4.1. Transitions and flow states

Torque measurements and visualization experiments have been performed for suspensions with concentrations $\phi = 0\,\%, 1\,\%, 3\,\%, 6\,\%, 9\,\%, 12\,\%, 15\,\%, 20\,\%$. For the concentration of 28 %, only the primary bifurcation could be captured, due to the limitations in achievable angular velocities of the rheometer and considering the relatively high suspension viscosity (not reported in figure 1b but estimated from the Krieger–Dougherty fitting). In figure 3 the torque exerted by the solution on the inner cylinder is shown for all the suspensions. The solid lines and the dashed lines represent RU/RD experiments, respectively. The critical Reynolds number corresponding to the primary bifurcation is shown, as well as those for second and third bifurcations, in RU (square) and RD (circles). Note that all experiments have been repeated at least three times. Curves for single experiments are reported here, but average values for $\mathcal {R}_c$ will be provided later.

Figure 3. Variation of torque as a function of Reynolds for suspensions with concentration $0\,\% \leq \phi \leq 20\,\%$.

Increasing particle concentration leads to the following effects. Firstly, a general increase in the torque applied to the inner cylinder is seen in the whole Reynolds range. This means that particles increase the power needed to rotate the inner cylinder at a given angular velocity. Secondly, a general increase in the slope of the torque versus $\mathcal {R}$ is observed, which means that the torque increases faster with $\mathcal {R}$ when the concentration is increased. This is caused by the increased effective viscosity of the system which is related to particle–particle and particle–fluid interaction (see figure 1b). In all torque data, a sensible change in slope with the transition to TVF can be noted, as described earlier. With the transition of TVF–WVF, a less distinct change in slope is seen (symbolic slope shows for $\phi =20\,\%$ in figure 3). For suspensions at $\phi$ above 6 %, it can be noted that transition to TVF is more complex, and the change in slope less sharp: fluctuation is observed in torque measurements, and it will be shown thereafter to be due to the presence of non-axisymmetric flow patterns.

For a few experiments at selected concentration of $\phi =3\,\%, 9\,\%, 15\,\%, 20\,\%$ the space–$\mathcal {R}$ and effective Nusselt number ($\mathcal {N}$) variation as a function of Reynolds number in RU/RD modes are shown along with the space–Reynolds diagram in RU mode in figure 4. In each figure, the critical Reynolds values corresponding to first, second and third bifurcation are marked by dashed lines, for RU tests. Identification of critical $\mathcal {R}_c$ is done by flow visualization and torque evaluation simultaneously in the manner described in § 3.2. A general decreasing trend for the first $\mathcal {R}_c$ is visible and will be discussed in detail in § 4.3. CCF, TVF and WVF can be described in a similar fashion than in the particle-free fluids case (see § 3.3). The previously mentioned non-axisymmetric state appears from suspension of 6 % (not shown here) and above, and is clearly visible in figure 4(b). It can be detected also in Nu–$\mathcal {R}$ plots (figure 3) from the more gradual and unclear changes in slope between CCF and TVF trends. The non-axisymmetric state (NAS) is evidenced by fluctuations in reflected light intensities occurring between CCF and the onset of TVF. Fluctuations here appear random when observed at the full experimental scale (full $\mathcal {R}$ range), however, when focusing on a narrower Reynolds range, a clear spiralling-like non-asymmetric structure can be observed (see figure 5). Further details about NAS will be given in § 4.2.

Figure 4. Space–Reynolds number diagram (RU) in addition to Nusselt–Reynolds evolution (RU/RD) showing the transitions from CCF to WVF for concentration of (a) $\phi =3\,\%$; (b) $\phi =9\,\%$; (c) $\phi =15\,\%$; (d) $\phi =20\,\%$.

Figure 5. Close-up on the SVF regime: (a) snapshot photo of SVF flow from suspension of 15 %; (b) space time for a period of 806 s ($\mathcal {R}$ between 140–145) taken by visualization in suspension of 15 %; (c) space time of 2 s (70 f.p.s.) taken from zooming in two different period of time in SVF flow in panel (b).

In the figure 4 flow maps, the effects of boundary conditions for lowest concentrations can be seen again in the form of Ekman vortices. As the concentration increases though this effect is reduced, and Taylor vortices are created just after the termination of non-axisymmetric flow, all at the same time. As pointed out by Benjamin (Reference Benjamin1978), in experimental investigations of TCF, CCF cannot satisfy the boundary conditions. Specifically, the bifurcation found in theoretical models with repeated period boundary conditions does not exist in experiments. However, it is obvious in the figure that the onset of Taylor cells is sharp even with an aspect ratio of 10. The origin of this sharpness has been explained by Cliffe, Mullin & Schaeffer (Reference Cliffe, Mullin and Schaeffer2012) and Mullin, Heise & Pfister (Reference Mullin, Heise and Pfister2017) as resulting from the breaking of an approximate symmetry between neighbouring states. In the case studied here, this would involve eight, 10 or 12 cells. It is interesting to note that almost all of the cases contain 10 cells with the exception of the 15 % (figure 4c) which contains eight cells. As mentioned before, the boundary conditions used in this experiment are slightly more complicated as the upper surface is free. However, the results appear to be in accord with the Cliffe et al. (Reference Cliffe, Mullin and Schaeffer2012) interpretation.

In parallel, in the figure 4 Nusselt plots, the primary bifurcation is captured in a clear way, either by a sharp (when transitioning to TVF) or smooth (when a non-axisymmetric state is involved) change of the Nusselt slope, regardless of the presence or absence of Ekman vortices. This additionally suggests that Ekman vortices and the boundary conditions do not significantly affect the effective friction forces exerted on the inner cylinder and thus the Nusselt values. Hence, Nusselt value fluctuations evidenced at higher particle concentrations cannot be ascribed to boundary effects in CCF. The inflection point related to TVF–WVF transition is still subtle, as it was in the $\mathcal {T}$–$\mathcal {R}$ plots.

The Nusselt number related to CCF ($\mathcal {R} < \mathcal {R}_{c_1}$) is almost constant and is around one. Rheologically speaking, this means that torque depends linearly on viscosity and viscosity does not depend on the shear rate variation. This linear behaviour is observed by Dash et al. (Reference Dash, Anantharaman and Poelma2020) ($\mathcal {N}$ with respect to $Ta$) for the concentration of less than 25 %, and Kang & Mirbod (Reference Kang and Mirbod2021) in their numerical results. On the other hand $\mathcal {N} \sim \mathcal {R}^{c}$ with an exponent $c$ that depends on concentration has been observed by Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019). They reported an increasing in effective Nusselt number with increasing $\mathcal {R}$ and proposed dimensionless torque $G \sim \dot {\gamma }\left (r=r_{i}\right ) / \mu (\phi ) \sim \mathcal {R}^{\alpha }$ with $\alpha =1$ for a particle-free fluids and $\alpha >1$ for a suspension with $\phi \geqslant 0.05$ in the CCF regime. Finally, Dash et al. (Reference Dash, Anantharaman and Poelma2020) also observed an increase of $\mathcal {N}$ with $Ta$ (Taylor number, alternative of $\mathcal {R}$), but for concentrations over 25 %. They argued that this could be due to anisotropy in the microstructure of non-colloidal suspension flow, and non-homogeneous distribution of particle concentration in the radial direction which comes from the migration of particles, and modifies the local shear rate and viscosity along the gap. In this study, particle concentrations remain below 25 % with the exception of one test case, and the particle size to gap size ratio is below the one used in Dash et al. (Reference Dash, Anantharaman and Poelma2020). Hence, microstructure anisotropy is not expected, which is consistent with the torque scaling observations in CCF. Torque scaling in TVF and VWF will be further discussed in § 4.6.

4.2. Non-axisymmetric states

A specific feature of the flow of particle suspensions for volume fractions at $\phi >6\,\%$ is the existence of non-axisymmetric flow states. A snapshot of non-axisymmetric state for suspension of 15 % in RU protocol is shown in figure 5(a). Non-axisymmetric states were generally evidenced in Dash et al. (Reference Dash, Anantharaman, Greidanus and Poelma2019), Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) and Majji et al. (Reference Majji, Banerjee and Morris2018) experimentally and in the numerical study of Kang & Mirbod (Reference Kang and Mirbod2021). The flow structure present here is a SVF regime such as the one reported by Majji & Morris (Reference Majji and Morris2018) and Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019). This regime consisted of axially travelling vortex pairs. Figure 5(b) shows a space–time diagram for a 806 s time span, corresponding to a $\mathcal {R}$ range of 140–145, recorded at 70 f.p.s. In the middle of this space–time plot, a change of travelling direction occurs. Similar changes in direction were observed in all experiments for concentrations of 6 % and above, seemingly randomly with no particular pattern in time or space. Figure 5(c) illustrates this behaviour by displaying close-ups of figure 5(b) on 2 s time spans before and after this changing direction.

The effect of SVF on $\mathcal {N}$ can be seen in figure 4 in the form of a slope drop in Nusselt values resulting in a smoother change of slope for CCF to TVF transition, through intermediate scaling. While Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) did not report any changes in the Nusselt scaling associated with SVF, Kang & Mirbod (Reference Kang and Mirbod2021) noticed a slight drop in Nusselt number for SVF and WSVF at a concentration of $10\,\%$ with spherical particle size of $\delta /d_p=30$. We here report a first experimental observation of how SVF is associated with $\mathcal {N}$ scaling variations.

Figure 6 shows evolution of Nusselt number normalized by Nusselt value at which the non-axisymmetric state starts (for each concentration separately) as a function of dimensionless $\mathcal {R}$ ($(\mathcal {R}-\mathcal {R}_{c_{(SVF)}})/(\mathcal {R}_{c_{(T V F)}}-\mathcal {R}_{c_{(SVF)}})$). In this way, every plot in the Nusselt axis starts from 1 and in the Reynolds axis the values remain between 0–1. This helps when comparing the evolution of Nusselt number with $\mathcal {R}$ within the SVF range.

Figure 6. Evolution of Nusselt number normalized by critical Nusselt number of SVF state for each concentration as a function of dimensionless Reynolds number in the SVF period for $\phi =6\,\%, 9\,\%, 12\,\%, 15\,\%, 20\,\%$. Solid and dashed lines represent the RU and RD protocol, respectively.

The first visible feature is that the $\mathcal {N}$ trend goes from mildly increasing $\mathcal {R}$ at $\phi =6\,\%$ to non-monotonic and globally constant or decreasing at higher $\phi$. In other words, the presence of particles ($\phi >6\,\%$), which cause SVF, also has a non-trivial effect on the Nusselt number and can reduce the torque (or the increase of torque with $\mathcal {R}$) applied to the inner cylinder.

A proposed explanation for this non-monotonic behaviour is the following. The presence of particles promotes axially travelling non-axisymmetric flow structures and causes a delay of the transition from unsteady (SVF) to stationary (TVF) instability. During this phenomenon, convective momentum transfer decays (as mentioned by Kang & Mirbod (Reference Kang and Mirbod2021)), resulting in a reduction of the torque exerted on the inner cylinder. Simultaneously, increasing the rotational velocity leads to an increase in kinetic energy, which ultimately overcomes the energy loss induced by the presence of particles. The Reynolds number at which this shift occurs corresponds to a local minimum of Nusselt number.

On the other hand, Kang & Mirbod (Reference Kang and Mirbod2021) showed that in an SVF state, particle migration induced by shear leads to the accumulation of particles inside the core of one of the two vortices of a counter-rotating pair (and not both of them) resulting in an heterogeneous particle distribution. This non-homogeneity is intensified by increasing Reynolds number and concentration. We believe that the latter also may cause the reduction and variation of Nusselt number as can be seen in figure 6.

4.3. Onset of primary and secondary instabilities

Let us now consider the global picture and report the critical Reynolds number values for transition to the various flow states encountered. The average $\mathcal {R}_c$ related to all bifurcations and for all concentrations are listed in table 1, and represented in figure 7(a). These values are obtained from the average on three times repetition for each concentration, and the uncertainty (error bars on the figure) is estimated as the standard deviation of $\mathcal {R}_c$ on the three tests. For concentrations less than 6 %, no SVF is encountered before TVF, and so there are only two bifurcations, the primary bifurcation being from CCF to TVF. For concentrations above 6 %, the primary bifurcation is CCF–SVF, the secondary bifurcation is SVF–TVF, and the tertiary bifurcation is TVF–WVF. For the concentration of 28 %, as mentioned before, just the primary instability is captured and shown.

Figure 7. (a) Variation of critical Reynolds $\mathcal {R}_c$ corresponding to primary (square symbol), secondary (circle symbol) and tertiary (triangle symbol) bifurcation as a function of concentration. Values have been averaged over three experiments and the standard deviation serves as error bars. The solid line represent RU and the dashed line represents RD test. (b) Critical Reynolds number correspond to the primary instability (TVF or SVF), normalized by the critical Reynolds number for $\phi =0\,\%$, as a function of concentration. Comparison with experimental data of Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019), Majji & Morris (Reference Majji and Morris2018) and theoretical data of Gillissen & Wilson (Reference Gillissen and Wilson2019), Ali et al. (Reference Ali, Mitra, Schwille and Lueptow2002) is provided. Solid lines represent RU and dashed line represent RD protocol. (c,d) difference between $\mathcal {R}_c$ at the onset and $\mathcal {R}_c$ at the end of (c) SVF ($\phi \geqslant 6\,\%$ only) and (d) TVF in RU (full squares) and RD (full circles), and data from the literature. Error bars are cumulative uncertainties derived from each $\mathcal {R}_c$ detection.

Table 1. Summary of critical Reynolds numbers corresponding to primary, secondary and tertiary bifurcation and their standard deviation ($\pm$ intervals) for the suspension with concentration $0\,\% \leq \phi \leq 28\,\%$. Due to the significant increase in viscosity of the solution for the concentration of 28 % and shear rate limitations of the rheometer, higher-order instabilities could not be attained for this fluid and the test was limited to the detection of the primary bifurcation. These data values are represented in figure 7.

Several observations can be made from figure 7(a).

1. (i) There is a clear trend for a decrease in the critical $\mathcal {R}$ for the primary bifurcation with increasing particle concentration (the primary bifurcation being either CCF–TVF or CCF–SVF), and thus for a destabilization of CCF.

2. (ii) The critical $\mathcal {R}$ for transition to WVF is globally constant for $\phi <15\,\%$ and sharply decreases for the higher concentration measured.

3. (iii) A weakly hysteretic behaviour appears for primary bifurcations from $\phi >12\,\%$, with a difference between $\mathcal {R}_c$ of RU/RD. For secondary and tertiary bifurcations, it can be observed even sooner, for $\phi >3\,\%$.

For the secondary and tertiary bifurcations, $\mathcal {R}_c^{(down)}$ $< \mathcal {R}_c^{(up)}$ at all concentrations above $3\,\%$, corresponding to subcritical bifurcations. This is consistent with the results of Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) and Dash et al. (Reference Dash, Anantharaman and Poelma2020) ($\phi = 0.10, 0.20$), who reported subcritical behaviours and degrees of subcriticality ($\mathcal {R}_{c}^{(up)}-\mathcal {R}_{c}^{(down)}$) increasing with increasing $\phi$. The hysteretic behaviour is not trivial as far as the primary bifurcation is concerned: no hysteresis is observed for $\phi <3\,\%$ where this bifurcation involves the CCF–TVF transition, and no hysteresis is observed for $3\,\%<\phi \leq 9\,\%$ where it involves the CCF–SVF transition. For $9\,\%<\phi \leq 20\,\%$, $\mathcal {R}_c^{(up)} > \mathcal {R}_c^{(down)}$. The trend is reversed in the $\phi =28\,\%$ case where the primary bifurcation CCF–SVF also becomes subcritical ($\mathcal {R}_c^{(down)} < \mathcal {R}_c^{(up)}$). About apparent hysteresis effects of the onset of time-dependence in TCF, it was shown by Pfister & Gerdts (Reference Pfister and Gerdts1981) that there is ‘critical slowing down’ of the dynamics at the onset of wavy motion. Hence, ramping through the Hopf bifurcation point would have to be carried out extremely slowly to avoid apparent up and down parameter shifts in the estimates of the bifurcation point. Fortunately, these effects usually occur over small ranges of $\mathcal {R}$ but could well explain this small apparent hysteresis for TVF–WVF transition. Nevertheless, for both possibilities (apparent hysteresis or real hysteresis) it is still related to the presence of the particles, as for $\phi >3\,\%$ we can see this hysteresis.

Focusing on the effect of particles on the CCF stability, figure 7(b) shows the $\mathcal {R}_c / \mathcal {R}_{c,0}$ for the primary bifurcation as a function of concentration and comparison with data from different studies in the literature, both experimental or theoretical. In particular, the first black solid line represents the theoretical result of Gillissen & Wilson (Reference Gillissen and Wilson2019) (with geometrical parameters identical to the experiments of Majji et al. (Reference Majji, Banerjee and Morris2018)). In this study, the stability was estimated based on a suspension stress theory which ignores all non-hydrodynamic forces but accounts for fluid–particles interactions in non-dilute suspensions. An added extra stress is induced by the lubrication forces between the spheres, but no direct contact (collision) between them is considered. The theory also assumes no density difference between particle and fluid (fluid and particle inertia are the same on a volumetric basis, and hydrodynamic force are negligible). The results of Gillissen & Wilson (Reference Gillissen and Wilson2019) qualitatively agree with the theoretical result of Ali et al. (Reference Ali, Mitra, Schwille and Lueptow2002) (black dashed line in figure 7), based on a two-fluids theory that ignores interactions between spheres and is subsequently restricted to $\phi <5\,\%$, in the sense that particles globally lead to a destabilization of CCF with increasing particle concentration. The trend and asymptotic behaviour at $\phi =0\,\%$ are yet quite different from each other. Experimental results of Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019), with control parameter of $\varGamma =11$, $\eta =0.914$ close to the present study but with a difference in boundary condition (lid on top) and particle size ratio, $d_p/\delta = 37.5$, seem to follow the theory of Ali et al. (Reference Ali, Mitra, Schwille and Lueptow2002) in an almost straight line. An interesting agreement between their results and the present study is that in the moderate concentration, the difference between $\mathcal {R}_c^{up}$ and $\mathcal {R}_c^{down}$ grows ($\mathcal {R}_c^{down}-\mathcal {R}_c^{up}>0$). The results of Majji et al. (Reference Majji, Banerjee and Morris2018) are comprised between the trends of the two theories and also follow linear variations.

The results of the present study are also well framed by the two theories, yet closer to the trend of Gillissen & Wilson (Reference Gillissen and Wilson2019), especially in the low particle volume fraction where the forces (non-hydrodynamic) acting on the particles are modelled more accurately. This agreement underlines the importance of hydrodynamic particle–particle interactions and particle induced fluid stress, even at relatively low particle volume fractions. The limitation when comparing our results with the theory of Gillissen & Wilson (Reference Gillissen and Wilson2019) is that they assumed that primary instability modes were necessarily axisymmetry, while the primary instability modes for $\phi >3\,\%$ are here non-axisymmetric.

4.4. $\mathcal {R}$ range for flow state existence

Beyond the estimation of critical $\mathcal {R}$ values, interesting information can be obtained by measuring the span of Reynolds number spent in each flow state: $\mathcal {R}_{c_{{TVF}}} - \mathcal {R}_{c_{{SVF}}}$ for SVF and $\mathcal {R}_{c_{{WVF}}} - \mathcal {R}_{c_{{TVF}}}$ for TVF. Figure 7(c) shows the evolution of this quantity for (a) SVF and (b) TVF, in accordance with the results reported in figure 7(a,b).

The range of $\mathcal {R}$ corresponding to SVF firstly increases until a 15 % concentration is reached, and then reduces, in both RU/RD cases. In other words, there is an optimum concentration value for which the existence of SVF is maximized, regardless of the hysteretic behaviour of bifurcations, and the implications of this result in terms of torque and potential for mixing will be discussed later. Studies of Dash et al. (Reference Dash, Anantharaman and Poelma2020), Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) and Majji et al. (Reference Majji, Banerjee and Morris2018), on the other hand, did not report this non-monotonic behaviour.

The range of TVF existence, unlike that of SVF, simply decreases with increasing concentration. This decrease is all the more pronounced as the concentration exceeds 15 %, in accordance with the sharp decrease in the critical $\mathcal {R}$ for the onset of WVF reported in this range in figure 7(a). This trend is in good qualitative agreement with those of Majji et al. (Reference Majji, Banerjee and Morris2018) ($\phi <20\,\%$) and Dash et al. (Reference Dash, Anantharaman and Poelma2020). The difference in magnitude is due to the variations in aspect ratio (larger for Majji et al. (Reference Majji, Banerjee and Morris2018) and even larger for Dash et al. (Reference Dash, Anantharaman and Poelma2020)), smaller aspect ratios that lead to a damping of the waviness of the Taylor vortex, a delaying of the onset of WVF, and thus an increased span for TVF. Note that Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) report values similar to those of this study for a very similar aspect ratio, but yet found a concentration-independent behaviour for TVF ($\phi <25\,\%$).

4.5. Frequency map

In order to describe the unsteady features of some flow states (SVF, WVF), we here present frequency maps of RU/RD experiments for selected $\phi =0\,\%, 9\,\%, 15\,\%, 20\,\%$ values, in figure 8. Frequency maps at other $\phi$ were also produced, but not shown here for the sake of brevity.

Figure 8. Frequency maps in RU (a,c,e,g) and RD (b,d,f,h) protocols at $\phi =0\,\%$ (a,b), $\phi =9\,\%$ (c,d), $\phi =15\,\%$ (e,f) and $\phi =20\,\%$ (g,h). Horizontal axis limits are set on the same Reynolds interval for better comparison, which leads to empty regions as concentration (and viscosity) increase and limit the Reynolds range that can be explored. Note that $\phi =20\,\%$; the recording frequency was 140 f.p.s., corresponding to a maximum detectable frequency of 70 Hz, versus 35 Hz for the other cases where the recording frequency was 70 f.p.s. The vertical axis has been scaled accordingly.

Characteristic frequencies can be captured as long as they are below the Nyquist criterion ($f< 1/2 \times f_{acq}$ where $f_{acq}$ is acquisition frequency), here equal to 35 Hz for the first three cases and 70 Hz for $\phi =20\,\%$. All frequency maps share a common feature: the presence of a clear line with a frequency linearly increasing with $\mathcal {R}$, that can be observed for example between $f=5$ Hz at $\mathcal {R}=150$ and $f=10$ Hz at $\mathcal {R}=250$ in the $\phi =0$ case. It can be shown that the characteristic frequency of those lines simply scale as $f=f_{\varOmega } / 2{\rm \pi}$ where $f_{\varOmega }$ is an inner cylinder rotation frequency. This line thus corresponds to the ability of the method to detect the rigid body rotation frequency of the inner cylinder, which varies linearly with $\varOmega$ and thus with $\mathcal {R}$. Harmonics $f_k=k \times f_{\varOmega } / 2{\rm \pi}$ can also be identified in some cases (Dutcher & Muller Reference Dutcher and Muller2009; Cagney & Balabani Reference Cagney and Balabani2019).

Now, CCF and TVF being steady laminar states, they are not expected to display any spectral signature other than the line associated with the rotation speed of the inner cylinder and its harmonics. This can be verified for example in figure 8 $\phi =0\,\%$ (RU/RD), where CCF and TVF are essentially indistinguishable for the frequency map alone.

In the SVF state ($\phi >6\,\%)$ a clear line appears at a frequency lower than $f_{\varOmega }$ during its lifetime, along with higher-order harmonics (additional fainter lines seen above), as seen in figure 8 for $\phi =9\,\%-20\,\%$. This frequency simply scales as the vortex travelling velocity divided by the spatial wavelength. In other words, the intensity signal probed at a given altitude fluctuates as vortices travel axially. The SVF lines disappear when the flow enters the TVF regime, thus allowing this transition to be evidenced on the frequency maps as well as on the flow maps reported previously. Interestingly, the frequency of SVF was always $f_{{SVF}}\approx 0.5 f_{\varOmega }$.

The onset of WVF finally corresponds to the appearance of one or several distinct additional lines, the frequency of which are not harmonics of $f_{\varOmega }$. For the particle-free fluids case ($\phi =0\,\%$, see figure 8), the main wavy frequency at the onset of WVF is such that $f_{{WVF}}>f_{\varOmega }$ , consistent with Cagney & Balabani (Reference Cagney and Balabani2019) and Lacassagne et al. (Reference Lacassagne, Cagney and Balabani2021), and more specifically $f_{{WVF}}/f_{\varOmega }=2.83$ close to the value of 2.5 reported by Coles (Reference Coles1965).

However, unlike what was observed in Cagney & Balabani (Reference Cagney and Balabani2019) and Lacassagne et al. (Reference Lacassagne, Cagney and Balabani2021), $f_{{WVF}}$ decreases with increasing $\mathcal {R}$, unlike $f_{\varOmega }$. This discrepancy is believed to be due to the slight difference in upper boundary condition (free surface here versus rigid lid in Cagney & Balabani (Reference Cagney and Balabani2019) and Lacassagne et al. (Reference Lacassagne, Cagney and Balabani2021)). Note that while Majji et al. (Reference Majji, Banerjee and Morris2018) did not produce frequency maps, the flow maps reported in their figure 5 suggest that they would also have observed a decrease in $f_{{WVF}}$ with similar boundary conditions, as in this work.

It can be noted that the dynamic frequency behaviour of flow regime is an analogue in RU/RD protocols (see figure 8), not only for $0\,\%$ but also for the suspension cases reported hereinafter. When the volume fraction is increased, the first effect observed is that of a decrease in the $f_{{{WVF}}}$ (see figure 8, $\phi = 0\,\%-15\,\%$). The decreasing behaviour is retrieved for the $20\,\%$ case, with a dramatic increase of $f_{{WVF}}$ (see figure 8, $\phi =20\,\%$). Unlike the WVF state, $f_{{SVF}} \simeq 0.5 f_{\varOmega }$ for all the concentration. On the other hand, the modulation of WVF by particle concentration with increasing Reynolds number ($\text {slope}_{f_{{WVF}}}/\text {slope}_{f_{\varOmega }}$) seems independent of particle concentration. Dash et al. (Reference Dash, Anantharaman and Poelma2020) previously reported values of $f_{{WVF}}/f_{\varOmega }$ at the onset of WVF between 3.5–3.9 for particle concentration between 3.4 % and 30 %, with yet a higher aspect ratio and lower gap to particle ratio. However, the wavy frequency for their particle-free case was not presented. Thus, the question of whether the origin of this difference is the aspect ratio, the gap-to-particle ratio, or combination of factors remains open.

The dynamics of SVF and WVF can be further described by comparing two indicators: the value of the dominant characteristic frequency at the onset of SVF and WVF (at the critical $\mathcal {R}_c$), here scaled by the $f_{\varOmega }$ at this same onset and written $f_{{SVF}}/f_{\varOmega }$ and $f_{{WVF}}/f_{\varOmega }$; and the ratio of the frequency slope of the SVF and WVF main line to the frequency slope of the inner cylinder. This is done for all $\phi$ values (included those illustrated in figure 8). The two indicators are reported in figure 9(a,b) (in absolute value), respectively. Note that the first indicator is also illustrated by empty circles plotted at the onset where the frequency is probed, in figure 8. In figure 9(a), arrows also indicate the trend for the evolution of the wavy frequency after the onset (increasing for upward pointing arrow, decreasing for downward pointing arrows). The RU/RD protocols are shown with square/circle symbols, respectively.

Figure 9. (a) Dominant characteristic frequency of wavy vortices at the onset of the WVF state normalized by the inner cylinder frequency ($f_{{WVF}}/f_{\varOmega }$), plotted as a function of concentration. The up and down arrows illustrate the increasing or decreasing trend of this wavy frequency with increasing $\mathcal {R}$ after the onset. (b) Absolute values of the ratio between the slope of the dominant wavy frequency with $\mathcal {R}$, and the slope of $f_{\varOmega }$. The $+$ (negative, respectively) symbols indicate that the ratio was positive (negative, respectively), i.e. that the frequency of the inner cylinder and WVF state have gradients of the same sign (opposite, respectively) along $\mathcal {R}$. Square and circle symbols in both (a,b) plots represent RU/RD protocols, respectively (note that the value of $f_{{WVF}}/f_{\varOmega }$for $\phi =12\,\%$ in left-hand figure is equal to 0.02).

Starting with the first indicator (frequency at the onset), a first observation is the fact that the trend for the evolution of $f_{{WVF}}$ with increasing $\mathcal {R}$ is a decreasing one for all concentrations except for $\phi =12\,\%$ and $15\,\%$. Suspension behaviour, in the WVF state, can be divided into three parts according to this indicator. In the first part for $\phi \leqslant 9\,\%$, the frequency is globally higher than the frequency of the inner cylinder (above the horizontal dashed line equal to 1 in figure 9a) and experiences a decreasing trend.

At $\phi =$ $9\,\%$, the frequency of the wavy state is approximately equal to the frequency of the inner cylinder, and a further increase in particle concentration leads to entering a second part where $f_{{WVF}} < f_{\varOmega }$ at the onset of the wavy state and the wavy frequency evolution trend reverses to become increasing with $\mathcal {R}$. The minimum onset frequency is found at $\phi =12\,\%$. The third part is for 15 % and above, where there is an abrupt increase in frequency but again the trend is declining. The third part, here illustrated by the single $\phi =20\,\%$ data point, corresponds to a dramatic increase in $f_{{WVF}}/ f_{\varOmega }$ and a return to a decreasing trend.

Figure 9(b) brings additional information on those trends, thanks to the second indicator which relates to the intensity of the decrease/increase of the WVF state frequency relative to the inner cylinder frequency ($\text {slope}_{f_{{WVF}}}/\text {slope}_{f_{\varOmega }}$). This indicator is reported in absolute values in order to better compare the magnitudes, but $+$ or $-$ signs indicate whether the ratio was increasing (stemming from an increasing wavy frequency) or decreasing (when the wavy frequency decreased with $\mathcal {R}$ in the previous figure). The most striking result is that the magnitude of this indicator remains roughly constant with concentration, regardless of the sign of the slopes, values varying between 4.2–5.6, i.e. on a $\pm 15\,\%$ interval around the average 4.9 value. For the SVF, this indicator is ${\approx }0.5$ for all concentrations.

It is finally worth noting that there is almost no difference in the RU/RD process for the two indicators, except for the slope ratio at $\phi =20\,\%$, indicating that the dynamics of WVF at and after the onset is independent of the potentially hysteretic nature of the bifurcation allowed to trigger it. The fact that increasing the concentration (and so increasing effective viscosity) first leads to a decrease of the frequency ratio at the onset of the WVF state (until $\phi =12\,\%$) seems consistent with the rise in effective viscosity. Yet after that, in the intermediate concentration range, the expected trend is reversed. It is believed that this non-trivial behaviour marks the appearance of significant interactions between particles, overcoming hydrodynamic forces. Further increasing particle concentration up to $\phi =20\,\%$, particle–particle interactions are expected to become dominant.

4.6. $\mathcal {N}$ versus $Ta$ scaling

In the previous sections, we have described the succession of flow states and their dynamic behaviour using flow visualization coupled with torque measurements in order to accurately detect flow transitions. In this section and the following, we will use only torque measurements to discuss the friction dynamic behaviour associated with each flow state. Figure 10 shows consolidated (average from three tests) plot of Nusselt number variation as a function of Taylor number for the inner cylinder for all suspensions in RU/RD experiments (some of the curves were previously reported in figure 4 for $\mathcal {N}$ as a function of $\mathcal {R}$). The solid lines and the dashed lines represent RU/RD experiments, respectively. Here the Taylor number is used rather than $\mathcal {R}$ in order to compare our scaling exponents with those reported in the literature. It should be noted that $Ta$ can be expressed as a function of $\mathcal {R}$ and of geometrical parameters, as described in § 1, so for a given geometry, both control parameters can equally be used.

Figure 10. (a) Consolidated plots (averaged on all repeats at identical $\phi$) for the Nusselt number as a function of the Taylor number. Panel (b) reports the evolution of $\alpha$ and (c) $\beta$ coefficients with $\phi$ where $\mathcal {N}\sim Ta^{\alpha }$ and $\mathcal {N}\sim Ta^{\beta }$ are power law scalings in TVF and WVF, respectively. Full lines and squares, and dashed lines and circles, illustrate RU/RD experiments, respectively. Insets in panel (a) show plots of $\mathcal {N} \times Ta^{-\alpha }$ (left-hand inset) and $\mathcal {N} \times Ta^{-\beta }$ (right-hand inset) as a function of Reynolds number, justifying the power law fittings in TVF (left-hand inset) and WVF (right-hand inset), respectively. The vertical dashes denote the TVF range and onset of WVF in the left-hand and right-hand insets, respectively.

For each curve, the TVF and WVF domains are then extracted and fitted by functions of the type $\mathcal {N}\sim Ta^p$, with $p=\alpha$ for TVF and $p=\beta$ for WVF. Values of the $\alpha$ and $\beta$ exponents are then represented as a function of $\phi$ in the two upper left-hand-corner insets. This allows us to compare the slopes of $\mathcal {N}$–Ta plots for selected flow states at all particle concentrations. In the $\phi =0\,\%$ case, $\alpha \simeq 0.6$, and $\beta \simeq 0.3$, with no obvious difference between RU/RD protocols. As $\phi$ increases up to $15\,\%$, the $\alpha$ slope for Nusselt number versus Reynolds number in the TVF state gradually decreases, down to a value of 0.49 in RU, and a sensible difference between RU/RD exponents appears. Then it is reversed after 15 %, with 0.53 for $\phi =20\,\%$. A similar decay with increased $\phi$ is observed for the $\beta$ exponent, this time until the $\phi =12\,\%$ concentration is reached 0.18. The trend is then reversed with a further increase in $\beta$ for $\phi =15\,\%$ and $20\,\%$. For higher $Ta$ values still in the WVF regime, attained by RU experiments, and with $\eta = 0.917$ and $\varGamma = 21.7$, Dash et al. (Reference Dash, Anantharaman and Poelma2020) reported that a $\mathcal {N}(\mu _{l})\sim Ta^{0.23}$ scaling (Nusselt number based on the solvent viscosity $\mu _l$, which is not expected to change the scaling exponent). Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) in their figure 27, on the other hand, report a $\mathcal {N} \sim Ta^{0.26}$ scaling for $\phi =10\,\%$ RD experiments and $\mathcal {N} \sim Ta^{0.35}$ for $\phi =20\,\%$. Our results thus agree well with the range observed in the literature.

The presence of suspended particles at concentrations below $\phi =15\,\%$ in the TVF state apparently reduces the transverse momentum transfer (and so there is added energy dissipation in the system). This behaviour is reversed at higher concentrations. It is believed that part of the explanation can be found in the existence of the shear induced migration of particles which comes from non-homogeneous microstructure between them in suspension and this may result in enhancing the transverse momentum transfer in the system and so on torque (in other words drag reduction). Likely the reason for this early slope change in the $\phi =12\,\%$ WVF state may be because of increasing interaction between particles, while the suspension is well mixed in this flow regime, so this enhancing in transverse momentum transfer begins earlier.

4.7. Friction coefficient

The above results can be completed by showing another parameter quantifying the torque exerted on the inner cylinder, namely the friction factor. It is a dimensionless measure of the torque that the fluid exerts on the inner rotating cylinder and is computed as the ratio of the wall shear stress to the dynamic pressure. This parameter relates quite well to a common requirement in practical applications, the need to quantify the power required to overcome the frictional drag of a rotating shaft. The friction coefficient for a rotating cylinder can be defined as follows (Kang & Mirbod Reference Kang and Mirbod2021):

(4.1)\begin{equation} C_{M_{z}}=\frac{\mathcal{\tau}}{\rho {\rm \pi}r_{i}^{2} h (r_{i} \varOmega_{i})^{2} / 2}. \end{equation}

This can be expressed as is such that

(4.2)\begin{equation} G=f(\eta)\times C_{M_z} \times \mathcal{R}^2 \end{equation}

with the suspension Reynolds number ($\mathcal {R}= \rho \left (r_{i} \varOmega _{i} \right ) \delta / \mu _s =$ $\rho \dot {\gamma } \delta ^{2} / \mu _s$). Thus

(4.3)\begin{equation} \mathcal{N} = G / G_{lam} \simeq g (\eta) \times C_{M_z} \times \mathcal{R}. \end{equation}

So $C_{M_z}$ is the coefficient of proportionality between $\mathcal {N}$ and $\mathcal {R}$. It is in some way related to the slope of $\mathcal {N} = f(Ta)$ curves since for these we write

(4.4)\begin{equation} \mathcal{N} \sim Ta^\alpha. \end{equation}

Figure 11 shows the evolution of friction coefficient for $\phi =0\,\%$ and $15\,\%$. In the $\phi =0\,\%$ case, CCF, TVF and WVF are identified by black, blue and red lines, respectively. In the $\phi =15\,\%$ case the additional SVF state is shown in purple. Friction coefficients corresponding to RU/RD experiments are plotted in solid and dashed lines, respectively.

Figure 11. Evolution of friction coefficient for $\phi =0\,\%$ and $15\,\%$. In the $\phi =0\,\%$, CCF, TVF and WVF are identified by black, blue and red lines, respectively. In the $\phi =15\,\%$ the additional SVF state is shown in purple. Friction coefficients corresponding to RU/RD experiments are plotted in solid and dashed lines, respectively.

In the $\phi =0\,\%$ case, the friction coefficient decreases quickly in CCF, down to the point at which the primary instability occurs in the system ($\mathcal {R}=150.8$ in RU and $\mathcal {R}=148$ in RD). This point is the local minimum value in terms of the friction coefficient. Afterwards, further increase in $\mathcal {R}$ leads to the development of counter-rotation vortices characteristic of TVF states, and the friction coefficient increases until a local maximum value is reached, just before the secondary bifurcation to WVF occurs ($\mathcal {R}=200$ in RU and $\mathcal {R}=148$ in RD). The friction coefficient then decreases again with increasing $\mathcal {R}$ in the WVF state. So, TVF thus corresponds to a state where the friction coefficient is maximized, and low values of $C_{M_z}$ are encountered for high $\mathcal {R}$ or locally just before the onset of TVF.

In the $\phi =15\,\%$ case, the CCF behaviour is similar to the one for $\phi =0\,\%$. When entering the SVF state (purple line, $\mathcal {R}=132$ in RU and $\mathcal {R}=138$ in RD), the decreasing $C_{M_z}$ decays even more rapidly with $\mathcal {R}$, allowing the friction coefficient to reach lower values than those encountered for $\phi =0\,\%$. A local minimum of $C_{M_z}$ is reached, this time not at the transition between CCF and TVF, but in the middle of the SVF regime. This minimum is preserved until the transition to the TVF state occurs right after this point at $\mathcal {R}=163$ in RU mode (and $\mathcal {R}=161$ in RD mode), and the friction coefficient increases again with increasing $\mathcal {R}$, moderately this time. When the WVF state is reached, $C_{M_z}$ begins to decrease, with a slope similar to the one observed for $\phi =0\,\%$.

The general behaviour of the friction coefficient with $\mathcal {R}$ in CCF, TVF and WVF is thus comparable in the two cases, but an additional interesting behaviour is introduced by SVF. Kang & Mirbod (Reference Kang and Mirbod2021) also reported in their numerical study that spiral patterns caused by the presence of particles tend to reduce the friction coefficient on the inner cylinder for the suspension of 10 %. The axial travelling of SVF reduces the convective momentum transfer in the radial direction; in consequence, it results in the reduction of the Nusselt number and torque acting on the inner cylinder.

Summarizing §§ 4.6 and 4.7, the effect of particle concentration on $\alpha$ and $\beta$ coefficients suggests that transverse momentum transfer is enhanced at concentrations above 15 % in the TVF state, and at concentrations above 12 % in the WVF state. On the another hand, in the semidilute suspension (concentrations below 12 %) there is (i) a significantly reduced friction coefficient in SVF and WVF states; and (ii) a weaker increase of friction coefficient in TVF state compared with particle-free solutions. This is an interesting input in terms of power consumption efficiency. Note that the Nusselt number and friction coefficient are related together: a reduction in transverse momentum transfer means a lower friction coefficient.

4.8. hydrodynamic concentration subregimes

Typically in the particle-suspension literature (Ancey & Vollm Reference Ancey and Vollm2005; Morris Reference Morris2009), particle suspensions are said to be in the dilute regime for $\phi <5\,\%$, semidilute regime for $5\,\%\leq \phi \leq 30\,\%$ and concentrated regime for $\phi >30\,\%$. According to this nomenclature, all of the particle suspensions used here belong to the dilute and semidilute regimes. It is worth noting that this classification was built on cases without inertia, at vanishing $\mathcal {R}$. The various parameters identified in this work allow us to identify three distinct concentration subregimes (separated by two critical concentrations), in which the hydrodynamic behaviours are quite different, and that we will thus label as ‘hydrodynamic concentration subregimes’ (HCR).

1. (i) The first concentration regime (HCR1) is found for $\phi <6\,\%$, and is included in the conventional dilute regime. It corresponds to a hydrodynamic behaviour very similar to the $\phi =0\,\%$ with only a minor increase in torque corresponding to the minor increase in suspension apparent viscosity: Nusselt scalings with $Ta$ are similar, unlike what was observed in Majji et al. (Reference Majji, Banerjee and Morris2018), there is no evident decrease in critical Reynolds numbers for the onset of primary and secondary instability, and subsequently no change in parameter range for flow states existence, or in the nature of such bifurcations. Transitions do not appear hysteretic. Only the frequency map analysis allows us to detect an effect of particles on unsteady flow dynamics: the wavy frequency at the onset of WVF decreases with increasing particle concentration, indicating that the presence of particles even at very low concentrations plays a role in the selection of the most unstable mode responsible for the development of WVF.

2. (ii) The second regime (HCR2) corresponds to the $6\,\%\leq \phi <15\,\%$ range and is also included in what would be called the dilute regime when looking only at the particle concentration value. However, from a hydrodynamic point of view, it is very different from the previous one (HCR1). The torque exerted on the inner cylinder significantly increases in magnitude owing to the increases suspension viscosity, but more importantly, a new unsteady non-axisymmetric flow state appears between CCF and TVF, SVF modifying the nature of the primary instability and a sensible decrease in its critical Reynolds number, in good qualitative agreement with the theory of Gillissen & Wilson (Reference Gillissen and Wilson2019). The $\mathcal {R}$ range of SVF increases with yet no particular effects on the $\mathcal {R}$ range of TVF, and hence on the onset of WVF. Yet, the higher-order transitions (SVF–TVF or TVF–WVF) become notably subcritical. Particle concentration also has an influence on the dynamics in the WVF regime, indeed both the frequency at the onset and the Nusselt scaling exponent vary non-monotonically with a local minimum around $12\,\%$ or $15\,\%$. The Nusselt scaling exponent in TVF also decreases in this HCR2 range. All of this results in a minimum friction coefficient found at $\phi =15\,\%$ in the middle on the SVF $\mathcal {R}$ range, and indicates a complex interplay between particles and fluid flows.

3. (iii) The last regime (HCR3) corresponds to the $\phi >15\,\%$ cases investigated here and would thus be equivalent to the ‘semidilute’ concentration regime. In that regime, there is a dramatic increase of viscosity and thus of the torques exerted on the inner cylinder, but also in Nusselt scaling exponent for TVF and WVF state. The $\mathcal {R}$ ranges of SVF and TVF are reduced in accordance with the global destabilization of all intermediate flow regimes (SVF, TVF). The frequency of the mode responsible for the onset of WVF is also greatly increased.

It is believed that the origin of this behaviour, and thus the nature of bifurcation, comes from the competition between particle–fluid and particle–particle interaction, the latter becoming gradually dominant over the former as the concentration of regime is changing.