Viscoelastic instabilities of Taylor–Couette flows with
different rotation regimes

Yang Bai, Noureddine Latrache, Fayçal Kelai, Olivier Crumeyrolle, Innocent Mutabazi

► To cite this version:

Yang Bai, Noureddine Latrache, Fayçal Kelai, Olivier Crumeyrolle, Innocent Mutabazi. Viscoelastic instabilities of Taylor–Couette flows with different rotation regimes. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2023, 381 (2246), 10.1098/rsta.2022.0133. hal-05339263

HAL Id: hal-05339263

https://hal.univ-brest.fr/hal-05339263v1

Submitted on 30 Oct 2025

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Research

<figure><img src="image_2.png" /></figure>

Check for
updates

**Cite this article:** Bai Y, Latrache N, Kelai F, Crumeyrolle O, Mutabazi I. 2023 Viscoelastic instabilities of Taylor–Couette flows with different rotation regimes. *Phil. Trans. R. Soc. A* **381**: 20220133.

https://doi.org/10.1098/rsta.2022.0133

Received: 8 August 2022

Accepted: 23 December 2022

One contribution of 16 to a theme issue
‘Taylor–Couette and related flows on the
centennial of Taylor’s seminal *Philosophical
Transactions* paper (part 2)’.

**Subject Areas:**

mechanical engineering, fluid mechanics

**Keywords:**

viscoelastic, instabilities, Taylor–Couette

**Author for correspondence:**

Innocent Mutabazi

e-mail: innocent.mutabazi@univ-lehavre.fr

Viscoelastic instabilities of
Taylor–Couette flows with
different rotation regimes

Yang Bai<fnref n="1" />, Noureddine Latrache<fnref n="2" />, Fayçal Kelai<fnref n="1" />,
Olivier Crumeyrolle<fnref n="1" /> and Innocent Mutabazi<fnref n="1" />

<figure><img src="image_3.png" /></figure>

NL, 0000-0003-1287-8796; IM, 0000-0001-9863-368X

The critical modes of the instabilities of viscoelastic Taylor–Couette flow are investigated using both experiments and linear stability analysis when only one cylinder rotates and the other is fixed. A viscoelastic Rayleigh circulation criterion highlights that the elasticity of the polymer solution can induce a flow instability even if the Newtonian counterpart is stable. When the inner cylinder solely rotates, experimental results show three critical modes: stationary axisymmetric vortices or Taylor vortices for small elasticity, standing waves, also called ribbons for intermediate values of elasticity, and disordered vortices (DV) for large elasticity values. When the outer cylinder rotates and the inner cylinder is fixed and for large values of elasticity, the critical modes appear in the form of DV. There is a good agreement between experimental and theoretical results provided that the elasticity of the polymer solution is accurately determined.

This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal *Philosophical Transactions* paper (part 2)’.

**1. Introduction**

In his seminal study on the stability of the flow between
two concentric cylinders, Taylor [1] has launched intense
activities on the investigation of complex fluids in
the so-called Taylor–Couette (TC) system. Nowadays,

<footnote n="1">¹Normandie Université, UNIHAVRE, Laboratoire Ondes et Milieux Complexes (LOMC), UMR 6294, CNRS, B.P. 540, Le Havre Cedex 76058, France</footnote>

<footnote n="2">²Université de Bretagne Occidentale, IRDL/UBO UMR CNRS 6027, Rue de Kergoat, Brest 29238, France</footnote>
besides the viscosimetric studies of complex fluids, the TC system is used to investigate the study of the ultimate turbulence [2] and the transition to turbulence in flows of suspensions [3] or viscoelastic fluids [4], among many others. The present work presents the results from the study of a viscoelastic flow in the differential rotating cylindrical annulus in two situations: the rotating inner cylinder while the outer cylinder is fixed and the rotating outer cylinder while the inner one is stationary.

In the sheared flows, the addition of a small amount of long-chain polymer molecules to Newtonian liquid can modify the threshold and the nature of the critical modes. Investigation of instabilities in viscoelastic flows has gained much interest in recent decades because of the discovery of new striking phenomena induced by the interplay between inertial and elastic stresses [4–6].

The TC system that consists of a flow confined in the gap between two coaxial differentially rotating cylinders is suitable to investigate the competition between inertial and elastic stresses in fluids. The flow in a viscoelastic TC system with a given radius ratio can be controlled by the rotation ratio $\mu = \Omega_o/\Omega_i$, the Taylor number $Ta = Re\sqrt{d/a}$ for the inertia and flow curvature effects on one hand and by the elasticity number $E = \tau/\tau_v$ on the other hand, and by the viscosity ratio $S = v_p/v$ for the viscoelastic effects. Here, $\Omega_{i,o}$ represent the angular frequencies of the inner and outer cylinders, respectively, $\tau$ is the relaxation time, $\tau_v$ is the viscous diffusion time, $d$ is the gap width, $a$ is the radius of the inner cylinder, $v$ is the solution kinematic viscosity, $v_s$ the solvent kinematic viscosity and $v_p = v - v_s$ represents the polymer contribution to the solution kinematic viscosity. We made the assumption that the densities of the solvent $\rho_s$ and the polymer $\rho_p$ are almost equal i.e. $\rho_s \simeq \rho_p = \rho$. The elasticity number can be written as the ratio between the Weissenberg number $Wi = \tau\dot{\gamma}$ and the Reynolds number $Re = \tau_v\dot{\gamma}$ i.e. $E = Wi/Re$. It characterizes the polymer solution in a flow configuration with a characteristic length $d$ under shear rate $\dot{\gamma}$. The TC system offers a good flow system to investigate viscoelastic instabilities depending on the rotation rate of cylinders. For very small values of the elasticity number $E$, the elastic effects are very weak compared to inertia effects. In this case, experiments with different polymer solutions and rotating only the inner cylinder [6–12] have shown a stabilization of the circular Couette flow (CCF), i.e. the threshold $Ta_c$ of the primary instability mode (Taylor vortex, TV) is increased compared to the Newtonian case. For moderate elasticity, inertial effects are not negligible and the critical state occurs via a supercritical Hopf bifurcation in the form of rotating standing waves, also known as ribbons [5–15]. For high elasticity, the elastic effects are very important compared to inertia effects. Pure elastic instability is observed in the form of disordered vortices (DV) for very small values of $Re$ or $Ta$ [4,6,7,10–12,16–18]. Theoretical analysis of the flow solutions using the Oldroyd-B model [4,19,20] and direct numerical simulations (DNS) of the FENE-P model [21–24] have reproduced some of the flow patterns observed in the experiments. Recent numerical studies have attempted to give a quantitative characterization of the behaviour of global quantities (angular momentum current and kinetic energy) of inertia-elastic turbulence [25] and of the elastic turbulence in the TC viscoelastic flows [26].

The purpose of this work is to detect the critical instability modes in viscoelastic TC flow in the plane ($E, Ta_c$) or ($ES, Ta_c$) using both experiments and the linear stability analysis (LSA) of the Oldroyd-B model of viscoelastic fluid when the inner cylinder rotates and the outer is fixed, or when the inner cylinder is fixed and the outer cylinder rotates, for a given finite radius ratio. By using a viscoelastic Rayleigh circulation criterion [27], we show that the elasticity of the polymer solution can induce a flow instability even if the Newtonian flow is stable. The experimental and theoretical (LSA) results are compared with those obtained in the previous works for both $\mu = 0$ and $\mu = \infty$ regimes.

The paper is organized as follows. In the next section, we first present the general theoretical model (Oldroyd-B model), we describe the base flow state and we present the generalized Rayleigh stability criterion. In §3, we describe the experimental system. Section 4 contains results on linear stability and experimental results for both $\mu = 0$ and $\mu = \infty$ regimes. Section 5 contains the conclusion.
## 2. Theoretical model

A viscoelastic fluid can be obtained by adding a polymer with high molecular mass to a Newtonian solvent. The contribution of polymer to the Newtonian solvent depends on its molecular mass, viscosity of the solvent and polymer concentration. The behaviour of the viscoelastic fluids is described by rheological laws which connect the stress to the rate of strain tensor [28]. There exist many models to describe the viscoelastic fluids; one of the simplest models used to describe viscoelastic fluids with constant viscosity (known as Boger fluids) is the Oldroyd-B model [28]. In this model, the polymer solution has a constant viscosity independent of the shear rate $\dot{\gamma}$. The total stress tensor $T$ is the superposition of the Newtonian solvent stress tensor $T_s$ and the polymeric stress tensor $T_p$: $T = T_s + T_p$.

### (a) Dimensionless Oldroyd-B fluid equations

The mass and momentum conservation for an Oldroyd-B viscoelastic fluid are described in [28]. By introducing $d$ and $U_0$ as characteristic length and characteristic velocity, respectively, in the Oldroyd-B model, the dimensionless equations for viscoelastic fluid read

$$ 
\left.
\begin{aligned}
\nabla \mathbf{U} &= 0, \\
Re \left( \frac{\partial \mathbf{U}}{\partial t} + \mathbf{u} \nabla \mathbf{U} \right) &= -\nabla \Pi + \mathbf{T}_p + (1-S)\nabla^2 \mathbf{U} \\
\mathbf{T}_p + Wi \left[ \frac{\partial \mathbf{T}_p}{\partial t} + \mathbf{u} \nabla \mathbf{T}_p - (\nabla \mathbf{U})_p - \mathbf{T}_p \cdot \nabla \mathbf{U} \right] &= S[\nabla \mathbf{U} + (\nabla \mathbf{U})^T].
\end{aligned}
\right\} \quad (2.1)
$$

In these equations, the pressure and the stress tensor are scaled by $\rho U_0^2$. Here, $\mathbf{U}$ and $\mathbf{T}_p$ represent the velocity field and the polymer stress tensor, respectively, and $\Pi$ is the dimensionless generalized pressure. Equations (2.1) contain the following control parameters:

$$ Re = \frac{dU_0}{\nu}, \quad E = \frac{\tau}{\tau_v} = \frac{Wi}{Re}, \quad ERe = \tau\dot{\gamma} = Wi \quad \text{and} \quad S = \frac{\nu_p}{\nu}, \quad (2.2) $$

where $\dot{\gamma} = U_0/d$ is the characteristic shear rate; $\tau_v$ is the viscous diffusion time. For a cylindrical annulus of inner radius $a$ and outer radius $b$, the characteristic length is the gap width $d = b - a$. The characteristic velocity $U_0$ is the velocity of the cylinders: $U_0 = \Omega_i a$ for the case where the inner cylinder rotates and the outer cylinder is fixed; $U_0 = \Omega_o b$ for the opposite case where the inner cylinder is fixed and the outer cylinder rotates.

### (b) Viscoelastic Couette flow elasto-rotational Rayleigh discriminant

In a TC system with a large aspect ratio ($\Gamma \gg 1$), the end effects can be neglected and the laminar flow known as the CCF has only the azimuthal velocity component: $\mathbf{U} = V(r)e_{\theta}$. The stress tensor $\mathbf{T}_p$ reduces to three non-vanishing components: $T_{r\theta} = T_{\theta r}$ and $T_{\theta\theta}$. The dimensionless equations of the viscoelastic CCF are:

$$ \frac{d\Pi}{dr} = \frac{V^2}{r} + \frac{T_{\theta\theta}}{r} \quad \text{and} \quad 0 = \frac{1}{r^2} \frac{d(rT_{r\theta})}{dr}, \quad T_{r\theta} = S\dot{\gamma}, \quad T_{\theta\theta} = 2WiS\dot{\gamma}^2, \quad (2.3) $$

where $\dot{\gamma}_{r\theta} = d(V/r)/dr$ is the local shear rate. The radial pressure gradient is balanced by the centrifugal force supplemented by the first normal stress difference $N_1 = T_{\theta\theta}$ since $T_{rr} = 0$. The velocity profile of the CCF is given in [4,27]

$$ V(r) = Ar + \frac{B}{r}, \quad A = \frac{\mu - \eta^2}{\eta(1 + \eta)} \quad \text{and} \quad B = \frac{(1 - \mu)\eta}{(1 - \eta)(1 - \eta^2)}, \quad (2.4) $$
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Rotational Rayleigh discriminant $\Phi_{cent}$ for Newtonian fluids ($E = 0$), where $x = (r - a)/d$: red discontinuous line for $\mu = 0$ and red solid line for $\mu = \infty$ regime. Elasto-rotational Rayleigh discriminant $\Phi_{er}$ for viscoelastic fluids $ES = 1$: blue pointed line for $\mu = 0$ and blue solid line for $\mu = \infty$ regime. The line $\Phi = 0$ separates the potentially stable zone with positive discriminant ($\Phi > 0$) from the potentially unstable zone with negative discriminant ($\Phi < 0$). (Online version in colour.)</figcaption></figure>

where $\eta = a/b$ is the radius ratio. The shear stress component and the first normal stress difference in the viscoelastic CCF are given by

$$T_{r\theta} = -\frac{2B}{r^2} \quad \text{and} \quad N_1 = T_{\theta\theta} = \frac{8WiS B^2}{r^4}, \qquad (2.5)$$

while the second normal stress difference vanishes identically in the Oldroyd-B model: $N_2 = 0$.

In a viscoelastic Couette flow, there are two radial forces that can drive perturbations from the equilibrium and lead to instability when they sum each other: the centrifugal force and the first normal stress difference. To investigate the stability of viscoelastic solutions, Bai et al. [27] have derived the generalized Rayleigh circulation criterion to include the elasticity effects. The resulting elasto-rotational Rayleigh discriminant reads [27,29]

$$\Phi_{er}(r) = \frac{4(\eta^2 - \mu)}{(\eta^2 - 1)^2} \left[ (\eta^2 - \mu) + (\mu - 1) + \left(\frac{\eta}{1-\eta}\right)^2 \frac{1}{r^2} \right] - \frac{32ES(\mu-1)^2}{(\eta^2-1)^2} \left(\frac{\eta}{1-\eta}\right)^4 \frac{1}{r^6}, \qquad (2.6)$$

where $E_p = ES$ represents the polymer contribution to the elasticity of the solution, which can be considered as the elasticity number for pure melt polymer. The particular case $ES = 0$ corresponds to the Newtonian flow [30]: $\Phi_{er} = \Phi_{cent}$. The flow is stable if $\Phi_{er} > 0$ and potentially unstable if $\Phi_{er} < 0$. In figure 1, we have plotted the rotation Rayleigh discriminant $\Phi_{cent}$ and the elasto-rotational Rayleigh discriminant $\Phi_{er}$ for a solution for $ES = 1$ with two rotation regimes: $\mu = 0$ and $\mu = \infty$. When $\mu = 0$, $\Phi_{er} < \Phi_{cent} < 0$, the flow of viscoelastic fluid is more unstable than that of the Newtonian fluid. When the inner cylinder is fixed and the outer cylinder rotates ($\mu = \infty$), the elasto-rotational Rayleigh discriminant is negative ($\Phi_{er} < 0$) in the inner part of the flow for $ES = 1$, thus the elasticity of the polymer solution can induce flow instability even if its Newtonian counterpart is stable ($\Phi_{cent} > 0$). In the case of the stationary inner cylinder and rotating outer cylinder, the epicyclic frequency $\omega_{ep}(x) = \sqrt{\Phi(r)}$ exists in the whole gap for the viscous flow while it occurs only in the zone close to the outer cylinder for viscoelastic flows.

### (c) Linear stability analysis

The elasto-rotational Rayleigh discriminant allows us to predict the potential instability in a TC system for inviscid fluids. The LSA of the flow permits us to determine the threshold properties of
the first instability of a viscoelastic laminar flow. Infinitesimally small perturbations are added to the base flow and the resulting field is substituted in to the dimensionless flow equations (2.1). The resulting equations are linearized about the base state (equation (2.4)). As the base state depends only on the radial coordinate and it is axisymmetric, i.e. it is invariant along the cylinder axis and around the azimuthal direction, the small perturbations can be expanded in normal modes of the form of $[st + i(m\theta + qz)]$. Here, $s = \sigma + i\omega$, in which $\sigma$ is the temporal growth rate and $\omega$ is the angular frequency of the perturbations; $m$ and $q$ are azimuthal number of modes and axial wavenumber, respectively. Thus the perturbed flow variables are written as follows:

$$ 
\left. 
\begin{aligned}
\mathbf{U}(r) &= \begin{pmatrix} 0 \\ V(r) \\ w(r) \end{pmatrix} + \begin{pmatrix} u(r) \\ v(r) \\ w(r) \end{pmatrix} e^{[st + i(m\theta + qz)]} \\
\Pi(\mathbf{r}) &= \Pi(r) + p(r) e^{[st + i(m\theta + qz)]} \\
\mathbf{T}_p(\mathbf{r}) &= \mathbf{T}_p(r) + \mathbf{t}_p(r) e^{[st + i(m\theta + qz)]},
\end{aligned}
\right\} \qquad (2.7)
$$

where

$$ \mathbf{T}_p(r) = \begin{pmatrix} 0 & T_{r\theta}(r) & 0 \\ T_{r\theta}(r) & T_{\theta\theta}(r) & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \text{and} \quad \mathbf{t}_p(r) = \begin{pmatrix} t_{rr}(r) & t_{r\theta}(r) & t_{rz}(r) \\ t_{r\theta}(r) & t_{\theta\theta}(r) & t_{\theta z}(r) \\ t_{rz}(r) & t_{\theta z}(r) & t_{zz}(r) \end{pmatrix}. \qquad (2.8) $$

Substitution of expressions (2.7) into the linearized system of equations of perturbed flow and the subtraction of the base flow yield the following system of equations for the structure functions (i.e. radial dependence) of perturbations:

$$
\left.
\begin{aligned}
& Du + \frac{u}{r} + \frac{imv}{r} + iqw = 0, \\
& Re [(s + im\Omega)u - 2\Omega v] = -Dp + \left(\frac{t_{rr}}{r} + \frac{imt_{r\theta}}{r} + iqtrz - \frac{t_{\theta\theta}}{r}\right) \\
& \qquad + (1-S) \left[ \left(D^2 - q^2 - \frac{1+m^2}{r^2} + \frac{D}{r}\right)u - \frac{2imv}{r^2} \right], \\
& Re [(s + im\Omega)v + (2\Omega + rD\Omega)u] = -\frac{imp}{r} + \left(\frac{2t_{r\theta}}{r} + \frac{imt_{\theta\theta}}{r} + iqt_{\theta z}\right) \\
& \qquad + (1-S) \left[ \left(D^2 - q^2 - \frac{1+m^2}{r^2} + \frac{D}{r}\right)v + \frac{2imu}{r^2} \right], \\
& Re(s + im\Omega)w = -ikp + \left(\frac{t_{rz}}{r} + Dt_{rz} + \frac{imt_{\theta z}}{r} + iqt_{zz}\right) + (1-S) \left[ \left(D^2 - q^2 - \frac{m^2}{r^2} + \frac{D}{r}\right)w \right], \\
& [1 + Wi(s + im\Omega)]t_{rr} - Wi \frac{2imT_{r\theta}u}{r} = 2SDu, \\
& [1 + Wi(s + im\Omega)]t_{r\theta} + Wi \left[ DT_{r\theta}u - rD\Omega t_{rr} - T_{r\theta}Du - \frac{imT_{\theta\theta}u}{r} - \frac{imT_{r\theta}v}{r} - \frac{uT_{r\theta}}{r} \right] \\
& = S \left( Dv + \frac{imu}{r} - \frac{v}{r} \right), \\
& [1 + Wi(s + im\Omega)]t_{\theta\theta} - Wi \frac{imT_{r\theta}w}{r} = S(Dw + iqu), \\
& [1 + Wi(s + im\Omega)]t_{\theta\theta} + Wi[DT_{\theta\theta}u - 2rD\Omega t_{r\theta} \\
& \qquad + \frac{2T_{r\theta}v}{r} - 2T_{r\theta}Dv - \frac{2imT_{\theta\theta}v}{r} - \frac{2T_{r\theta}u}{r}] = 2S \left( \frac{u}{r} + \frac{imv}{r} \right), \\
& [1 + Wi(s + im\Omega)]t_{\theta z} + Wi \left[ -rD\Omega t_{rz} - T_{r\theta}Dw - \frac{imT_{\theta\theta}w}{r} \right] = S \left( \frac{imw}{r} + iqv \right), \\
& [1 + Wi(s + im\Omega)]t_{zz} = 2S iqw,
\end{aligned}
\right\} \qquad (2.9)
$$
where $\Omega = V/r$ is a function of $r$. The boundary conditions require that the perturbations vanish at the cylindrical surfaces $r = \eta/(1 - \eta)$ and $r = 1/(1 - \eta) : u = Du = v = w = 0$ where $D = d/dr$.

The linearized system can be written in the form [27]

$$AX = sBX, \quad (2.10)$$

where $X = (u, v, w, t_{rr}, t_{r\theta}, t_{rz}, t_{\theta\theta}, t_{\theta z}, t_{zz})^T$ is the vector of the structure functions of the perturbations. A and B are square matrices. This eigenvalue problem is solved by the Chebyshev collocation method described in more detail in [27]. The vector X is projected on a discrete Chebyshev series and then transformed into a mesh vector $S_r$ which is concentrated on the boundaries of the domain. To ensure a good convergence of results with a reasonable computation time, after several tests, we have chosen the number of mesh grids to be $N = 100$. The numerical code was validated by the comparison with existing results for the same sets of parameters [19,31,32]

### 3. Experimental set-up

The experiments were conducted in a TC system that consists of two vertical coaxial cylinders with the length $L = 45$ cm. The inner cylinder is made of plexiglas with the radius $a = 4$ cm and the outer cylinder is made of Delrin with the radius $b = 5$ cm. So the gap width of our TC system $d = 1$ cm, the radius ratio $\eta = a/b = 0.8$ and the aspect ratio $\Gamma = L/d = 45$. Each cylinder is connected to a step motor so that the rotation of the two cylinders can be controlled separately. The rotational velocity is limited to 200 r.p.m. in order to avoid the system vibration. The end plates of the top and the bottom rotate together with the outer cylinder. As we have used cylinders of large aspect ratio $\Gamma$, the influence of the end effects on the flow in the large part of the flow can be ignored as long as we are limited to low values of the rotation rates. The TC system is immersed in a water tank of $30 \times 30 \times 45$ cm connected by tubes to a thermostatic bath for temperature control. Temperature tests show that the working temperature is $20^{\circ}C \pm 0.4^{\circ}C$. The whole working environment is the same as the one used in [27,29] and it is illustrated in figure 2. The experiments were conducted by increasing step by step, the rotation frequency of the inner cylinder for $\mu = 0$ and of the outer cylinder for $\mu = \infty$ and then waiting at least 20 min at each step before recording the flow. The ramping rate of the angular frequency of each cylinder was chosen such that $dT_a/dt < 0.1$. The increase of the rotation velocity was done until a consistent pattern was formed in the flow. A ramping up and a ramping down were also performed to check the supercritical or subcritical nature of the transition.

The working fluid used in our experiments is the polymer solution of Polyoxyethylene (PEO) dissolved in a mixture of Polyoxyethylene (PEG) and water. The PEO and the PEG are both neutral polymers with flexible linear chains of similar chemical properties, i.e. $(-CH_2-O-CH_2-)_n$, and they differ only by the degree of polymerization $n: n_{POE} \sim 1.8 \times 10^5$ with an average molar mass of $8 \times 10^6$ g mol$^{-1}$ while $n_{PEG} \sim 2 \times 10^4$ with a molar mass of $2 \times 10^4$ g mol$^{-1}$. In the final solutions, PEO works as the source of elasticity and viscosity while PEG contributes only to the viscosity. However, the interaction between PEO and PEG affects also the final elasticity of the solution. The mixture of water and PEG is considered as the solvent of the PEO as it has constant viscosity independent of the shear rate. In order to get large values of the elasticity $E$, we use 1000 ppm of PEO for all solutions, which is the maximum dissolvable quantity of PEO. The concentration of the PEG is changed from 2.5% to 25% by weight for different solutions. The variation of the solution elasticity with the PEG concentration is shown in figure 3a. We have also added 2.5% of isopropyl alcohol to facilitate the dissolution of PEO and its stabilization. The shear viscosity was measured with a cone-plane cell of the shear stress rheometer from Anton Paar MC300.

For visualization, we have added 2% aqueous Kalliroscope suspension to the working solutions. The flow patterns can be observed with light spots while their cross-section in the meridional plane $(r, z)$ is recorded by a CCD camera supported by the streampix software and resulting films are analysed with MATLAB. We recorded at regular time intervals the intensity
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Experimental Taylor–Couette system and its working environment: optical system (with laser, CCD camera, spot light) and temperature control system (water tank, water circulation through the inner cylinder, thermostatic bath). (Online version in colour.)</figcaption></figure>

distribution of the light reflected by Kalliroscope flakes along the cross-section of 35 cm of the height in the central part of the annulus. The intensity was sampled into 256 grey levels and the recorded lines were superimposed along the time axis to form a space–time diagram I(z, t).

Rheology measurements have shown that such a concentration of aqueous Kalliroscope suspension does not affect the viscosity of the working solution. We have tested the validity of the Oldroyd B model for our working solutions by measuring their viscosity for different values of the shear rate. The solution shows a weak shear-thinning over three decades, which is due to the large amount of PEO; the maximum shear is $\dot{\gamma} = 10 \text{ s}^{-1}$. The variation of the viscosity over this range of shear rates is below 10%. Reported results were obtained from experiments designed to determine the critical states so that for each experiment, the range of the variation of the viscosity is narrow and the decrease of the viscosity is below 5% so we estimate that the Oldroyd-B model can be applied to our working solutions.

Relaxation times have been determined with three different methods. The relaxation time $\tau_{N1} = N_1/2\nu_p\dot{\gamma}^2$ was determined from the data of the first normal stress difference $N_1$ [8]. The extensional relaxation $\tau_e$ was measured by the extensional rheometer CABER in which the diameter of polymer filaments between two plates is fitted by the exponential law $D(t)/D_0 = Ce^{-t/3\tau_e}$, where $C = 2(GD_0/4\sigma)^{1/3}$, $D_0$ is the filament diameter at the cessation of the stretching, $G$ and $\sigma$ represent, respectively, the elastic modulus and the surface tension of the filament [33]. The molar mass relaxation time $\tau_m$ was determined from the formula [28]: $\tau_m = \rho\nu_p/nRT$, where $n$ is the mole number of the PEO per unit volume of the solution, $R$ is the gas constant and $T$ is the temperature. For all tested polymer solutions, we observed that the shear relaxation time is the smallest, i.e. $\tau_{N1} < \tau_m < \tau_e$. These relaxation times allowed us to define three elasticity numbers: $E_e = \tau_e/\tau_v$, $E_m = \tau_m/\tau_v$ and $E_{N1}/\tau_v$. The shear viscosity of the different polymer solutions was measured in the interval of shear rate $\dot{\gamma} \in [10^{-1}, 10^3] \text{ s}^{-1}$. The variation of the elasticity solution with the PEG concentration is shown in figure 3b. The increase of the elasticity with PEG for a
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) Example of measured viscosity of the polymer solution (1000 ppm of PEO and 25% of PEG), (b) variation of the elasticities *E* with the PEG concentration. (Online version in colour.)</figcaption></figure>

fixed concentration of PEO (1000 ppm) illustrates the effects of interaction of PEO and PEG in the
final value of the solution elasticity.

**4. Results**

In this section, we present the results from LSA and from experiments for two cases: (a) the outer
cylinder is fixed and only the inner is rotating (μ = 0) and (b) the inner cylinder is fixed and the
outer cylinder is rotating (μ = ∞). For Newtonian flows, the first case is centrifugally unstable
and the first instability mode is the stationary TV flow while the second one is centrifugally stable.
Thus, we can determine the effects induced by the viscoelasticity on these two well-documented
flows.

(a) Critical flows for μ = 0 regime

We performed the LSA for different values of *E* and *S* which are accessible in our experiments. From marginal stability curves we have extracted the critical parameters (*Ta*<sub>c</sub>, *q*<sub>c</sub>, *m*<sub>c</sub>, *ω*<sub>c</sub>) and analysed their dependence on *E* and *S*. We have fixed the viscosity ratio *S* and have computed the critical parameters *Ta*<sub>c</sub> and *q*<sub>c</sub> for different values of *m*. The state diagram *Ta*<sub>c</sub>(*E*) (figure 4a) for
$S = 0.6$ shows that the most unstable mode is stationary and axisymmetric ($\omega_c = 0, m = 0$), i.e. we retrieve the Taylor vortices (TVF) at $E < 0.03$ (figure 4a), and oscillatory non-axisymmetric modes ($m = 1, \omega \neq 0$) at large $E > 0.03$. For $0.03 < E < 0.38$, the critical mode is prograde ($\omega_c < 0$). The cross-section of the azimuthal vorticity shows that the non-axisymmetric vortices have different sizes in contrast to the stationary case. For $E > 0.38$, the mode is retrograde ($\omega_c > 0$), it consists of the vortices of equal size inclined in the $r,z$ plane. The colour maps of vorticity in figure 4a show flow patterns of different critical modes. The variations of the wavenumber and frequency of the critical modes are presented in figure 4b,c, respectively. The axial wavenumber increases with the solution elasticity. The oscillatory modes occur in solutions with $E > 0.03$ and their frequency increases with $E$. In solutions with $E > 0.38$, the variation of frequency with $E$ saturates to $\omega_c = 0.02$ for $E > 1$.

In figure 5, we present one-by-one the gap view and the front view, and the space-time diagram (figure 6) of the three critical modes observed in the experiments: stationary axisymmetric vortices or TV for small elasticity $E$, standing waves also called ribbons for intermediate values of $E$ and DV for large values of $E$. In figures 5 and 6 are shown only the central parts of the flow patterns away from the end plates. The vertical shadows in the front view are nothing but a reflection of a support pillar of the set-up. The gap view is captured by the reflection of a laser sheet which sweeps the ($r, z$) plane. The space-time diagram is created by superposition of recorded vertical lines at regular time intervals. The TV mode has the same structure as in the Newtonian case (figures 5 and 6a) with $m_c = 0, \omega_c = 0$. This agrees with the LSA which predicted axisymmetric and stationary modes for small values of ES. The Ribbons mode is composed of non-axisymmetric ($m_c \neq 0, \omega_c \neq 0$) counter propagating waves (figures 5b and 6b). The ribbon patterns contain some spatio-temporal defects, the density of which increases with $Ta$. The counter propagating waves of the ribbon mode can be obtained by complex demodulation [5,15]. The LSA has predicted critical mode of ($m_c = 1, \omega_c < 0$) for intermediate ES. Although it was difficult to determine the azimuthal wavenumber $m_c$ and the sign of $\omega_c$ experimentally, we may say the non-axisymmetric and oscillatory mode agrees with the LSA prediction for intermediate ES.

Unlike the ribbons mode, the mode of DV is composed of non-regular and non-stationary waves (figures 5c and 6c) ($m_c \neq 0, \omega_c \neq 0$). The fine black lines in front view of the cylinder (figure 5c-2) are the centres of counter rotating pairs of vortices. These central lines of the counter rotating vortices are inclined, meaning that this mode is not axisymmetric. The space-time diagram (figure 6c) exhibits the chaotic behaviour of the pattern. We compare the long-time acquisition of space-time diagrams of the three modes in figure 6 to provide their general spatio-temporal behaviour. The central lines of the counter rotating vortices pairs tend to approach each other and merge into new vortices, while new vortices appear randomly from the left. Thus, the disordered waves mode is composed of non-axisymmetric and unsteady vortices which are created and vanish randomly.

To compare the experimental results to those obtained by LSA, we use the three different relaxation times ($\tau_e, \tau_m, \tau_{N1}$) to define the experimental values of the solution elasticity $E$. We plotted in figure 7 the experimental critical Taylor number $Ta_c$ against three values of ES corresponding to different relaxation times. The critical curves $Ta_c(ES)$ from LSA lie between the data corresponding to the elasticity defined with molar mass and the extensional relaxation time (figure 7). The curves of the threshold predicted by LSA are plotted in figure 7 for values of ES ranging from 0.005 to 10. The experimental values of $Ta_c$ decrease with ES in the same way as the theoretical values of $Ta_c$ from LSA: they decrease slightly at small ES corresponding to the stationary TV then decrease quickly at large ES for the ribbons and disordered waves. At larger ES, the LSA critical curves are closer to those with elasticity defined with the extensional time.

In our experiments, the TV mode is only observed for solutions with low elasticity (ES < 0.001). Ribons, i.e. standing waves, have been observed in the solutions with $ES \in [0.001, 0.035]$ and for solutions with ES > 0.035, we have observed the DV mode. Compared to the purely elastic mode first observed by Larson *et al.* [4], the DV mode has the same flow pattern as the purely elastic mode. However, the purely elastic mode is observed for fluids of very large $E$ and at negligible $Ta_c$. The same flow pattern corresponding to the purely elastic mode were also observed for fluid
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 4.** (a) Variation of the instability threshold with the elasticity number *E* and vorticity *ω*θ of the critical modes for a solution with *S* = 0.6, *μ* = 0, (b) *q*<sub>c</sub>(*E*); (c) *ω*<sub>c</sub>(*E*). (Online version in colour.)</figcaption></figure>

of *E* = 15, *S* = 0.13 at *Ta*<sub>c</sub> = 0.056 by Baumert & Muller [11] in a TC system of *η* = 0.827. In the same system of *η* = 0.827, Baumert & Muller [11] have also observed stationary counter-rotating vortices for less elastic fluid (*E* = 0.0562, *S* = 0.13) at *Ta*<sub>c</sub> = 15.5. In a system of *η* = 0.829, Groisman & Steinberg [6] have observed the TV mode for solutions with *E* < 0.08 and disordered oscillations mode for 0.08 < *E* < 30 when increasing the rotation velocity. The critical Weissenberg number for the first transition is comparable to our experimental results. Results in figure 7 show some discrepancies due to the difficulty in the identification of the right parameter that best controls
<figure><img src="image_10.png" /><figcaption>**Figure 5.** Pictures (front view and cross-section) of the critical modes observed in viscoelastic solutions for the $\mu = 0$ regime: (a) stationary vortices $Ta_c = 28.8$, $E_m = 0.011$, $S = 0.87$, (b) ribbons $Ta_c = 28.4$, $E = 0.0168$, $S = 0.81$, (c) disordered vortices $Ta_c = 15.4$, $E_m = 0.131$, $S = 0.61$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 6.** Space-time diagram of 600 s for three different modes: (a) Taylor vortex mode observed in a solution of 1000 ppm PEO and 2.58% PEG ($Ta_c = 28.8$, $E_m = 0.011$, $S = 0.87$) in the $\mu = 0$ regime, (b) ribbons at ($Ta_c = 28.4$, $E_m = 0.0168$, $S = 0.81$) in the $\mu = 0$ regime for a solution of 1000 ppm PEO and 5% PEG, (c) disordered vortices at $Ta_c = 12.1$, $E_m = 0.131$, $S = 0.61$) in the $\mu = 0$ regime for a solution of 1000 ppm PEO and 15%.</figcaption></figure>

the dynamics of the polymer solution: E, S or ES. The $Ta_c$ of the less elastic fluid of Baumert is found to be smaller than our results and far away from the LSA critical curves. This may be due to the sensitivity of the flow stability to the radius ratio $\eta$ [17].
<figure><img src="image_12.png" /><figcaption>**Figure 7.** Critical values on the regime of $\mu = 0$. Continuous lines-LSA critical curves. Solid points-Kelai's experiments with *E* defined using the relaxation time defined by the molar mass [34]. Open points: Bai experiments with *E* defined with extensional relaxation time [35]. Violet triangle-Baumert's experiment [11]. (Online version in colour.)</figcaption></figure>

### (b) Critical flows for $\mu = \infty$ regime

When we rotate the outer cylinder while fixing the inner cylinder, the fluid in the outer part rotates faster than the inner part and the centrifugal force is no longer destabilizing, only the elastic force may destabilize the flow according to the elasto-rotational Rayleigh discriminant. The LSA of the base flow in this regime has shown that the most unstable mode is an oscillatory ($\omega_c \neq 0$) but always axisymmetric with $m_c = 0$. We observe that the critical curves of $Ta_c(E)$ for different $S \in [0.5, 0.8]$ are very close each other (figure 8a), which means that for the $\mu = \infty$ regime, the viscosity ratio $S$ has very weak effects on the viscoelastic instability. Indeed, as the centrifugal force is no longer destabilizing in the $\mu = \infty$ regime, the only destabilizing force is the elastic force which is independent of the polymer viscosity ratio $S$. The critical curve can be fitted by a power law $Ta_c(E) = AE^{-\alpha}$ with $\alpha = 1.048$, $A = 10.295$, and the coefficient of determination $R^2 = 0.9996$. To provide a general view of the critical modes in the $\mu = \infty$ regime, we plot in figure 8b the azimuthal vorticity of the flow patterns for solution of $S = 0.7$, $E = 1$, $Ta_c = 4.73$ with $m_c = 0$, $q_c = 14.4$, $\omega_c = 0.050$. We observe that the vortices of the critical mode are located near the inner cylinder in agreement with the prediction of the elastorotational Rayleigh discriminant (figure 1).

The variation of the wavenumber and of the frequency of critical modes with the elasticity are given in figure 8c,d, respectively, for four values of the viscosity ratio $S$. The critical wavenumber decreases with the solution elasticity $E$ while the frequency increases and saturates to values that depend on the viscosity ratio $S$. In experiments, we have observed only one critical mode in the form of the DV which have the same dynamics as those observed for $\mu = 0$. We present in figure 9 the gap view, the front view and the space-time diagrams of the critical DV mode. So we found that the viscosity of the viscoelastic solution does not intervene in the threshold of the elastic instability but it modifies the wavenumber and the frequency of the waves forming the critical patterns.

To compare the experimental results to the LSA predictions, we resume the critical values of $Ta_c(E)$ measured experimentally by three different relaxation times together with the LSA critical curves in figure 10. We use $E$ instead of $ES$ as the control parameter because the LSA critical curves of different $S$ are close to each other on $Ta_c(E)$ (figure 10). In our experiments, the viscosity ratio $S$ of the solutions that can be experimentally destabilized in the $\mu = \infty$ regime lies in the range
<figure><img src="image_13.png" /><figcaption>**Figure 8.** (a) Variation of threshold $Ta_c$ with the elasticity number $E$ for critical curves, $Ta_c(E)$ for different $S$ in the $\mu = \infty$ regime, (b) vorticity field of the critical mode for $S = 0.7$, $E = 1$: $Ta_c = 4.73$, $m = 0$, $q_c = 14.4$, $\omega_c = 0.050$, (c) axial wavenumber $q_c$, (d) angular frequency $\omega_c$. (Online version in colour.)</figcaption></figure>

of $S \in [0.5, 0.7]$. We observe that the experimental critical values $Ta_c$ decrease with $E$ defined by three relaxation times. This follows the trend of the LSA critical curves. The points corresponding to the elasticity $E_e$ defined by the extensional relaxation time fit perfectly with the LSA critical curves. In rotating the outer cylinder while fixing the inner cylinder, Baumert & Muller [12] and Kelai [34] have observed the purely elastic mode. The experimental elasticity for all the authors is estimated by the molar mass. We can conclude that the critical values of all these experiments show good coherence between each other and parallel to the LSA critical curves.

## 5. Discussion

The present study provides new insight into the critical modes of the viscoelastic TC flows. It is complementary to previous investigations made since the seminal paper of Larson *et al.* [4] which has shed light on the occurrence of a purely elastic instability for very low values of $Ta$. Their studies were made in a small gap TC cell and they considered only axisymmetric modes. In a later work [17], they included the effects of the gap and showed that the threshold of the instability induced by axisymmetric perturbations increases with the radius ratio. Other experimental works
<figure><img src="image_14.png" /><figcaption>**Figure 9.** Critical mode of disordered vortices for $\mu = \infty$ with a solution of 1000 ppm PEO and 15% PEG ($E_m = 0.131$, $S = 0.61$): $Ta_c = 15.4$: (a) front view, (b) gap view, (c) space-time diagram.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 10.** Critical values of the transition from laminar flow to unstable flow on regime of $\mu = \infty$. Solid lines-LSA critical curves for different $S$. Solid points-experimental transient values, three colours represent three different definitions of elasticity. Within each definition of elasticity, each point represents one solution tested. (Online version in colour.)</figcaption></figure>

[6,7,10,36,37] have extended the investigation of viscoelastic instabilities in the TC system with various polymer solutions. Theoretical and numerical works have also been performed for the TC and the Dean configurations [18] and non-axisymmetric patterns have been analysed [38].

We have revisited the LSA of the viscoelastic flow in the TC system with a finite radius ratio ($\eta = 0.8$) in order to determine the variation of the instability threshold with the properties of the viscoelatic solution (viscosity ratio $S$ and elasticity $E$) over a wide range of elasticity values from $E = 0$ to $E = 10$. The analyses have taken into account the non-axisymmetric modes ($m \neq 0$) and the effect of viscosity ratio $S$.

We have tested the hysteretic behaviour of the transition from laminar flow to flow patterns by slowly increasing the rotation rate above the threshold and then decreasing it below to detect the
rotation rate at which the pattern disappears. We found that the TV mode and the ribbons mode have no hysteresis and thus they occur via a supercritical bifurcation while the transition to DV modes has a hysteresis and thus it occurs via a subcritical bifurcation.

In particular, for the case when the inner cylinder is rotating, we have determined two codimension-two points where the stationary modes coexist with the time oscillatory modes ($E = 0.03, Ta_c = 37$ and $E = 0.38, Ta_c = 13.5$). The coexistence of patterns near codimension two points in the TC flow and the weakly nonlinear behaviour of the patterns near the threshold have been well documented [39]. Indeed, these authors have given a detailed description of the stationary TV mode, the spiral vortex mode and the ribbon mode like those observed in the present work although with a viscoelastic fluid. The stationary mode and the oscillatory modes near the codimension two point ($E = 0.03, Ta_c = 37$) can be represented as follows:

$$A(t, z) e^{iqz}, B(t, z) e^{i(\omega t - qz - m\theta)}, C(t, z) e^{i(\omega t + qz - m\theta)}, \quad (5.1)$$

where $A(t, z)$ is the amplitude of the stationary mode and $B(t, z)$ and $C(t, z)$ are the amplitudes of the oscillatory modes ($B$ for right travelling mode and $C$ for left travelling mode). We can write the following system of coupled Ginzburg-Landau equations near the codimension two point for the amplitudes $A, B$ and $C$:

$$\left. \begin{aligned} \frac{\partial A}{\partial t} &= \epsilon A + \xi_{0A}^2 \frac{\partial^2 A}{\partial z^2} - l_A \left[ |A|^2 + \delta \left( |B|^2 + |C|^2 \right) \right] A \\ \frac{\partial B}{\partial t} - c \frac{\partial B}{\partial z} &= \epsilon B + \xi_{0B}^2 (1 + ic_1) \frac{\partial^2 B}{\partial z^2} \\ & \quad - l_B (1 + ic_2) (|B|^2 + |C|^2 + \delta |A|^2) B \\ \frac{\partial C}{\partial t} + c \frac{\partial C}{\partial z} &= \epsilon C + \xi_{0C}^2 (1 + ic_1) \frac{\partial^2 C}{\partial z^2} \\ & \quad - l_B (1 + ic_2) (|C|^2 + |B|^2 + \delta |A|^2) C, \end{aligned} \right\} \quad (5.2)$$

and

where $\epsilon = (Ta - Ta_c)/Ta_c, \xi_{0A}, \xi_{0B}, \xi_{0C}$ represent the coherence lengths of the modes $A, B$ and $C, l_A$ and $l_B$ are the Landau saturation constants, $c$ is the group velocity, $c_1$ and $c_2$ are the dispersion constants and $\delta$ is the coupling between stationary mode $A$ and the oscillatory modes $B$ and $C$. We have assumed that the waves forming the ribbons have equal amplitudes and the same magnitude of the group velocities, the same coherence lengths ($\xi_{0B} = \xi_{0C}$), the same dispersion constants and the same nonlinear saturation constants ($l_C = l_B$). Away from the codimension two point, the amplitude of the stationary pattern is described by the first equation of (5.2) and that of the ribbon pattern is described by the second and third equations of (5.2) in which $\delta = 0$. Such a system of equations has been subject of many studies [40,41]. Near the second codimension two point ($E = 0.38, Ta_c = 13.5$), the description with the amplitude equations would be cumbersome because of the subcritical nature of the bifurcation, but more importantly, because the marginal stability curves are very flat near their minima, and there is a large band of wavenumbers that are almost critical at the same time. Moreover, the modes $m = 1$ and $m = 2$ are very close each other. Thus, it is the interaction between all these almost critical modes with different axial and azimuthal wavenumbers that leads to the disorder pattern observed in experiments.

A discussion using the symmetries of the viscoelastic TC flow has been developed by Avgousti & Beris [19].

The dynamics of viscoelastic flows away from the threshold should preferably be described by DNS using more elaborated models. Indeed, recent numerical simulations of the viscoelastic fluid using the FENE-P constitutive equation for the polymer contribution, have reproduced some results on inertia-elastic modes and elastic turbulence observed in the experiments [23,25,42,43]. New recent experiments have been performed to investigate inertia-elastic transitions in Taylor-Couette flows [43].

In the experiments, we used three types of relaxation times to find the elasticity number which allow the best comparison between experimental and theoretical results. The elongational
relaxation time better describes the experimental observations for the DV mode because the elastic instability is mainly driven by the radial stretching in the vortex structure [4,6] due to the competition between the elastic force and the centrifugal force. Liu & Khomami [23] have analysed the pdf of the radial velocity component for the inertia-elastic turbulence and found that the main driving forces for the radial fluid motion are distributed differently: the elastic force dominates near the outer cylinder when the centrifugal force almost vanishes while the centrifugal force is more active than the elastic force near the inner cylinder.

## 6. Conclusion

The stability of viscoelastic CCF has been revisited both with LSA and experiments for rotating solely either the inner or the outer cylinder. The generalization of the Rayleigh discriminant to include elasticity effects has allowed us to predict the destabilization of viscoelastic CCF even when their Newtonian counterparts are stable. The elasticity of solutions enhances the centrifugal instability and can induce the purely elasticity-induced instability when the flow is centrifugally stable. The results from the LSA agree qualitatively well with those from experiments performed near the threshold. When only the inner cylinder solely rotates, three types of modes have been detected: TV flow, ribbons and DV, depending on the solution polymer elasticity. When the outer cylinder solely rotates, only the DV were observed. In all cases, the threshold decreases with the solution elasticity. The disordered patterns are peculiar to large elastic stresses which dominate the inertia stresses in the fluid.

**Data accessibility.** This article has no additional data.

**Authors' contributions.** Y.B.: formal analysis, investigation, methodology, writing—original draft, writing—review and editing; N.L.: formal analysis, validation, visualization, writing—original draft, writing—review and editing; F.K.: formal analysis, investigation, visualization, writing—original draft; O.C.: formal analysis, methodology, software, validation, writing—original draft.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

**Conflict of interest declaration.** We declare we have no competing interests.

**Funding.** The present study has benefited from the French National Research Agency (ANR) through the programme Investissements d'Avenir (grant no. ANR LABX 09-01) LABEX EMC³ (project TUVECO).

## References

1. Taylor GI. 1923 Stability of a viscous liquid contained between two rotating cylinders. *Phil. Trans. R. Soc. Lond. A* **223**, 289–343. (doi:10.1098/rsta.1923.0008)

2. van Gils DPM, Huisman SG, Burggert GW, Sun C, Lohse D. 2011 Torque scaling in Turbulent Taylor-Couette flow with co-and counterrotating cylinders. *Phys. Rev. Lett.* **106**, 24502. (doi:10.1103/PhysRevLett.106.024502)

3. Ali ME, Mitra D, Schwille JA, Lueptow RM. 2002 Hydrodynamic stability of a suspension in cylindrical Couette flow. *Phys. Fluids* **14**, 1236–1243. (doi:10.1063/1.1449468)

4. Larson R, Shaqfeh E, Muller S. 1990 A purely elastic instability in Taylor-Couette flow. *J. Fluid Mech.* **218**, 573. (doi:10.1017/S0022112090001124)

5. Latrache N, Mutabazi I. 2021 Transition to turbulence via flame patterns in viscoelastic Taylor-Couette flow. *Eur. Phys. J. E* **44**, 63. (doi:10.1140/epje/s10189-021-00067-0)

6. Groisman A, Steinberg V. 1998 Mechanism of elastic instability in Couette flow of polymer solutions: experiment. *Phys. Fluids* **10**, 2451–2463. (doi:10.1063/1.869764)

7. Groisman A, Steinberg V. 1996 Couette-Taylor flow in a dilute polymer solution. *Phys. Rev. Lett.* **77**, 1480–1483. (doi:10.1103/PhysRevLett.77.1480)

8. Crumeyrolle O, Latrache N, Ezersky A, Mutabazi I. 2003 Instability modes observed in a viscoelastic Couette-Taylor flow. *Mech. Ind.* **4**, 397–409. (doi:10.1016/S1296-2139(03)00072-1)

9. Crumeyrolle O, Mutabazi I, Grisel M. 2002 Experimental study of inertia-elastic Couette-Taylor instability modes in dilute and semidilute polymer solutions. *Phys. Fluids* **14**, 1681–1688. (doi:10.1063/1.1466837)
10. Baumert B, Muller S. 1995 Flow visualization of the elastic Taylor-Couette in stability in Boger fluids. *Rheol. Acta* **34**, 147–159. (doi:10.1007/BF00398434)

11. Baumert B, Muller S. 1999 Axisymmetric and non-axisymmetric elastic and inertia-elastic instabilities in Taylor-Couette flow. *J. Non-Newton. Fluid Mech.* **83**, 33–69. (doi:10.1016/S0377-0257(98)00132-3)

12. Baumert B, Muller S. 1997 Flow regimes in model viscoelastic fluids in a circular Couette system with independently rotating cylinders. *Phys. Fluids* **9**, 566–586. (doi:10.1063/1.869209)

13. Latrache N, Crumeyrolle O, Abcha N, Mutabazi I. 2008 Destabilization of inertia-elastic mode via spatiotemporal intermittency in a Couette-Taylor viscoelastic flow. *J. Phys. Conf. Ser.* **137**, 012022. (doi:10.1088/1742-6596/137/1/012022)

14. Latrache N, Crumeyrolle O, Mutabazi I. 2012 Transition to turbulence in a flow of a shear-thinning viscoelastic solution in a Taylor-Couette cell. *Phys. Rev. E* **86**, 056305. (doi:10.1103/PhysRevE.86.056305)

15. Latrache N, Abcha N, Crumeyrolle O, Mutabazi I. 2016 Defect-mediated turbulence in ribbons of viscoelastic Taylor-Couette flow. *Phys. Rev. E* **93**, 043126. (doi:10.1103/PhysRevE.93.043126)

16. Muller SJ, Larson RG, Shaqfeh ESG. 1989 A purely elastic transition in Taylor-Couette flow. *Rheol. Acta* **28**, 499–503. (doi:10.1007/BF01332920)

17. Shaqfeh ESG, Muller SJ, Larson RG. 1992 The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability. *J. Fluid Mech.* **235**, 285–317. (doi:10.1017/S0022112092001113)

18. Joo YL, Shaqfeh ESG. 1994 The effects of inertia on the viscoelastic Dean and the Taylor-Couette flow instabilities with application to coating flows. *Phys. Fluids A* **4**, 2415–2431. (doi:10.1063/1.858483)

19. Avgousti M, Beris AN. 1993 Non-axisymmetric modes in viscoelastic Taylor-Couette flow. *J. Non-Newton. Fluid Mech.* **50**, 225–251. (doi:10.1016/0377-0257(93)80033-8)

20. Sureshkumar R, Beris AN, Avgousti M. 1994 Nonaxisymmetric subcritical bifurcations in viscoelastic Taylor-Couette flow. *Proc. R. Soc. Lond. A* **447**, 135–153. (doi:10.1098/rspa.1994.0132)

21. Thomas DG, Sureshkumar R, Khomami B. 2006 Pattern formation in Taylor-Couette flow of dilute polymer solutions: dynamical simulations and mechanism. *Phys. Rev. Lett.* **97**, 054501. (doi:10.1103/PhysRevLett.97.054501)

22. Thomas DG, Khomami B, Sureshkumar R. 2009 Nonlinear dynamics of viscoelastic Taylor-Couette flow: effect of elasticity on pattern selection, molecular conformation and drag. *J. Fluid Mech.* **620**, 353–382. (doi:10.1017/S0022112008004710)

23. Liu N, Khomami B. 2013 Elastically induced turbulence in Taylor-Couette flow: direct numerical simulation and mechanistic insight. *J. Fluid Mech.* **737**, R4. (doi:10.1017/jfm.2013.544)

24. Lopez JM. 2022 Vortex merging and splitting events in viscoelastic Taylor Couette flow. *J. Fluid Mech.* **946**, A27. (doi:10.1017/jfm.2022.579)

25. Song J, Lin F, Liu N, Lu X-Y, Khomami B. 2021 Direct numerical simulations of inertia-elastic turbulent Taylor-Couette flow. *J. Fluid Mech.* **926**, A37. (doi:10.1017/jfm.2021.757)

26. Song J, Lin F, Liu N, Lu X-Y, Khomami B. 2022 Direct numerical simulations of elastic turbulent Taylor-Couette flow: transition pathway and mechanistic insight. *J. Fluid Mech.* **949**, A49. (doi:10.1017/jfm.2022.801)

27. Bai Y, Vieu T, Crumeyrolle O, Mutabazi I. 2021 Viscoelastic Taylor-Couette instability in the Keplerian regime. *Geophys. Astrophys. Fluid Dyn.* **115**, 322–344. (doi:10.1080/03091929.2021.1873319)

28. Bird RB, Armstrong RC, Hassaguer O, Curtiss CF. 1977 *Dynamics of polymer liquids*. New York, NY: Wiley.

29. Bai Y, Crumeyrolle O, Mutabazi I. 2015 Viscoelastic Taylor-Couette instability as analog of the magnetorotational instability. *Phys. Rev. E* **92**, 031001(R). (doi:10.1103/PhysRevE.92.031001)

30. Chandrasekhar S. 1960 The stability of non-dissipative Couette flow in hydromagnetics. *Proc. Natl Acad. Sci. USA* **46**, 253–257. (doi:10.1073/pnas.46.2.253)

31. Shaqfeh ES. 1996 Purely elastic instabilities in viscosimetric flows. *Annu. Rev. Fluid Mech.* **28**, 129–185. (doi:10.1146/annurev.fl.28.010196.001021)

32. Ogilvie GI, Potter AT. 2008 Magnetorotational-type instability in Couette-Taylor flow of a viscoelastic polymer liquid. *Phys. Rev. Lett.* **100**, 074503. (doi:10.1103/PhysRevLett.100.074503)
33. Rodd LE, Scott TP, Cooper-White JJ. 2005 Capillary break-up rheometry of low viscosity elastic fluids. *Appl. Rheol.* **15**, 12–27. (doi:10.1515/arh-2005-0001)

34. Kelai F. 2011 Etude expérimentale des instabilités viscoélastiques dans le système de Taylor-Couette. Thèse de doctorat, Université du Havre.

35. Bai Y. 2015 Study of the viscoelastic instability in Taylor-Couette system as an analog of the magnetorotational instability. Thèse de doctorat, Université du Havre.

36. Groisman A, Steinberg V. 1997 Solitary vortex pairs in viscoelastic Couette flow. *Phys. Rev. Lett.* **78**, 1460–1463. (doi:10.1103/PhysRevLett.78.1460)

37. Dutcher CS, Muller SJ. 2013 Effects of moderate elasticity on the stability of co-and counter-rotating Taylor-Couette flows. *J. Rheol.* **57**, 791–812. (doi:10.1122/1.4798549)

38. Thomas DG, Al Mubaiyedh UA, Sureshkumar R, Khomami B. 2006 Time-dependent simulations of non-axisymmetric patterns in Taylor-Couette flow of dilute polymer solutions. *J. Non-Newton Fluid* **138**, 111–133. (doi:10.1016/j.jnnfm.2006.04.013)

39. Chossat P, Iooss G. 1994 *The Couette-Taylor problem*. New York, NY: Springer.

40. Cross MC, Hohenberg PC. 1993 Pattern formation outside equilibrium. *Rev. Mod. Phys.* **65**, 851–1112. (doi:10.1103/RevModPhys.65.851)

41. Laure P, Mutabazi I. 1994 Nonlinear analysis of instability modes in the Taylor-Dean system. *Phys. Fluids* **6**, 3630–3642. (doi:10.1063/1.868420)

42. Song J, Zhu Y, Lin F, Khomami B. 2023 Turbulent Taylor-Couette flow of dilute polymeric solutions: a 10-year retrospective. *Phil. Trans. R. Soc. A* **381**, 20220132. (doi:10.1098/rsta.2022.0132)

43. Boulafentis T, Lacasagne T, Cagney N, Balabani S. 2023 Experimental insights into ealsto-inertial transitions in Taylor–Couette flows. *Phil. Trans. R. Soc. A* **381**, 20220131. (doi:10.1098/rsta.2022.0131)