# Elastically modulated wavy vortex flow

T. Boulafentis<sup>a</sup>, T. Lacassagne<sup>b</sup>, N. Cagney<sup>c</sup>, S. Balabani<sup>a,*</sup>

## ARTICLE INFO

### Keywords:

Taylor-couette flow

Polymers

Viscoelasticity

Absolute/convective instability

## ABSTRACT

We investigate the transition pathway of low elasticity fluids ($El = 0.003 – 0.008$) in a Taylor-Couette configuration using low-molecular-weight polyacrylamide (PAAM) and visualisation experiments in the Reynolds range from 0 to 300. We report here for the first time an elastically modified wavy vortex flow state with altered spectral and structural characteristics, that precedes the onset of the traditional (inelastic) Newtonian wavy instability. This new wavy regime is characterised by oscillations of both the inflow and outflow boundaries, associated with a weakening of the outflow regions due to low hoop stresses. The modification of the boundaries persists at higher Reynolds numbers, where the spectral characteristics are unaltered compared to the inelastic, Newtonian case. In addition, a hysteretic behaviour is observed for increasing elasticity, as instabilities are shifted towards lower critical Reynolds numbers, confirming the importance of even vanishing elasticity on the stability of Taylor-Couette flows. At higher fluid elasticity ($El = 0.06$), the amplitude of inflows/outflows oscillations increases, and momentum is transferred axially between adjacent vortices, which may contribute to the emergence of Rotating Standing Waves.

## 1. Introduction

The instabilities of Newtonian fluids subjected to shear between two concentric cylinders, with one or both rotating, namely Taylor-Couette flow (TCF), have been well-documented [17,22,24]. The most commonly used control parameter for these instabilities in the simplest case of a stationary outer and rotating inner cylinder is the Reynolds number, $Re = \rho\Omega_t r_t d/\eta$, where $\rho$ is the fluid density, $\Omega_t$ is the rotational speed of the inner cylinder, $r_t$ is the radius of the inner cylinder, $d$ is the gap between the two cylinders, $\eta$ is the dynamic shear viscosity of the fluid. The primary instability leads to the formation of equally-spaced, axisymmetric vortices, termed Taylor Vortex Flow (TVF), named after the pioneering work of Taylor [42]. With increasing inertia, such secondary flows undergo a transition into non-axisymmetric, axially oscillating vortices, a regime termed Wavy Vortex Flow (WVF). WVF has been studied for a wide range of geometrical parameters, both analytically and experimentally; an abundance of wave modes have been revealed, attributed to an azimuthally moving wave whose speed depends on $Re$ [2,28,47]. A second travelling wave emerges as $Re$ increases further, leading to a quasi-periodic instability termed Modulated Wavy Vortex Flow (MWVF) [13,27,40,41], prior to the appearance of the

so-called broadband spectral component, signifying the onset of chaos [22]. The wavy and modulated wavy instabilities are associated with enhanced intercellular mixing compared to TVF due to the presence of the azimuthal wave which distorts the vortices periodically, allowing intervortex momentum transfer [1,16].

Viscoelasticity, induced by the addition of polymer chains, modulates the aforementioned Newtonian TCF sequence, leading to a variety of more complex transition pathways that depend on the Elasticity Number, $El = t_e\eta/\rho d^2$, where $t_e$ is the relaxation time. Most studies in the literature, focus on the transitions of solutions with Elasticity numbers in the range $El = 0.023 – 27$ which are highly unstable [6,44] and are characterised by the appearance of distinct flow states like Rotating Standing Waves (RSW), Ribbons (RB) and Flame Pattern (FP) [7,32,39].

However, in the limit of vanishing elasticities ($El \sim 0$), published literature reports Newtonian-like transitions in dilute solutions of long-chained polymers or solutions with high polymer entanglement, which most of the time also exhibit shear-thinning behaviour ([9–11]; [14, 18]). Yet, some observations, such as the high-frequency WVF with reduced amplitude oscillations in [14], suggest that even in cases where the transition appears Newtonian-like, the flow instabilities can be modified by the smallest amount of elasticity. Performing long

<footnote><sup>a</sup> Department of Mechanical Engineering, University College London, WC1E 6BT, UK</footnote>

<footnote><sup>b</sup> IMT Nord Europe, Institut Mines-Télécom, Univ. Lille, Centre for Energy and Environment, Lille, F-59000, France</footnote>

<footnote><sup>c</sup> School of Engineering and Materials Science, Queen Mary University of London, E1 4NS, UK</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: s.balabani@ucl.ac.uk (S. Balabani).</footnote>
experiments in the presence of low-molecular-weight polyacrylamide (PAAM) at a constant imposed shear rate, [36] have reported a modification of time-dependent Newtonian flows leading to oscillations at the boundaries with varying amplitude, which were eventually dampened after 20 h. Surprisingly, this was not the case after decreasing the rotational speed of the inner cylinder, indicating a dependence of the flow on the shear history. Dutcher and Muller [18] reported a similar example of altered time-dependent flows, where a MWVF flow state was interposed for a short Re range between two WVF regimes in the transition path. Surprisingly, this was accompanied by a delay on the onset of TVF as the elasticity increased, also reported in [10,14,46], possibly attributed to shear-thinning effects. At high Re, around Re ≈ 800, well-above the Newtonian WVF stability limit and after the appearance of MWVF, a flow state reminiscent to WVF with altered structural characteristics appeared [18].

The presence of fluid shear-thinning in experimental [9,19,48,20] and numerical [45,48] studies makes it difficult to disentangle the effects of elasticity and shear-thinning on observed transitions and flow phenomena.

Nevertheless, the experimental studies outlined above indicate a strong effect of even low elasticities on time-dependent and weakly turbulent flow states, but it remains unclear how the flow transitions are affected in the case of vanishingly small elasticity.

In this work, experiments with low molecular weight polyacrylamide (PAAM) polymer solutions were performed, with the following two objectives: firstly, to refine our understanding of elastic instabilities in the case of vanishing elasticities and absence of shear-thinning-achieved through short-chained polymers- for increasing or decreasing inertia; secondly to ascertain whether the increasing entanglement of polymer chains is sufficient to trigger higher-order instabilities like the RSW/FP. To this end, an experiment with a higher elasticity fluid, i.e. using longer chain polymers, has also been performed to link RSW to the Newtonian-like, wavy instabilities.

## 2. Materials and methods

We employ the same TC geometry (Fig. 1) as in our previous work [11,29,29,31]. It has a height of $H = 135$ mm, inner and outer cylinders radii of $r_i = 21.66$ mm and $r_o = 27.92$ mm, respectively, resulting in a gap width of $d = 6.26$ mm, an aspect ratio of $\Gamma = H/d = 21.56$, a radius ratio of $\eta_{cell} = r_i/r_o = 0.776$ and a curvature of $d/r_i = 0.289$. The TC was placed in a recirculating water tank to maintain the fluid temperature constant throughout the experiments at $20 \pm 0.1^{\circ}$C. Both the TC cell and the outer water container are made of transparent material for optical flow measurements. The inner cylinder was rotated by means of a stepper motor (SmartDrive Ltd, Cambridge, UK), with micro-stepping function of 52,000 microsteps/revolution as described in [29,31].

A highly viscous solvent of 72 % glycerol/water (WGL-72) and a low-molecular-weight ($M_n = 20, 000 \text{ g} \cdot \text{mol}^{-1}$) PAAM polymer were selected to introduce low values of elasticities and negligible shear-thinning. Polymer solutions with concentrations varying in the range of 500–7000 ppm were prepared and their shear rheology was measured in an ARES rheometer at $20 \,^{\circ}$C, using a Couette geometry with $r_i=32$ mm and $r_o=34$ mm. The resulting shear viscosity curves are illustrated in

<figure><img src="image_1.png" /><figcaption>**Fig. 2.** (a) Viscosity curves for polymer solutions of low-molecular-weight PAAM. The solvent case is noted as WGL-72 and the high-molecular-weight PAAM solution as HMW. All solutions have almost constant viscosity and can be considered Boger fluids. (b) Fitting of Huggins equation to the specific viscosity data, $\eta_{sp} = (\eta_0 - \eta_s)/\eta_s$, where $\eta_0$ is the zero-shear viscosity and $\eta_s$ the solvent viscosity.</figcaption></figure>

**Fig. 2a.** All solutions exhibit almost constant viscosity for the range of shear rates studied and thus can be considered Boger fluids. This can be quantified using the average scaling exponent of shear stress with strain rate, $\bar{n}_e$, which tends to unity for Boger fluids (Table 1). By fitting the Huggins equation [8] in the viscosity data (Fig. 2b), the critical overlap concentration (i.e. the polymer concentration above which polymer chains entanglement becomes significant) was estimated to be $c^* = 3967$ ppm ($|\eta| = 0.0002, k_H = 0.3695$).

An additional solution was prepared with the same solvent using 200 ppm of high-molecular-weight PAAM ($M_w = 5.5 \times 10^6 \text{ g} \cdot \text{mol}^{-1}$), to achieve higher fluid elasticity. The viscosity curve of this solution is noted as HMW in Fig. 2a. This solution lies also in the dilute regime ($c^* = 718$ ppm) (the analysis for the estimation of $c^*$ for this polymer is not shown here for brevity).

Among all the fluids characterized in Fig. 2, two polymer concentrations corresponding to the dilute and a semi-dilute regime (1500 ppm and 5000 ppm, respectively) were employed in the present TC experiments, alongside the pure solvent case (WGL-72, Newtonian inelastic) and the HMW solution. The relaxation time of the two low-molecular-weight polymer solutions was estimated using extensional rheology implementing the Slow Retraction Method (SRM) [12,38] and was found equal to $t_e = 2.4$ and 4.4 ms for the 1500 and 5000 ppm, respectively, corresponding to $El = 0.003$ and 0.008. The

<table><caption>**Table 1**
Rheological characteristics of the solutions used in the TC experiments.</caption><thead><tr><th>Solution</th><th>State</th><th>η<br>[Pa.s]</th><th>n&#773;<sub>e</sub></th><th>t<sub>e</sub> [ms]</th><th>β = η<sub>p</sub>/η<sub>0</sub></th><th>El</th></tr></thead><tbody><tr><td colspan="7"><strong>Newtonian</strong></td></tr><tr><td>WGL-72</td><td>-</td><td>0.0372</td><td>1</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan="7"><strong>Low molecular weight polymer</strong></td></tr><tr><td>1500 ppm</td><td>Dilute</td><td>0.0469</td><td>1</td><td>2.4</td><td>0.261</td><td>0.003</td></tr><tr><td>5000 ppm</td><td>Semi-dilute</td><td>0.0856</td><td>0.997</td><td>4.4</td><td>1.3</td><td>0.008</td></tr><tr><td colspan="7"><strong>High molecular weight polymer</strong></td></tr><tr><td>HMW</td><td>Dilute</td><td>0.0436</td><td>0.987</td><td>65.6</td><td>0.172</td><td>0.06</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** The Taylor-Couette cell and visualisation setup used in this work.</figcaption></figure>
high-molecular-weight solution relaxation time was estimated using the same method at $t_e = 65.6$ ms, resulting in $El = 0.06$. All fluid properties used in the TC experiments are summarised in Table 1.

Experiments for the low-molecular-weight solutions were performed for (a) increasing *Re* (ramp-up) (b) decreasing *Re* (ramp-down) and (c) fixed *Re* (steady-states). Measurements covered the range *Re* = 0 – 300. Polymer degradation was considered unlikely for the short-chained polymers used in this work. The absence of degradation was further validated by repeating a ramp-up experiment to ascertain that the transitional sequence remains unaltered. The inner cylinder was accelerated/decelerated sufficiently slowly to ensure quasi-static ramp-up/down conditions. The non-dimensional acceleration $\Gamma_0$, which takes into account the variation of *Re* compared to the viscous time scale, was kept well-below the quasi-static limit, $\Gamma_0 = 0.3 < 1$ [11,17,29,30](Table 2). For the high-molecular-weight solution, a steady-state experiment at *Re* = 105 was performed.

The Taylor-Couette flow was visualized using mica flakes (10–100 µm, Cornelissen & Son, Pearl Lustre Pigments), added to the solution before each experiment [5,11,29] at low enough volume fraction (0.01% v/v) to not affect the rheological properties of the fluids [23]. A flow strip along the height of the cell (2176 × 16 pixels) was imaged using a high-speed camera (Phantom Miro M340, Vision Research). The acquisition frequency was $f_s = 80$ Hz for the Newtonian case and $f_s = 130$ Hz and 180 Hz for $El = 0.003$ and $El = 0.008$, respectively. More than 300 frames/*Re* (1/Δ*Re* > 300), were acquired in all cases (Table 2), where Δ*Re* is the variation of Reynolds number between two consecutive frames. A much higher acquisition frequency of $f_s = 2000$ Hz was selected for the steady-state experiments to provide high temporal resolution of the wavy and RSW flows.

Following the acquisition, the image intensity was averaged to obtain a single column of intensity *I* values, and spatiotemporal maps *I*(*z*, *t*) were assembled by stacking different time points horizontally (where *z*- and *t*- axes indicate spatial and temporal dimensions, respectively). The intensity was then inverted to highlight the inflows/outflow boundaries. In the ramp-up/down experiments, the *t*-axis was converted to *Re*. Fast Fourier Transform was applied on the spatiotemporal intensity maps, both temporally (along the *t*-axis) and spatially (along the *z*-axis), to reveal the dominant frequencies and wavelengths of the flow, respectively. Frequency maps were thus constructed by separating the ramp-up/down maps into sections with 512 columns and 50 % overlap, as described in [29].

### 3. Results and discussion

The transition sequence for the solvent case (not shown here for brevity) is non-hysteretic (i.e. the same transition sequence occurs for ramp-up and ramp-down) and well documented in the literature [3,17,47]. Namely, the laminar Couette flow transitions firstly into TVF at $Re_{TVF}^{ru} = 100$ and secondly to WVF at $Re_{WVF}^{ru} = 200$, where the subscript

signifies the flow state at which the flow transitions and the superscript either the ramp-up or ramp-down. No transition to MWVF was observed in the range of *Re* studied here. The axial wavelength of the flow remains the same for both TVF and WVF and was found equal to $\lambda = 1.7d$ (Fig. 3b). The wavy frequency in the WVF scales linearly with the rotational frequency of the inner cylinder, $f = 1.3f_i$, where $f_i = \Omega_i/2\pi$.

The dilute low molecular weight polymer mixture (1500 ppm) has an elasticity value $El = 0.003$ and exhibits a transition sequence during the ramp-up that closely resembles the Newtonian case (Fig. 4a). The critical Reynolds numbers for the transition to TVF and WVF remain unaffected ($Re_{TVF}^{ru} = 100$ and $Re_{WVF}^{ru} = 200$). However, just before the appearance of WVF, a non-axisymmetric instability is observed for a small range of Reynolds numbers $Re = 158 – 200$, as evidenced by a distinct ridge on the frequency maps (Fig. 4c, d). This instability has the same wavelength (Fig. 3b) as TVF and WVF but occurs at a lower frequency compared to WVF (Fig. 3a). The wave frequency is not directly proportional to the cylinder oscillation frequency, with $f = 1.15 f_i$ at the onset of the instability ($Re_{WVF}^{ru} = 158$) and $f = 1.08 f_i$ at the transition to WVF ($Re_{WVF}^{ru} = 200$). Within this range the frequency increases linearly.

We term this low-frequency instability Elastic Wavy Vortex Flow (EWVF) and to the best of our knowledge, it has not been reported previously. Elastic modifications of WTV flows have been reported by a number of researchers. For example, Crumeyrolle et al. [14] observed

<figure><img src="image_3.png" /><figcaption>Fig. 3. (a) Temporal and (b) spatial spectra for the inelastic case $El = 0$ ($Re = 180$ - TVF) and $El = 0.003$ ($Re = 180$ - EWVF, $Re = 250$ - WVF). TVF, EWVF, WVF have the same spatial wavelength but different temporal characteristics.</figcaption></figure>

### Table 2

Experimental settings for ramp-up/down and steady-state experiments presented later in the results section.

<table><thead><tr><th>Solution</th><th>Experiment Type</th><th>Reynolds Range</th><th>f<sub>s</sub> [Hz]</th><th>1/ΔRe</th><th>Ω<sub>max</sub> [s<sup>-1</sup>]</th><th>dΩ/dt [s<sup>-2</sup>]</th><th>Γ<sub>0</sub></th></tr></thead><tbody><tr><td colspan="8"><strong>Newtonian</strong></td></tr><tr><td>WGL-72</td><td>RU/RD</td><td>0 – 300</td><td>80</td><td>336.54</td><td>68.7</td><td>0.054</td><td>0.3</td></tr><tr><td>WGL-72</td><td>Steady-State</td><td>180</td><td>2000</td><td>–</td><td>34.35</td><td>–</td><td>–</td></tr><tr><td>WGL-72</td><td>Steady-State</td><td>250</td><td>2000</td><td>–</td><td>60</td><td>–</td><td>–</td></tr><tr><td colspan="8"><strong>Low molecular weight polymer</strong></td></tr><tr><td>1500 ppm</td><td>RU/RD</td><td>0–300</td><td>130</td><td>351.6</td><td>90.42</td><td>0.0943</td><td>0.2</td></tr><tr><td>5000 ppm</td><td>RU/RD</td><td>0–300</td><td>180</td><td>317.84</td><td>163.67</td><td>0.309</td><td>0.3</td></tr><tr><td>1500 ppm</td><td>Steady-State</td><td>180</td><td>2000</td><td>–</td><td>54.29</td><td>–</td><td>–</td></tr><tr><td>1500 ppm</td><td>Steady-State</td><td>250</td><td>2000</td><td>–</td><td>75.33</td><td>–</td><td>–</td></tr><tr><td>5000 ppm</td><td>Steady-State</td><td>180</td><td>2000</td><td>–</td><td>98.2</td><td>–</td><td>–</td></tr><tr><td>5000 ppm</td><td>Steady-State</td><td>250</td><td>2000</td><td>–</td><td>136.4</td><td>–</td><td>–</td></tr><tr><td colspan="8"><strong>High molecular weight polymer</strong></td></tr><tr><td>HMW (200 ppm)</td><td>Steady-State</td><td>105</td><td>2000</td><td>–</td><td>26.05</td><td>–</td><td>–</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>**Fig. 4.** Spatiotemporal maps for ramp-up (a) and ramp-down (b) experiments and corresponding frequency maps (c) and (d) for *El* = 0.003. The orange line in the frequency maps (c), (d) denotes the rotational frequency. *f*<sub>max</sub> in (c) and (d) corresponds to the maximum rotational speed of the inner cylinder reached throughout the experiment.</figcaption></figure>

that the travelling waves had a higher frequency compared to the Newtonian case, and the frequency further increased at higher elasticities. Dutcher et al. [18] reported that elastic effects in wavy flows at high Reynolds numbers (*Re* ≈ 800), modified the structure of inflows and outflows. Lacassagne et al. [29] reported the sudden appearance of a dominant periodic frequency for a highly elastic and shear-thinning fluid, resembling the mode described in this work. However, unlike present findings, the observed frequency of the wavy instability mode in [29] was found to be constant, remained lower than the rotational frequency at all times and was associated with the persistence of previously existing RSW structures. The EWVF observed in the present work differs from the aforementioned ones; it appears in low Reynolds numbers at a purely Newtonian transition of a Boger fluid and alters the basic WVF without adding any additional peak on the frequency spectra. This early Hopf bifurcation is attributed to weak fluctuations caused by elastic stresses in the flow as it does not appear in the non-elastic case. Unlike Modulated Wavy Vortex Flow, in which a second frequency appears due to the modulation of the basic frequency and the instability is quasi-periodic in nature [41], the observed elastic instability is periodic, resembling WVF, but altering its characteristic frequency (Fig. 3a).

The argument that EWVF is elastic in nature is also suggested by the presence of hysteretic effects in the flow. Although the critical Reynolds number for the primary bifurcation to TVF from CF during ramp-down stays constant (*Re* = 100), the Reynolds range for the appearance of EWVF becomes smaller and is shifted to lower *Re* numbers (*Re* = 143 – 161) in ramp-down experiments (Fig. 4b). However, EWVF has identical spatial and temporal spectra at both ramp-up and ramp-down.

Following EWVF, WVF onsets at *Re* = 161, i.e., at a critical *Re* lower than the equivalent for the ramp-up or the Newtonian case, signifying that the transition is subcritical. This also implies destabilization of the flow during the ramp-down by the presence of polymers even though WVF is considered a purely inertial instability.

Increasing the polymer concentration above the critical overlap concentration c⁺ (*El* = 0.008), does not alter the transition observed for the less elastic case, *El* = 0.003 (Fig. 5, the maps are omitted for brevity). This implies that the appearance of elastic transitions is highly dependent on the extensibility of the polymer chains [33,37] rather than the entanglement of polymer chains.

However, the critical Reynolds numbers and the Reynolds span for each regime vary with *El*. Both the primary and secondary bifurcation onset earlier ($Re_{TVF}^u$ = 95 and $Re_{EWVF}^u$ = 140) for *El* = 0.008, compared to the lower *El* transitions (Fig. 5) for the case of cylinder acceleration, in agreement with the analytical work of [43]. On the other hand, the onset of WVF is delayed ($Re_{WVF}^u$ = 210 for *El* = 0.003, compared to $Re_{cr}$ = 200 for the Newtonian case). After this point, the elastic wavy instability is replaced by the inelastic WVF. However, the vortex pairs drift axially (Fig. 4b in the WVF regime), in contrast to the Newtonian case, illustrating the impact of the elasticity on the stability of the flow even within the WVF regime.

For a decelerating inner cylinder, WVF extends to a lower Reynolds number $Re = 133$ in the case of $El = 0.008$. Similarly, EWVF shifts to lower Reynolds numbers $Re = 121 – 133$ and persists for a smaller $Re$ range. The lower critical Reynolds number in ramp-down compared to ramp-up are evidence of hysteretic flow behaviour.

<figure><img src="image_5.png" /><figcaption>**Fig. 5.** Overview of the effect of Elasticity number (*El*) on the critical Reynolds number for flow transitions in ramp-up (left) and ramp-down (right) experiments, *El* = 0 – 0.008.</figcaption></figure>
Detailed spatiotemporal maps from steady-state experiments, 1 s in duration, are illustrated in Fig. 6 for the inelastic case at TVF (Re = 180) and WVF (Re = 250). To further elucidate the nature of EWVF, maps have been produced for the two elastic cases at the same Re.

In all cases, the inflows and outflows are highlighted and noted with arrows. Outflows appear stronger for Newtonian-like fluids due to centrifugal forces [21,47]. We fit a generalized sinusoidal function Asin(ωt + φ) to capture the periodicity of these boundaries (black and magenta lines in Fig. 6). In the inelastic case, inflows and outflows are stable in the TVF regime, whereas in WVF only the inflows oscillate (Fig. 6, first column). In the low elasticity fluids, both inflows and outflows oscillate in the EWVF and WVF regimes, at the same frequency. Interestingly, the oscillations are out of phase Δφ = 2π/5 (Fig. 6, second and third column).

The ratio between the oscillation amplitude of the outflows and inflows is found $A_r = A_{out}/A_{in} = 0.7$ for EWVF, independent of the fluid elasticity. This is in good agreement with ([9,10]) who reported that while in Newtonian fluids only the inflows oscillate, both inflows and outflows tend to oscillate with the same amplitude for shear-thinning fluids with increased elastic properties. The stronger inflow jet oscillations in Newtonian fluids have been attributed to the increased strength of the outflow jets [47], which hinders the axial flow between adjacent vortices.

Similarly, the oscillations of the outflow boundaries in elastic fluids can be attributed to the weakening of the outflow strength due to the opposing inward-acting elastic forces. In WVF, the amplitude ratio decreases to $A_r = 0.5$, with the amplitude of inflows remaining the same and that of outflows decreasing relative to EWVF; this is due to the outflow jets becoming stronger due to the increasing inertia, in agreement with [47]. The amplitude ratio $A_r$ seems thus, highly dependent on the Reynolds number, decreasing as the role of inertia increases.

Although the fluid elasticity does not alter the ratio $A_r$, it has a strong effect on the amplitude of the oscillations of both inflows and outflows. For $El = 0.003$, the amplitude of outflows and inflows are $A_{out} = 0.07d$ and $A_{in} = 0.1d$ for EWVF, $A_{out} = 0.05d$ and $A_{in} = 0.1d$ for WVF (Table 3). For $El = 0.008$ the amplitudes are more than double these values ($A_{out} = 0.15d$ and $A_{in} = 0.22d$ for EWVF, $A_{out} = 0.11d$ and $A_{in} = 0.22d$ for WVF (Table 3)). The opposite effect is reported in [14] for low-elasticity fluids, where increased elasticity leads to a decrease in the amplitude of oscillations. This can be attributed either to the significant shear-thinning of the fluids used in their study or the larger radius ratio ($η = 0.88$).

This increase in the amplitude of oscillations might be connected to the emergence of RSW in the case of more elastic fluids, $El > 0.01$ as reported by [4], even at much lower Re [31]. This flow state, illustrated in Fig. 7a for the high-molecular-weight fluid ($El = 0.06$) at Re = 105, has been previously stated to be inertia driven but modified by elasticity [39] without however making any direct connection to the Newtonian

instabilities. RSW has a characteristic frequency equal to $f_{el}/3$, where $f_{el} = 2c_e/λ$ is the elastic wave frequency and $c_e = \sqrt{η/ρt_e}$ is the wave celerity, in agreement with previously reported values in the literature [31,33]. This frequency is significantly lower than the inelastic and slightly elastic wave frequencies (Fig. 7b). The similarity of the inflows and outflows in the spatiotemporal map in Fig. 7a, implies a radial inflow/outflow symmetry as suggested by [25,26].

Fitting the generalized sinusoidal function used previously to the RSW boundaries (Fig. 7a) also reveals an inflow/outflow boundary oscillatory behavior similar to that of the wavy instabilities. However, the amplitudes of inflow and outflow oscillations are equal ($A_{out} = A_{in} = 0.22d$, $A_r = 1$) and the phase lag increases to $Δφ = π$ (Table 3).

On close inspection, one can observe that the waves are not perfectly sinusoidal; they are distorted every half period (Fig. 7a). At these points, momentum is transferred between adjacent vortices, allowing the propagation of the axial, elastic wave described in [31,33]. A similar mechanism is not possible at lower elasticities as (i) $A_r$ is not equal to unity and (ii) the phase lag is significantly lower; hence the two boundaries do not approach each other.

In light of these observations, we can consider RSW an extension of the aforementioned elastically modulated wavy instabilities at the limit of high amplitude, large phase lag boundary oscillations. Since wavy instabilities are attributed to an azimuthally propagating wave [2,28, 47], we postulate that RSW and its characteristic frequency should not be considered solely the result of an axial elastic wave but rather as a highly three-dimensional flow state, emerging as a combination of both an azimuthal inertial and an axial, elastic wave. Further work is however needed to fully comprehend its exact mechanism.

## 4. Summary

In this study, two solutions of low molecular weight PAAM have been used to elucidate the effect of very small elasticities and increasing polymer entanglement on the Newtonian transitions in Taylor Couette Flow.

It has been found that increasing polymer concentration from the dilute ($1500$ ppm, $c/c^* = 0.38$) to a semi-dilute ($5000$ ppm, $c/c^* = 1.26$) regime is not a sufficient condition for the appearance of highly elastic instabilities like RSW or FP; the extensibility of the polymer chains rather appears to be the key factor. However, a low elasticity signature, termed Elastic Wavy Vortex Flow (EWVF) has been observed for both solutions, which has not been reported in the literature hitherto. This instability was observed prior to the appearance of WVF and has similar time-dependent, single-frequency, non-axisymmetric characteristics. The characteristic feature of this new mode is the lower frequency compared to the typical WVF frequency in the Taylor-Couette system used.

Using steady-state experiments with both inelastic and elastic fluids,

<figure><img src="image_6.png" /><figcaption>**Fig. 6.** Spatiotemporal maps during steady-state experiments for the inelastic case ($El = 0$) and the two elastic cases ($El = 0.003$ and $El = 0.008$) for Re = 180 and Re = 250. The sinusoidal fitting to the oscillatory inflow (black lines) and outflow regions (magenta lines) is shown in all cases. The steady-state maps capture 5 and 9 ($El = 0$), 8 and 15 ($El = 0.003$), 15 and 21 ($El = 0.008$) revolutions of the inner cylinder at Re = 180 and Re = 250 respectively.</figcaption></figure>
<table><caption>**Table 3**
Amplitude characteristics of the oscillating outflow and inflow boundaries ($A_{out}$, $A_{in}$) for $El = 0 - 0.06$</caption><thead><tr><th rowspan="2">El</th><th colspan="3">RSW (Re = 105)</th><th colspan="3">EWVF (Re = 180)</th><th colspan="3">WVF(Re = 250)</th></tr><tr><th>A<sub>in</sub>/d</th><th>A<sub>out</sub>/d</th><th>A<sub>r</sub></th><th>A<sub>in</sub>/d</th><th>A<sub>out</sub>/d</th><th>A<sub>r</sub></th><th>A<sub>in</sub>/d</th><th>A<sub>out</sub>/d</th><th>A<sub>r</sub></th></tr></thead><tbody><tr><td>0</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.07</td><td>0</td><td>0</td></tr><tr><td>0.003</td><td>-</td><td>-</td><td>-</td><td>0.10</td><td>0.07</td><td>0.7</td><td>0.10</td><td>0.05</td><td>0.5</td></tr><tr><td>0.008</td><td>-</td><td>-</td><td>-</td><td>0.22</td><td>0.15</td><td>0.7</td><td>0.22</td><td>0.11</td><td>0.5</td></tr><tr><td>0.06</td><td>0.22</td><td>0.22</td><td>1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 7.** (a) Steady-state spatiotemporal map of Rotating Standing Waves (RSW) for a polymeric fluid with $El = 0.06$ at $Re = 105$. Sinusoidal fits to the oscillating inflow (black lines) and outflow regions (magenta lines) are superimposed. (b) Temporal spectra of the RSW in comparison with wavy instabilities ($Re = 180$ - TVF, $Re = 250$ - WVF) for a less elastic solution ($El = 0.008$).</figcaption></figure>

we concluded that (i) the value of $El$ (and thus here the entanglement state) modifies the frequency of the additional EWVF state, but (ii) the frequency of WVF is not altered, only its structure.

The waviness appears on both outflow and inflow boundaries as described by ([9,10]) and can thus, be attributed to elasticity and not the shear-thinning nature of the fluids used in their study.

For increasing elasticity, the $Re$ range at which TVF is observed decreases while for EWVF increases during ramp-up. At the same time, WVF is delayed (appears at higher Reynolds number). For ramp-down, a slightly hysteretic behaviour is observed as all flow states are sustained to lower $Re$.

By fitting a sinusoidal function to the wavy oscillations, we conclude that increased elasticity leads to an increase in the amplitude of oscillations of both inflow and outflow boundaries. This effect along with the increase in the phase lag of the two oscillating boundaries, potentially links the elastic RSW flow state to the Newtonian-like instabilities.

The present study highlights the importance of elastic phenomena on flow transitions and provides evidence that the additions of even small amounts of polymers may enhance the intercellular mixing, as both inflow and outflow boundaries oscillate, even at $Re$ lower than the in-elastic Newtonian case. However, caution may be exercised on the use of low molecular weight polymers to increase the viscosity of fluids, which is a common technique [15,34,35], as they can potentially induce undesirable elastic instabilities. In addition, considering RSW as an extension of the wavy instabilities may bear significant changes to our understanding of the phenomenon as both an axial and an azimuthal

wave may contribute to its mechanism.

### CRediT authorship contribution statement

**T. Boulafentis:** Writing – original draft, Visualization, Methodology, Investigation, Data curation. **T. Lacassagne:** Writing – review & editing, Visualization, Methodology, Investigation, Data curation. **N. Cagney:** Writing – review & editing, Visualization, Methodology, Investigation, Data curation. **S. Balabani:** Writing – review & editing, Supervision, Resources, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### Funding

Financial support for this work from the Engineering and Physical Sciences Research Council (EPSRC) Manufacturing the Future programme (grant no. EP/N024915/1) as well as the EPSRC DTP award EP/R513143/1 is gratefully acknowledged.

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