}}$ for
instability (note that, when inertial effects are negligible, the limit
$Wi\,\epsilon^{1/2}\ll 1$ corresponds to plane Couette flow which, as already
seen in section 2, is linearly stable [62, 68, 72]). The above $Wi-\epsilon$
scaling, and similar scalings found for the other curvilinear flows (the
parallel-plate and cone-and-plate geometries, with $\epsilon$ being replaced
by the pertinent geometric factor), are discussed below in Sec. 4.9, in the
context of the Pakdel-McKinley criterion.

The restriction to axisymmetric disturbances, for $\epsilon\ll 1$, is often
justified based on the assumption of the azimuthal variation of the
perturbation being on scales comparable to the cylinder radii; as a result,
for $\epsilon\to 0$, both axisymmetric and nonaxisymmetric modes are governed
by the same leading order eigenvalue problem. Working within the axisymmetric
disturbance framework, Shaqfeh et al. [156] showed that, as $\epsilon$
increases, $Wi_{c}$ from both experiments and the finite-gap theory decreases
much more slowly with $\epsilon$ than the aforementioned prediction
($Wi_{c}\propto\epsilon^{-1/2}$) of small-gap theory. Significant (positive)
deviations of the threshold are already predicted for $\epsilon=0.05$,
pointing to the restrictive range of validity of the small-gap assumption,
with $Wi_{c}$ becoming nearly independent of $\epsilon$ in the range 0.1 to
0.25, this being consistent with experimental observations (Fig. 11 of Ref.
[146]).

As pointed out above, the effect of nonaxisymmetry is only expected to enter
the analysis at a higher order for $\epsilon\ll 1$. Indeed, Joo and Shaqfeh
[146] showed that the terms differentiating the various nonaxisymmetric modes
are $O(\epsilon^{2}Wi^{3})$. Since $\epsilon Wi^{2}\sim O(1)$ for curvature
effects to remain finite, these additional terms are seen to be
$O(\epsilon^{1/2})$, and therefore, asymptotically small compared to the
leading order axisymmetric ones. However, detailed calculations showed the
thin gap assumption to again be very restrictive. Thus, Joo and Shaqfeh, even
for $\epsilon\sim 10^{-4}$, found the additional terms (characterizing
departure from axisymmetry) to be important; nonaxisymmetric modes, in fact,
turned out to be more unstable than the axisymmetric mode. Joo and Shaqfeh
[146] found the results obtained using nonaxisymmetric disturbances to be in
good qualitative agreement with experimental observations of the variation of
$Wi_{c}$ with $\beta$ and $\epsilon$. Further, the wavenumber of the axial
vortices developing at onset, determined using an image analysis, was found to
be in good agreement with the prediction for the most unstable nonaxisymmetric
mode (about 5.5 inverse gap thicknesses). The results of Joo and Shaqfeh were
also consistent with the earlier findings of Avgousti and Beris [160], who
again found a nonaxisymmetric oscillatory mode in viscoelastic Taylor-Couette
flow to be more unstable than the aforementioned axisymmetric mode, even for
$\epsilon=0.1$.

###  4.4 The dominant instability in the $Wi-Re$ plane

The discussion above began with a brief description of the Newtonian Taylor-
Couette instability ($Wi=0$), and thereafter, shifted to an examination of the
purely elastic instability ($Re=0$) arising in the same flow configuration.
The initial efforts referred to above [147, 150, 151, 152] emphasized the
destabilizing role of weak elasticity on the Newtonian (centrifugal)
instability (for $N_{2}/N_{1}\rightarrow 0$). The destabilizing nature of weak
elasticity on the Newtonian instability holds even for the Dean and Taylor-
Dean configurations [153]. A study of the opposite limit, that is, the role of
weak inertia on the purely elastic instability [155], was first carried out by
Joo and Shaqfeh [153]. Expectedly, the rotation of the outer cylinder was
found to have a stabilizing effect, and that of the inner cylinder a
destabilizing one; note that this breaks the symmetry present in the purely
elastic limit, for $\epsilon\ll 1$, where the instability is independent of
the particular cylinder being rotated777It is perhaps worth noting that the
analogous question for the other viscometric flow configurations viz. the
cone-and-plate and parallel-plate geometries, is complicated by the fact that
the purely azimuthal flow is no longer an exact solution for finite $Re$; see
section 4.5..

It is also of interest to examine the nature of the dominant unstable mode in
the $Re-Wi$ plane as a whole (Fig. 14). Recall from section 2 that, within the
Oldroyd-B framework of a shear-independent viscosity and first normal stress
coefficient, the elasticity number $E=Wi/Re$ is a convenient measure of the
relative magnitudes of elasticity and inertia, with $E=0$ and $\infty$
corresponding to the purely inertial and purely elastic limits, respectively.
Avgousti and Beris [161], using both linear stability analyses and symmetry
arguments (that remain valid beyond the linear regime), reported the existence
of three different eigenmodes in the $Re-Wi$ plane for Taylor-Couette flow of
a UCM fluid subject to axisymmetric perturbations. For $E\ll 1$, the unstable
mode is stationary and its structure analogous to the well-known Taylor
vortices in the Newtonian limit; for $E\gg 1$, the unstable mode is
oscillatory akin to the purely elastic mode analyzed by [155]. For finite
values of $E$, however, a distinct inertio-elastic oscillatory mode becomes
most unstable with a wavelength and flow structure intermediate between the
purely elastic and inertial modes. Avgousti and Beris [160] showed that the
above trend, of new modes becoming dominant with increasing $E$, persisted for
non-axisymmetric disturbances [161], although this finding was restricted to
the UCM limit ($\beta=0$). Moreover, the unstable regions in the $Re$-$Wi$
plane were mapped out only for fixed wavenumbers, and for the specific case
$\Omega_{out}/\Omega_{in}=1/2$.

Figure 14: Dominant modes in the $Re$-$Wi$ plane for viscoelastic Taylor-
Couette with outer cylinder stationary or rotating at half the angular
velocity of the inner cylinder (main schematic). The continuous lines are for
the UCM limit and the dashed lines are for $\beta=0.8$. The two insets show
the dominant mode(s) in Taylor-Couette flow with a stationary inner cylinder
and in Dean flow. For the former configuration, the elastic mode is stabilized
by inertia. In contrast to Taylor-Couette flow, a single unstable mode
continuously spans the $Re$-$Wi$ plane for Dean flow.

In contrast, Joo and Shaqfeh [153], in addition to the aforementioned angular
velocity ratio, also considered the cases $\Omega_{out}=0,\Omega_{in}\neq 0$
(only inner cylinder rotating), and $\Omega_{out}\neq 0,\Omega_{in}=0$ (only
outer cylinder rotating), with the restriction of axisymmetric disturbances.
For $\Omega_{out}/\Omega_{in}=1/2$ and $\Omega_{out}/\Omega_{in}=0$, the said
authors presented results for the critical Reynolds number, minimized over the
entire range of axial wavenumbers, as a function of $Wi$, and confirmed the
existence of three distinct modes in the $Re$-$Wi$ plane in the UCM limit,
similar to the results of Avgousti and Beris discussed above. However, the
authors also showed that the nature of the dominant mode was sensitively
dependent on $\beta$. Thus, while the inertio-elastic mode was the most
unstable over a small, intermediate range of $Wi$ for $\beta=0$, this range
disappears entirely for $\beta$ slightly greater than zero, suggesting that
the inertio-elastic mode may not be relevant for polymer solutions (see Fig.
14). Thus, for $\beta=0.8$ and $\Omega_{out}/\Omega_{in}=0$, there are only
two dominant modes in the $Re$-$Wi$ plane, viz., the continuation of the
inertial mode from the Newtonian limit to finite $Wi$, and the continuation of
the purely-elastic mode from the inertialess limit to finite $Re$ (see Fig.
14). For $\Omega_{in}=0$, there is no instability in the Newtonian limit.
Thus, the purely elastic instability, present for $Re=0$, is the only unstable
mode in $Re$-$Wi$ plane, and the critical $Wi$ for this instability increases
as $Re$ is increased from zero. In contrast to this rather complicated
scenario in Taylor-Couette flow, wherein different modes become dominant as
$E$ is increased, the same unstable axisymmetric mode continues all the way
from $E=0$ to $E=\infty$ for Dean flow [153]; moreover, this scenario holds
for different values $\beta$ (Fig. 14). This simple picture may not, however,
hold for nonaxisymmetric disturbances, although this remains to be
investigated.

Interestingly, the nature of the dominant instability in the $Re-Wi$ plane in
the limit $Re,Wi\rightarrow\infty$, with $E$ fixed, is of relevance to the
astrophysical scenario. Note that this distinguished limiting scenario arises
only for the Oldroyd-B model since shear thinning in the nonlinear
constitutive models leads to elastic stresses scaling sub-quadratically, and
thereby, becoming asymptotically small in relation to the inertial ones in the
aforementioned limit. The limit may be termed the elastic Rayleigh limit since
the equation governing linearized evolution is a second order ODE similar to
the Rayleigh equation in the inviscid limit [50], but with $E(1-\beta)$ as a
parameter that measures the importance of elastic stresses (the reduction of
order points clearly to the singular nature of this limit). It has been shown
[162] that the equations governing polymer solutions in the elastic Rayleigh
limit are identical to the magnetohydrodynamics (MHD) equations in the
relaxationless limit (that of infinite magnetic Reynolds number), with the
dumbbell end-to-end vector field being analogous to the magnetic field. A
consequence is that the instabilities of a polymer solution in this limit have
corresponding astrophysical analogs; in particular, an inertio-elastic
instability in this limit should map onto the so-called magnetorotational
instability (MRI). The axisymmetric version of the latter instability was
originally predicted by Vehlikov [163] and Chandrasekhar [164], and
rediscovered much later by Balbus and Hawley [165]. In the polymer solution
literature, the elastic Rayleigh limit was first considered by Azaiez and
Homsy [110], in the context of a viscoelastic shear layer, although elasticity
had a stabilizing influence in this case, as already discussed in section 3.
Rallison and Hinch [166] were the first to discover an inertio-elastic
instability for a submerged jet configuration, arising from a novel mechanism
involving a resonant interaction of elastic shear waves (the analogs of Alfven
waves in the MHD context) in the aforementioned limit. More recently, an
analogous shear-wave-driven instability has been shown to destabilize an
elastic vortex column [33]; unlike the jet, this inertio-elastic instability
exists in isolation for the vortex case, and is therefore of greater
relevance. Importantly, the vortex column configuration bears an obvious
relation to the Taylor-Couette geometry. There have indeed been experiments in
the Taylor-Couette setup motivated by the above analogy [167], and also
numerical stability calculations for relatively modest $Re$ and $Wi$ [168]. It
is certainly of interest to theoretically investigate the instabilities of the
Taylor-Couette configuration in the elastic Rayleigh limit, and in particular,
examine the relation between any unstable modes arising from the
aforementioned shear-wave-resonance mechanism, and the inertio-elastic mode,
at finite $Re$ and $Wi$, that has been discussed in the elastic instability
literature[160, 153].

###  4.5 Purely elastic instability in cone-and-plate and parallel-plate flows

The mechanism underlying the purely elastic Taylor-Couette instability is
suggestive of a similar instability in the two other rheometric flow
configurations, viz., the cone-and-plate and parallel-plate geometries. Magda
and Larson [169] observed an anomalous increase in the shear stress above a
threshold rate of shear, in both these geometries, even for a vanishingly
small $Re$. Although similar observations had been made by earlier researchers
[170, 171], the aforementioned authors were the first to ascribe the anomalous
increase to a flow transition, rather than to the intrinsic rheological
character of the fluid. Subsequent visualization experiments by McKinley et
al. [172] showed the onset of a secondary flow at the critical shear rate,
corroborating the inference of Magda and Larson.

The first theoretical studies to demonstrate an instability of the flow of an
Oldroyd-B fluid, in the above geometries, were due to Phan-Thien [173, 174].
Motivated by the form of the Newtonian flow in these geometries, the author
assumed a self-similar so-called von-Karman ansatz for the base-state flow at
finite $Re$ and $De$, with $De=\lambda\Omega$. Despite the base-state flow
reducing to a purely azimuthal one for zero $Re$ (regardless of $De$) 888In
the Newtonian case, the (inertial) centrifugal forces driving the meridional
secondary flow scale with the velocity, which is larger near the rotating
plate, and therefore give rise to a secondary flow. In contrast, the elastic
stresses scale with the velocity gradient (the square of the gradient for
Oldroyd-B), and are therefore uniform across the gap. The divergence of these
stresses may be balanced with a radial pressure gradient, allowing the base-
state flow to have only an azimuthal component. This argument is applicable,
however, only in the limit of small cone angles, which is when the velocity
gradient across the gap is a constant. The existence of an elastic
instability, of course, implies solution multiplicity, and the azimuthal flow
is not the only possible axisymmetric flow beyond a threshold $De$., the
author again invoked the Karman ansatz for the perturbation, and demonstrated
the onset of an stationary and axisymmetric elastic linear instability beyond
a critical $De$. The absence of a length scale in the assumed ansatz implied a
disturbance flow field in the form of a single ‘roll’, in the meridional
plane, that ‘closes at infinity’. Thus, the only relevant dimensionless group
is $De$ above, with the threshold criterion therefore coming out to be
independent of the gap-angle ($\theta$) for the cone-and-plate geometry, and
of the non-dimensional plate spacing ($H/R$, $H$ and $R$ being the inter-plate
spacing and plate radii) for the parallel-plate geometry; note that
$Wi=De/\theta$, and $De\,(R/H)$ for the two geometries.

The aforementioned visualization experiments of McKinley et al. [172], for the
parallel plate geometry, showed that the growing disturbance flow field was
not of the similarity form assumed in [173]; instead, the observed patterns
were characterized by a radial scale of $O(H)$. The first analysis to account
for this scale was that of Oztekin and Brown [175], who used the Oldroyd-B
model to conduct a local stability analysis of the parallel-plate flow in the
vicinity of a particular radius ($R^{*}$). While the analysis correctly showed
the unstable modes to be time dependent non-axisymmetric spiraling patterns,
the prediction of the threshold was qualitatively incorrect. The threshold
$De$, obtained from the Oldroyd-B analysis, bore an inverse relation to
$R^{*}$, so the torsional flow configuration was predicted to be unstable for
any finite $De$ beyond a certain $R^{*}$ (in contrast to the Phan-Thien
analysis above). This, however, contradicted the later experiments of McKinley
and coworkers [176], who showed that the unsteady secondary motion, just
beyond the threshold, was restricted to an annular region between a pair of
critical radii. The restabilization of the flow beyond the second critical
radius arises from effects of shear thinning which decrease the effective
relaxation time at higher shear rates. This was confirmed by a linear
stability analysis using the FENE-CR model [176]. The aforementioned
restabilization also translates to the stability of the torsional flow
configuration below a threshold inter-plate spacing; the resulting shape of
the unstable region in parameter space is discussed in section 4.9 below in
the context of the Pakdel-Mckinley criterion.

An analogous scenario was shown to hold for the cone-and-plate geometry [177].
Use of the Oldroyd-B model again reproduced the unstable spiral patterns in a
qualitative sense; unlike the parallel-plate case, the homogeneity of the
base-state shearing flow and the resulting absence of a characteristic length
scale implies that the spirals in this case are no longer radially localized.
Use of the FENE-CR model was needed to capture the correct nature of the
threshold condition, which involved a stabilization of the flow for small
enough cone angles; this is discussed in section 4.9. The use of a multimode
Giesekus model, in fact, yielded quantitative agreement with experimental
observations [178]

###  4.6 Experiment vs Theory (Non-isothermal effects)

The comparison between theory and experiment, even for the primary transition
in the viscoelastic case, is nowhere as quantitative as for the Newtonian
case. The first and perhaps obvious reason is the uncertainty surrounding the
constitutive equation used. The Oldroyd-B equation, used in the stability
analyses, is only an approximation for the Boger fluids used in the
experiments. Boger fluids have a nontrivial spectrum of relaxation times, and
consistent modeling of both their steady state and transient rheology requires
a nonlinear multimode constitutive equation [158]. The underlying relaxation
spectrum manifests as a dependence of the apparent relaxation time on the
method of measurement. The resulting arbitrariness in the choice of time
scale, to be used in the Oldroyd-B model, leads one to expect an ambiguity in
the theory-experiment comparison. A second reason has already been mentioned
above in the context of the first experiments: the shallow nature of the
viscoelastic neutral curve, in comparison to the Newtonian one (compare
Figs.III-2 and III-11 of Ref. [3]), has been speculated to lead to strong
nonlinear interactions even close to onset, that then manifest as a slow drift
of the observed pattern with the appearance of progressively smaller-scaled
structure over times much longer than the nominal relaxation time [155, 179].
A second factor favoring the aforementioned importance of nonlinearity is that
the bifurcation to the primary non-axisymmetric mode is likely a subcritical
one [180], in contrast to the Newtonian case. Thus, as briefly mentioned in
Sec. 4.2, while the theoretical predictions based on axisymmetric disturbances
over-predict the threshold for instability, consideration of nonaxisymmetric
disturbances and the associated subcritical nature of the bifurcation leads to
a narrowing of the gap between the theoretical and experimental thresholds.

(a) left panel: inner cylinder rotating; right panel: outer cylinder rotating

(b) Critical $De$ vs Nahme number.

Figure 15: (a) Space-time plots of Taylor-Couette flow of a viscoelastic fluid
at $De=2.7$, $Re=2.7$, Brinkmann number $=6.5\times 10^{-3}$, and Prandtl
number $\approx 24000$. The extent of the z axis is 5.82 gap widths.
Reproduced with permission from Ref. [181]. (b) $De_{c}$ vs. Nahme number,
reproduced with permission from Ref. [182]. These experiments differed from
earlier efforts in adopting the so-called adiabatic (quasi-static) path,
involving a slow ramp-up (as opposed to a step increase) to the rotation rate
of interest.

A third and rather unexpected reason that has led to a stark difference
between theory, and some of the experiments using Boger fluids, is the
deviation from isothermal conditions arising due to viscous heating (Boger
fluids have very high viscosities). Despite the critical parameters for
instability onset being sensitive to the particular rheological model used,
the elastic instability for Taylor-Couette flow, within the usual isothermal
formulation, is predicted to be always non-axisymmetric and oscillatory at
onset. However, many of the experimental observations [183, 184, 181, 182]
have revealed a primary transition to a weak stationary axisymmetric mode on
time scales much longer than the polymer relaxation time, and at $Wi$ much
lower than the theoretical threshold. This discrepancy was addressed by Al-
Mubaiyedh et al. [185, 186] who showed that the inclusion of viscous heating
in the analysis, and the resulting prediction of a thermoelastic instability,
leads to a good agreement between experiment and theory. That viscous heating
leads to destabilization is somewhat counter-intuitive since both viscosity
and the relaxation time should decrease with an increase in temperature, and
this ought to lead to a higher threshold rotation rate for a given $Wi_{c}$.
This expected stabilizing effect has indeed been found for other curvilinear
geometries such as the parallel-plate and cone-and-plate configurations [187,
188]. The destabilizing mechanism [185, 186] for Taylor-Couette flow is argued
to arise due to the stratification of the hoop stress in the gap between the
cylinders, which drives a radial secondary flow, and the convection of base-
state temperature gradients by the radial perturbation velocity then leads to
the instability. Figure 15 shows the slow development of the stationary
vortices on a space-time plot [181], with the adjoining plot showing the
experimentally observed decrease in $De_{c}$ (to be interpreted as $Wi_{c}$)
with the Nahme number, the latter being a dimensionless measure of viscous
heating [182].

###  4.7 Secondary instabilities and elastic turbulence

Transition from the laminar state, in the Newtonian case, occurs in
contrasting fashions for the canonical (and non-inflectional) rectilinear
shearing flows examined in section 2, and for the curvilinear shearing flows
examined here (the Taylor-Couette geometry in particular). In the former case,
with increasing $Re$, turbulence appears abruptly and in its entire
complexity, albeit in spatially localized forms [189], while there is a
gradual but global increase in the spatiotemporal complexity for the Taylor-
Couette configuration. Thus, the topic of secondary or higher-order
transitions following the primary instability, and that lead to the eventual
turbulent state, is more pertinent to curvilinear shearing flows; indeed, the
various flow regimes for Newtonian Taylor-Couette flow have been extensively
studied and are well documented, as already mentioned in section 4.1 [144,
190, 191].

At the other extreme of elasticity being dominant ($E\gg 1$), a
spatiotemporally disordered state has been shown to arise for sufficiently
large $Wi$, and has appropriately been termed elastic turbulence (ET). The
experiments of Groisman and Steinberg [8] were the first to characterize ET in
a wide-gap parallel-plate geometry ($\epsilon=0.263$ and $0.526$), with the
shear stress in this state being up to $20$ times larger than the
(hypothetical) laminar flow at the same rotation rate. The transition was
found to be subcritical, although the detailed pathway and the associated rich
sequence of intermediate states (more readily accessed in a narrow-gap
setting) were characterized in later experiments, in the same geometry
($\epsilon=0.2$), by Schiamberg et al. [192]. Groisman and Steinberg [8] also
examined the power spectrum of velocity fluctuations in the ET state via
Doppler velocimetry; the plots decayed as $\omega^{-3.5}$ for large
frequencies ($\omega$), and thence, for sufficiently small length scales (via
Taylor’s frozen turbulence hypothesis). The rapid decay is in contrast to the
Kolmogorov $(-5/3)$-law known for the inertial range in Newtonian turbulence,
but is consistent with theoretical constraints obtained based on the Oldroyd-B
model [193]. The implication is that, in contrast to Newtonian turbulence, the
velocity gradient is the maximum at the largest scales, and ET therefore
corresponds to a stochastic but spatially smooth chaotic flow controlled by
the large scales [10]. The ET state has also been accessed in a curvilinear
channel [9], the analog of the Dean-flow configuration discussed earlier but
one that does not conform to the infinite (large) depth assumption underlying
the derivation of the theoretical stability threshold. The curvature ratio was
large to facilitate transition, which was now found to be supercritical [194],
but with the power spectra again pointing to the large scales being dominant.

Groisman and Steinberg have also accessed an ET state in a wide-gap
($\epsilon\approx 1$) Taylor-Couette geometry [194], the transition being
hysteretic similar to the parallel-plate configuration above. The velocity
fluctuation power spectra, however, exhibited a large-$\omega$ decay exponent
a little greater than $2$, thereby deviating from the scalings for the other
geometries above, in turn implying the possibly non-universal geometry-
dependent nature of the ET state. In earlier experiments involving a Taylor-
Couette configuration with a narrower gap ($\epsilon=0.5$), Groisman and
Steinberg have examined the higher-order transitions en-route to turbulence,
as a function of $E$, although the aforementioned ET state itself was not
identified [195, 196, 197, 184]. For $E<0.15$, the primary instability of the
azimuthal flow gave rise to a Newtonian-like Taylor-Vortex flow (TVF) which,
with increasing $Re$, became unstable to a new oscillatory state termed a
‘rotating standing wave’. At a higher $Re$, this state transitioned to yet
another new oscillatory state termed ‘disordered oscillations’, characterized
by broad peaks in the frequency spectra and by the appearance of patches of
both standing and travelling waves in space-time plots. For $E>0.15$, the TVF
directly transitioned to disordered oscillations, while for $E>0.22$,
disordered oscillations emerged directly from the purely azimuthal laminar
base flow. At higher elasticities $(E\sim 20)$, the transition to disordered
oscillations became increasingly hysteretic. Thus, along a path of decreasing
$Re$, these oscillations persisted down to an $Re$ much lower than that
marking the onset of the forward transition, eventually giving way to
stationary so-called ‘diwhirls’; solitary diwhirls have since been identified
via fully nonlinear computations [198]. Many of the aforementioned
transitions, for small but finite $E$, have been reproduced in simulations
(for $E=1/3$) by Khomami and co-workers, using the Oldroyd-B and FENE-P
models, with $L$ in the latter case being large enough for shear-thinning to
be unimportant [199].

Detailed experiments, with the intent of mapping both the primary and higher-
order transitions in the Taylor-Couette geometry, have also been conducted by
Muller and co-workers. While the early experiments using high-viscosity Boger
fluids were likely corrupted by the emergence of a thermoelastic instability
(the later experiments of White and Muller pertaining to the thermoelastic
instability are discussed in more detail in section 4.6), and possibly, its
interaction with the elastic instability on account of adopting a non-
quasistatic ramping protocol [183, 200], later experiments [201, 202], based
on a quasi-static ramping protocol, have extended the known mapping of flow
states in the $Re_{in}-Re_{out}$ plane (see section 4.1) to non-zero $E$. The
general pattern is that of elasticity affecting the higher-order less
symmetric flow states at a lower $E$, which is expected on account of the time
scales of oscillation involved being comparable to the polymer relaxation
time. For the highest $E$ examined ($\approx 0.2$), the authors identify an
elastically dominated turbulent state based on the emergence of a broadband
frequency spectrum, although the relation with the ET-state, identified by
Steinberg and coworkers above, is not clear [202].

###  4.8 Beyond Oldroyd-B

Here, we comment briefly on how features not captured by the Oldroyd-B model
affect the prediction of the purely elastic instabilities in curvilinear
flows. These include (i) a nonzero $N_{2}$, (ii) the existence of a nontrivial
relaxation time spectrum, (iii) the shear-rate dependence of the viscosity and
first normal stress difference, (iv) the use of a multi-scale simulation
approach that avoids closure approximations, and (v) going beyond the dilute
regime.

Most entangled polymer solutions have negative $N_{2}$’s [148], which are
predicted to strongly stabilize the purely elastic instability in Taylor-
Couette flow [156], especially for small gap widths ($\epsilon\ll 1$); as
alluded to in section 4.1, in the context of the second-order fluid model, the
effects of $N_{2}$ enter at a lower order in $\epsilon$ (compared to $N_{1}$)
in this limit. Indeed, Shaqfeh et al. [156] noted a significant discrepancy
between the prediction from the Oldroyd-B model ($Wi_{c}=47$) and experiment
($Wi_{c}=71$) for the smallest gap ratio ($\epsilon=0.03$), where the small-
gap theory should otherwise have been accurate. To address this discrepancy,
Shaqfeh et al. used a modified Oldroyd-B model with an additional contribution
to the stress tensor that, albeit of the second-order fluid form, gave rise to
a nonzero $N_{2}$, and showed that this indeed had a stabilizing effect.
Similarly, using the Giesekus model, Beris et al. [203] also concluded that
negative second normal stress differences were strongly stabilizing. Indeed,
for $-N_{2}/N_{1}>0.1$ (typical for entangled polymer solutions),
$Wi_{c}>100$, and this could be one reason why the purely elastic Taylor-
Couette instability is not reported for the flow of entangled polymeric
solutions [3].

Larson et al. [204] used the K-KBZ equation to incorporate both shear thinning
and a relaxation time spectrum to analyze the elastic Taylor-Couette
instability, and concluded that the critical conditions are a function of both
the longest (the Oldroyd-B) and the average relaxation times. This immediately
points to the arbitrariness inherent in the choice of a single-time-scale
(either linear or nonlinear) model, as has already been pointed out in section
4.6. In a later effort pertaining to instability onset in the cone-and-plate
and parallel-plate geometries, a multi-mode Giesekus model, with parameters
tuned so as to best fit the rheological properties of the Boger fluid used in
the experiments, has been shown to quantitatively predict the critical $Wi$ as
a function of gap width or cone angle [178]. Notwithstanding the subtle issue
of stress-conformation hysteresis that might come into play for instabilities
in non-viscometric flows (see discussion at the end of Sec. 5.3), multimode
versions of nonlinear constitutive models appear necessary for a quantitative
match with experiments.

Larson et al. [204] found shear thinning to have a monotonically stabilizing
effect on the instability, on account of the decrease in the first normal
stress coefficient with the shear rate. As mentioned in Sec. 4.5, shear-
thinning has, in fact, a profound effect on the neutral boundary demarcating
the stable and unstable regions [172, 177, 175, 176], and this is discussed in
more detail in the following subsection, in the context of the Pakdel-Mckinley
criterion.

Although the commonly used constitutive equations like the Oldroyd-B, the
FENE-P or the Giesekus models, have the obvious advantage of needing only
modest computational resources, they are limited either by the simplicity of
the microscopic picture (Oldroyd-B), by closure approximations (FENE-P) or by
lack of a rigorous connection to an underlying kinetic theory framework (FENE-
CR, Giesekus). There have been some efforts, in the context of hydrodynamic
stability, that have attempted to go beyond the said limitations by adopting a
‘micro-macro’ approach. For instance, Somasi and Khomami [205, 206] calculate
the polymer stress field from Brownian dynamics simulations using Hookean or
FENE dumbbells, which is subsequently coupled with a finite-element solution
of the Cauchy momentum equations. Using this approach, the authors examined
the linear stability of plane- and Taylor-Couette flows, finding good
agreement with the results of a stability analysis carried out using closed-
form constitutive equations. However, very large computational times were
required (approximately 100 times that required for a regular stability
computation) even for the simple elastic dumbbell-based models used.

Finally, the results obtained using the Oldroyd-B model are strictly valid
only in the truly dilute regime. However, the polymer solutions used in
experiments are not always in this regime, and it is important to include the
concentration dependence of the fluid rheology in a systematic manner. In this
regard, as a first step, it seems appropriate to use the Giesekus model
(discussed earlier in Sec. 1), with the anisotropic drag coefficient in the
model acting as a proxy for polymer concentration. Recent efforts [207] have
attempted to account for, in a more rigorous manner, hydrodynamic interactions
between chains that come into play close to and beyond the overlap
concentration. Although the role of such inter-chain interactions has been
examined in the context of the CABER technique, now commonly used to measure
the relaxation time [208, 209], in future, one could also envisage a stability
analysis, based on the rigorous models above, that extends across the overlap
concentration, spanning both the dilute and semi-dilute regimes.

###  4.9 The Pakdel-McKinley criterion

We now discuss a heuristic argument developed by Pakdel and McKinley [65, 210]
which unifies the threshold criteria for instability onset in the different
curvilinear shearing flows examined above. As will be seen below and in
section 5, it allows one to arrive at sensible stability thresholds (to within
a numerical factor of order unity) even when closed form results from linear
stability analysis are not available (owing, for instance, to the geometrical
complexity of the flow configuration). The criterion incorporates the two key
ingredients required for a purely elastic instability viz. streamline
curvature in the base-state laminar flow as well as the magnitude of the
streamwise (tensile) normal stresses, and expresses the threshold for elastic
instability in terms a product of the Deborah and Weissenberg numbers.
Recalling the definition $De=\lambda/T$, $T={\cal R}/U$ now being the
residence time with ${\cal R}$ the radius of curvature (for the curvilinear
shearing flows in question), and writing $Wi$ in terms of the ratio of the
normal and shear stresses as $Wi=N_{1}/2\tau$, the Pakdel-Mckinley criterion
for elastic instability may be written as:

$[(\lambda U/{\cal R})(N_{1}/\tau)]_{c}>M^{2},$ (2)

$M$ being an order unity number in the simplest cases (curvilinear viscometric
flows), or a field (when the flow configuration is more complicated, and in
particular, inhomogeneous, as is the case for the non-viscometric examples
considered in section 5). Note that the ratio $\lambda U/{\cal R}$ may also be
interpreted as the ratio of the distance (along a streamline), over which
disturbances relax elastically, to the characteristic radius of curvature.
Replacing $\lambda U$ by an appropriate viscous boundary layer length scale
recovers the known stability criteria associated with Newtonian instabilities
driven by curvature; these include the classical Taylor-Couette instability
itself, and the Gortler instability associated with laminar boundary layers on
curved surfaces [211]. Note, however, that the application of this criterion
relies on the assumption that the instability mechanism is local and can be
predictive only within this scenario.

We now obtain the explicit form for this criterion for the Taylor-Couette and
cone-and-plate configurations. For the former, $U=\Omega R_{in}$,
corresponding to the inner cylinder (say) rotating with angular velocity
$\Omega$, and an obvious choice for the radius of curvature is the (inner)
cylinder radius, viz., ${\cal R}=R_{in}$. The streamwise normal stress for the
Oldroyd-B model is $\tau_{\theta\theta}=2\eta_{p}\lambda\dot{\gamma}^{2}$, and
the total shear stress $\tau=\eta_{t}\dot{\gamma}$, $\eta_{t}$ being the
solution viscosity. Substituting these expressions leads to

$(\sqrt{DeWi})_{c}\geq\frac{M}{\sqrt{2(1-\beta)}}\,,$ (3)

for instability in the thin-gap limit, where $De=\lambda\Omega$,
$Wi=\lambda\dot{\gamma}$, and $\eta_{p}/\eta_{t}=(1-\beta)$. Using
$\dot{\gamma}=\Omega R_{in}/d$ and the dimensionless gap width
$\epsilon=d/R_{in}$ with $Wi=De/\epsilon$, the above equation reduces to

$Wi_{c}\epsilon^{\frac{1}{2}}\geq\frac{M}{\sqrt{2(1-\beta)}},$ (4)

in agreement with the original thin-gap analysis of Larson et al. [155] (note
that these authors refer to $Wi$ in (4) as $De$). In Eq. 4, $M$ is in general
a complicated function of $\beta$, and asymptotes to a constant only in the
limits $\beta\rightarrow 0$ and $1$. In the former limit, the stability
analysis yields $M\approx 6$ for axisymmetric disturbances [155], and for the
latter case, the threshold $Wi$ diverges as the reciprocal square root of
$(1-\beta)$ in the limit of vanishing elasticity.

For the cone-and-plate configuration, the velocity at any radial position is