# A continuous pathway between the elasto-inertial and elastic turbulent
states in viscoelastic channel flow

Mohammad Khalid  Department of Chemical Engineering, Indian Institute of
Technology, Kanpur 208016, India  V. Shankar<EMAIL_ADDRESS>Department
of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India
Ganesh Subramanian<EMAIL_ADDRESS>Engineering Mechanics Unit,
Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore 560084,
India

###### Abstract

We show that viscoelastic plane Poiseuille flow becomes linearly unstable in
the absence of inertia, in the limit of high elasticities, for ultra-dilute
polymer solutions. While inertialess elastic instabilities have been predicted
for curvilinear shear flows, this is the first ever report of a purely elastic
linear instability in a rectilinear shear flow. The novel instability
continues upto a Reynolds number ($Re$) of $O(1000)$, corresponding to the
recently identified elasto-inertial turbulent state believed to underlie the
maximum-drag-reduced regime. Thus, for highly elastic ultra-dilute polymer
solutions, a single linearly unstable modal branch may underlie transition to
elastic turbulence at zero $Re$, and to elasto-inertial turbulence at moderate
$Re$, implying the existence of continuous pathways connecting the turbulent
states to each other, and to the laminar base state.

###### pacs:

Dilute polymer solutions undergo two different transitions to novel turbulent
states, both driven by viscoelasticity, and thence, fundamentally distinct
from the now well-understood Newtonian transition (Kerswell, 2005; Eckhardt
_et al._ , 2007; Avila  _et al._ , 2011). Rectilinear shearing flows of
sufficiently elastic polymer solutions (e.g., pipe and channel flow)
transition from the laminar state at Reynolds numbers substantially lower than
the typical Newtonian threshold, the ensuing flow state dubbed ‘elastoinertial
turbulence’ (EIT) to emphasize the importance of both elastic and inertial
effects underlying the turbulent dynamics (Samanta  _et al._ , 2013; Choueiri
_et al._ , 2018; Chandra  _et al._ , 2018; Choueiri  _et al._ , 2021; Lopez
_et al._ , 2019; Shekar  _et al._ , 2019, 2020). The EIT state is dominated by
spanwise oriented 2D structures (Sid  _et al._ , 2018; Lopez  _et al._ , 2019)
in sharp contrast to the Newtonian scenario (Eckhardt  _et al._ , 2007), and
for plane-Poiseuille flow, has recently been shown (Page  _et al._ , 2020;
Dubief  _et al._ , 2020) to be connected subcritically to a linear center-mode
instability (Garg  _et al._ , 2018; Chaudhary  _et al._ , 2021; Khalid  _et
al._ , 2020); recent pipe-flow experiments have, in fact, found remarkable
agreement between the observed structures at EIT onset (Choueiri  _et al._ ,
2021) and the center-mode eigenfunction (Garg  _et al._ , 2018; Chaudhary  _et
al._ , 2021). On the other hand, curvilinear shearing flows of dilute polymer
solutions transition to ‘elastic turbulence’ (ET), with inertial effects being
negligibly small (Groisman and Steinberg, 2000, 2001; Steinberg, 2021). This
latter transition has been known for a longer time, and its origin may be
traced to a hoop-stress-driven linear instability in such flows (Larson  _et
al._ , 1990; McKinley  _et al._ , 1991; Oztekin and Brown, 1993; Joo and
Shaqfeh, 1994; Pakdel and McKinley, 1996; Shaqfeh, 1996; Schiamberg  _et al._
, 2006; Jun and Steinberg, 2011); the eventual disorderly ET state that arises
has been well characterized experimentally (Groisman and Steinberg, 2000,
2001; Jun and Steinberg, 2011), and to a limited extent, theoretically as well
(Fouxon and Lebedev, 2003).

Unlike their curvilinear counterparts, inertialess rectilinear shear flows of
dilute polymer solutions (where shear thinning effects are insignificant),
have hitherto been regarded as linearly stable (Ho and Denn, 1977; Larson,
1992; Wilson  _et al._ , 1999). Transition in these flows has been proposed to
occur via a subcritical mechanism, but one that nevertheless involves a hoop
stress that now arises at a nonlinear order due to the curvature of the
perturbed streamlines (Bertola  _et al._ , 2003; Meulenbroek  _et al._ , 2004;
Morozov and van Saarloos, 2005, 2007, 2019). There is some experimental
evidence of an inertialess finite-amplitude transition (Bertola  _et al._ ,
2003; Pan  _et al._ , 2013; Qin and Arratia, 2017; Qin  _et al._ , 2019), to
what might be an ET state similar to that observed for curvilinear flows; the
absence of a linear instability has also motivated examination of non-modal
growth mechanisms, both in the absence (Jovanovic and Kumar, 2010, 2011) and
presence (Zhang  _et al._ , 2013) of inertia. The aforementioned EIT and ET
states might seem unrelated at first sight, on account of fluid inertia
playing a fundamental role in the former while being irrelevant in the latter.
It has nevertheless been speculated (Samanta  _et al._ , 2013; Page  _et al._
, 2020; Qin  _et al._ , 2019; Steinberg, 2021; Choueiri  _et al._ , 2021) that
the two states may be linked, although there exist no concrete hypotheses or
proposals in this regard.

In sharp contrast to the prevalent view above, in this Letter, we show that:
(i) pressure-driven viscoelastic channel flow is linearly unstable even in the
absence of inertia, making this the first ever report of a purely elastic
linear instability not dependent on base-state streamline curvature; (2) the
instability smoothly continues, with increasing $Re$, to the aforementioned
elasto-inertial linear instability (Garg  _et al._ , 2018; Khalid  _et al._ ,
2020). Since the latter instability has been shown to subcritically continue
to nonlinear elastoinertial coherent structures (Page  _et al._ , 2020; Dubief
_et al._ , 2020), the implication is the existence of a continuous pathway
(the underlying unstable modal branch) connecting the EIT and ET states, one
that might provide a template for nonlinear coherent structures acting as
possible bridges between these states, thereby forming the framework for a
dynamical systems based interpretation of turbulence outside the Newtonian
realm.

Figure 1:  The spectrum for inertialess plane Poiseuille flow of an Oldroyd-B
fluid shows five discrete modes for $\beta=0.997$, $k=0.75$, $W=2500$, 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 (labelled CM).
Inset B shows the variation of $c_{r}$ and $c_{i}$ with $W$ for $\beta=0.997$
and $k=0.75$.

We consider pressure-driven flow of an incompressible viscoelastic fluid in a
channel of width $2H$. The governing mass and momentum equations Bird  _et
al._ (1977); Larson (1988), in dimensionless form, are

$\boldsymbol{\nabla}\boldsymbol{\cdot}\mathbf{u}=0,\,Re\,\Big{(}\frac{\partial\mathbf{u}}{\partial
t}+(\mathbf{u}\boldsymbol{\cdot}\boldsymbol{\nabla})\,\mathbf{u}\Big{)}=-\boldsymbol{\nabla}{p}+{\beta}\,\nabla^{2}\,\mathbf{u}+\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\tau},$
(1)

where $\mathbf{u}$, $p$, and $\boldsymbol{\tau}$, are the velocity, pressure,
and the polymer stress fields respectively. Here, lengths are
nondimensionalized using the channel half-width $H$, velocities using the
base-flow maximum $U_{max}$, and $\boldsymbol{\tau}$ is assumed to be governed
by the Oldroyd-B constitutive relation

$\displaystyle\boldsymbol{\tau}+{W}\,\Big{(}\frac{\partial\boldsymbol{\tau}}{\partial
t}+(\mathbf{u}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{\tau}-$
$\displaystyle(\boldsymbol{\nabla}\mathbf{u})^{T}\boldsymbol{\cdot}\boldsymbol{\tau}-\,\boldsymbol{\tau}\boldsymbol{\cdot}(\boldsymbol{\nabla}\mathbf{u})\Big{)}$
$\displaystyle=\Big{(}1-\beta\Big{)}\Big{(}\boldsymbol{\nabla}\mathbf{u}+\boldsymbol{\nabla}\mathbf{u}^{T}\Big{)}.$
(2)

The relevant dimensionless groups are the Reynolds number $Re=\rho
U_{max}H/\eta$ ($\rho$ and $\eta$ being the density and viscosity of the
polymer solution, respectively), the ratio of solvent to solution viscosities
$\beta$, and the Weissenberg number $W=\lambda U_{max}/H$, where $\lambda$ is
the polymer relaxation time; below, we also use the elasticity number $E=W/Re$
in lieu of $W$. The Oldroyd-B model regards the polymer molecules as non-
interacting Hookean dumbbells, resulting in a shear-rate-independent viscosity
and first normal stress coefficient in viscometric flows, and has been
successfully used to predict linear and nonlinear instabilities in rectilinear
(Sureshkumar and Beris, 1995; Morozov and van Saarloos, 2005; Garg  _et al._ ,
2018), and both curvilinear viscometric (Larson  _et al._ , 1990; McKinley
_et al._ , 1991; Joo and Shaqfeh, 1994), and non-viscometric Poole  _et al._
(2007) flows. For plane Poiseuille flow, whose stability is examined here, the
laminar state is given by $U(z)=(1-z^{2})$, with a first normal stress
difference $T_{xx}-T_{zz}=8(1-\beta)W^{2}z^{2}$. We analyze the temporal
stability of this state by imposing infinitesimal 2D perturbations (justified
by the Squire’s theorem (Bistagnino  _et al._ , 2007)) of the form
$f^{\prime}(x,z,t)=\tilde{f}(z)\exp[ik(x-ct)]$, where $f^{\prime}(x,z,t)$ is
the perturbation to any dynamical variable, $\tilde{f}(z)$ is the
eigenfunction, $k$ is the streamwise wavenumber, and $c=c_{r}+ic_{i}$ is the
complex wavespeed with the flow being temporally unstable when $c_{i}>0$.
Linearization of the governing equations results in a generalized eigenvalue
problem (Sureshkumar and Beris, 1995; Khalid  _et al._ , 2020) which is solved
using spectral collocation and shooting methods, both of which have been
extensively benchmarked in our earlier efforts (Chaudhary  _et al._ , 2019,
2021; Khalid  _et al._ , 2020).

Figure 2:  Neutral curves in the $W$–$k$ plane for different $\beta$’s in the
creeping-flow limit. Inset A shows the collapse of neutral curves for
$\beta\rightarrow 1$ when plotted as $Wk(1-\beta)$ vs $k$, and the results
obtained from the reduced equations in that limit. Inset B shows the variation
of the critical Weissenberg number $W_{c}$ and wavenumber $k_{c}$ with
$(1-\beta)$.

We first consider the creeping-flow limit ($Re=0$) where the relevant
dimensionless groups are $W$ and $\beta$. In the absence of the solvent
($\beta=0$; the upper-convected Maxwell (UCM) limit), the inertialess plane
Poiseuille eigenspectrum is known (Wilson  _et al._ , 1999) to contain six
discrete modes for $W,k\sim O(1)$, in addition to the continuous spectrum
(CS). As $W$ is increased for a fixed $k\sim O(1)$ (or, as $k$ is increased at
a fixed $W\sim O(1)$), two of the modes merge into the CS, while two
transition into ‘wall modes’. The remaining two transition into ‘center modes’
with phase speeds approaching the base-flow maximum. Regardless of $W$ and
$k$, however, plane Poiseuille flow of a UCM fluid always remains stable
(Wilson  _et al._ , 1999). For $\beta\neq 0$, the number of discrete modes
depends on $\beta$, $k$ and $W$. For ultra-dilute solutions with $\beta>0.99$,
there are two center modes out of a total of five discrete modes (Fig. 1).
Interestingly, for $\beta>0.9905$, at sufficiently high $W\sim 2500$, one of
the two center modes becomes unstable (inset A of Fig. 1). The instability
runs counter to the prevailing view (Wilson  _et al._ , 1999), and the
unexpected destabilizing role of the solvent viscosity is similar to the
recently discovered elastoinertial center-mode instability (Garg  _et al._ ,
2018; Chaudhary  _et al._ , 2021; Khalid  _et al._ , 2020). The regime $W\sim
O(10^{3})$, $\beta>0.99$ corresponds to highly-elastic ultra-dilute polymer
solutions (with concentrations around 1% of the overlap value), one that was
not explored in earlier theoretical work (Wilson  _et al._ , 1999), but has
recently been shown to be experimentally accessible in microscale flows
(Varshney and Steinberg, 2018; Jha and Steinberg, 2020).

Neutral curves demarcate unstable ‘tongues’ in the $W$-$k$ plane (Fig. 2),
with the tongues ceasing to exist beyond a critical $k$ and below a threshold
$W$. However, the instability continues to exist at arbitrarily small $k$,
with $W\sim 1/k$ for $k\ll 1$ along both the branches of the tongue. A plot of
$W_{c}$ (the critical $W$ corresponding to the minimum along a neutral curve)
with $(1-\beta)$ (inset B of Fig. 2) shows that the lowest $W_{c}$ is $\sim
973.8$ for $\beta=0.994$, and that $W_{c}\propto(1-\beta)^{-1}$, $k_{c}\sim
O(1)$ for $\beta\rightarrow 1$; expectedly, $W(1-\beta)k$ is the threshold
parameter for $k,(1-\beta)\ll 1$; see inset A of Fig. 2 and 111The lower and
upper threshold values of $Wk(1-\beta)$ for the instability are captured very
well by a numerical solution of the reduced set of governing equations in this
limit. The instability ceases to exist below $\beta=0.9905$.

Figure 3(a) shows contour plots for the streamwise velocity field alongside
the polymeric stress eigenfunctions in Fig. 3(b) for $W(1-\beta)$’s near the
lower threshold and close to the maximum growth rate (inset A of Fig. 2). 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 close to the
centerline, owing to the phase speed of the eigenmode closely approaching the
base-state maximum. 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 ($\tau_{xz}$)
drives a reinforcing flow, leading to exponential growth. Close to neutrality,
the stress ($\tau_{xx}$ and $\tau_{xz}$) eigenfunctions in particular (panel A
of Fig. 3(b)), are seen to develop singular features in an $O(1/W)$ elastic
critical layer, where the phase speed equals the local laminar velocity, and
this leads to additional rolls close to the centerline. The complicated
eigenfunction structure near the lower threshold, as also evident from the
cusps close to the centerline (see magnified view in Fig. 3(a)), arises from
competing influences of $\tau_{xz}$ (destabilizing, as mentioned above) and
$\tau_{xx}$ (stabilizing). Note that the existence of the singular elastic
critical layer above, while emphasizing a hoop-stress-independent pathway to
instability, also contrasts with the Newtonian scenario where the unstable
inflectional mode approaches a regular neutral mode at the critical $Re$
Drazin and Reid (1981).

(a) Contours of ${v}_{x}$ for $W(1-\beta)=43$ (A) and $80$ (B)

(b) $\tilde{\tau}_{xx}$ and $\tilde{\tau}_{xz}$ for $W(1-\beta)=43$ (A) and
$80$ (B)

Figure 3: Eigenfunctions for the center mode for $W(1-\beta)=43$, $80$ and
$k=0.08$, corresponding to neighborhood of the lower threshold and the center
of the unstable region in the collapsed neutral curves shown in inset (A) of
Fig. 2. The constant-amplitude contours of the streamwise velocity are shown
in subfigure (a), while the eigenfunctions for $\tau_{xz}$ and $\tau_{xx}$ are
shown in subfigure (b). The expanded region near the centerline for panel (A)
in subfigure (a) is also shown.

The results above highlight a hitherto unknown center-mode instability in
inertialess channel flow of an Oldroyd-B fluid for $\beta>0.9905$, whereas in
earlier efforts (Garg  _et al._ , 2018; Chaudhary  _et al._ , 2021; Khalid
_et al._ , 2020) we have identified a center-mode instability for $Re\sim
O(100)$ and $\beta\sim 0.9$, both for viscoelastic channel and pipe flows, and
that may underlie the EIT state in these geometries (Samanta  _et al._ , 2013;
Choueiri  _et al._ , 2018; Srinivas and Kumaran, 2017). The question arises,
naturally, as to whether there is any connection between the elastic center
mode identified here, and the elasto-inertial one identified in Refs. (Garg
_et al._ , 2018; Chaudhary  _et al._ , 2021; Khalid  _et al._ , 2020). Figure
4 shows the critical Reynolds number $Re_{c}$ as a function of $E(1-\beta)$
for different $\beta$, and confirms such a connection. For $\beta\leq 0.9905$,
the elasto-inertial instability (Garg  _et al._ , 2018) exhibits the scaling
$Re_{c}\propto(E(1-\beta))^{-3/2}$ for $E(1-\beta)\ll 1$, reflecting the
simultaneous importance of inertia, elasticity and viscous effects in a
boundary layer near the channel centerline (Garg  _et al._ , 2018), with
$Re_{c}$ increasing sharply beyond a threshold $E(1-\beta)$. In stark
contrast, for $\beta>0.9905$, while $Re_{c}$ initially decreases as
$(E(1-\beta))^{-3/2}$, there is a crossover to $Re\propto(E(1-\beta))^{-1}$
for higher $E(1-\beta)$, with this scaling persisting down to arbitrarily
small $Re$. The $Re\propto E^{-1}$ scaling translates to an independence with
respect to fluid inertia, and indeed corresponds to the creeping-flow
instability identified above, with $W(1-\beta)$ being the threshold parameter.
While a connection between EIT and ET states has been conjectured (Samanta
_et al._ , 2013; Page  _et al._ , 2020; Steinberg, 2021), and the possibility
of a common instability underlying these states speculated upon (Qin  _et al._
, 2019), ours is the first explicit demonstration of the same. Interestingly,
as seen in Fig. 4, an intermediate scaling regime with
$Re_{c}\propto(E(1-\beta))^{-1/2}$ begins to emerge for $\beta>0.99$. This
scaling implies that the flow velocity $V\propto\frac{1}{(1-\beta)}V_{shear}$,
$V_{shear}=\sqrt{\frac{\eta(1-\beta)}{\lambda\rho}}$ being the elastic shear
wave speed. The shear wave signature in this intermediate regime of relatively
low $Re$ and high $E$ could be relevant to the observation of ‘waves’ in
experiments involving the flow of highly elastic polymer solutions (Varshney
and Steinberg, 2018, 2019; Jha and Steinberg, 2020). Continuity considerations
imply that the $\beta$’s where $Re_{c}$ increases sharply, and those where
$Re_{c}$ transitions to an $E^{-1}$ scaling for $E$’s beyond the intermediate
regime, are likely separated by a critical $\beta$ where, remarkably, the
aforesaid shear-wave scaling must persist in the limit $Re\rightarrow 0$,
$E\rightarrow\infty$. The relevance of a shear wave at low $Re$, in the flow
of highly dilute solutions, is intriguing especially in light of the fact that
the class of shear waves originally analyzed in plane shear flow of a UCM
fluid for $Re\ll 1$ (Gorodtsov and Leonov, 1967) are heavily damped for small
but finite $\beta$ (Chaudhary  _et al._ , 2019).

Figure 4:  The variation of critical Reynolds number $Re_{c}$ with
$E(1-\beta)$ for different $\beta$’s for $\beta\geq 0.9$. For $\beta$ up to
$0.9905$, $Re_{c}$ decreases as $Re_{c}\propto(E(1-\beta))^{-3/2}$ for small
$E(1-\beta)$, but increases sharply beyond a critical $E(1-\beta)$. For
$\beta>0.9905$, while $Re_{c}$ initially decreases as
$Re_{c}\propto(E(1-\beta))^{-3/2}$, beyond a critical $E(1-\beta)$, there is a
crossover to the scaling $Re_{c}\propto(E(1-\beta))^{-1}$, with the
instability persisting down to the creeping-flow limit. For
$0.98<\beta<0.9905$, there is an intermediate scaling regime with
$Re_{c}\propto(E(1-\beta))^{-1/2}$.

Traditional research in Newtonian turbulence has been rooted in a statistical
description aiming to explain, for instance, the subtle nature of small-scale
universality at high $Re$, in a homogeneous isotropic setting (Frisch, 1995;
Sreenivasan and Antonia, 1997; Sreenivasan, 1999). The approach at moderate
$Re$, especially from the perspective of understanding transition, has,
however, undergone a paradigm shift with the advent of a dynamical systems
perspective (Barkley, 2016; Budanur  _et al._ , 2017). For the canonical
rectilinear shearing flows, this latter approach is based on the dynamics of
so-called exact coherent structures that account for both an ambient shear and
the presence of boundaries (Waleffe, 1998, 2001; Wedin and Kerswell, 2004;
Willis  _et al._ , 2016; Budanur  _et al._ , 2017). In contrast, theoretical
efforts aimed at understanding the effect of elasticity on, for instance, the
energy cascade in Newtonian turbulence, mirror the statistical approach above
(Balkovsky  _et al._ , 2000, 2001). For elastic turbulence, in particular,
such efforts have predicted a rapid decay, at least as fast as $k^{-3}$ ($k$
being the wavenumber), of the kinetic energy spectrum pointing to the
spatially smooth character of small-scale elastic turbulent fluctuations, an
aspect that has found experimental confirmation (Groisman and Steinberg, 2000,
2001); there remains a dearth of information on the structural front, however.
Recent experimental efforts, although still limited to curvilinear geometries,
have begun to explore this latter aspect (Jun and Steinberg, 2011; Steinberg,
2021); a recent theoretical work has also explored the near-wall boundary
layer structure in the ET state (Belan  _et al._ , 2018).

Our discovery of a linear instability that spans the ET and EIT regimes (Fig.
4) helps significantly expand the above picture, opening up multiple avenues
for future research. A linear elastic instability, with a physical origin
genuinely different from the hoop-stress-based pathway in curvilinear shearing
flows, offers a template for novel nonlinear elastic coherent structures that
could shed light on the large-scale dynamics of the ET state; this would
complement the aforementioned statistical approach tailored to the smallest
scales. Such an approach also offers an alternative to prevailing efforts that
analyze the elastic transition in rectilinear shearing flows, based on a
bifurcation-from-infinity perspective (Meulenbroek  _et al._ , 2004; Morozov
and van Saarloos, 2005, 2007), by expanding about a linearly stable eigenmode.
The physical basis of the nonlinear expansion is intimately tied to the hoop-
stress-based mechanism that leads to a linear instability in the curvilinear
geometries. It is also worth noting that these nonlinear analyses were
restricted to the $\beta\rightarrow 0$ limit (Morozov and van Saarloos, 2005,
2019). As evident from Fig. 1, and the description of the eigenspectrum above
(the stability of the UCM limit in particular), the elastic eigenspectrum is
sensitively dependent on $\beta$, and the validity of a $\beta\rightarrow 0$
analysis, for the experimentally relevant case of dilute solutions with
$\beta\rightarrow 1$, is not obvious. An expansion about, or a numerical
continuation from, the unstable eigenmode reported here offers access to a
much larger region in the viscoelastic parameter space comprising $W$, $Re$,
and $\beta$. The recent finding of a connection between an EIT structure (Page
_et al._ , 2020; Dubief  _et al._ , 2020) and the underlying linear
eigenfunction (Garg  _et al._ , 2018) points to the likely success of a
numerical continuation approach. Finally, the nonlinear coherent structures
might be relevant to a wider range of polymer concentrations and Weissenberg
numbers than the restricted range ($\beta>0.9905$, $W\sim O(1000)$)
corresponding to the actual linear instability.

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