3.1 Calibration with pure fluid

To calibrate the experimental apparatus, the annulus was filled with pure fluid for which the diffusion time scale is $t_{d}\approx 2.1~\text{s}$ . The inner cylinder rotation was increased quickly from rest to $Re=90$ and held at this condition for $20$ minutes to establish a steady circular Couette flow (CCF) in the annulus. The $Re$ was then increased slowly at a rate of $\text{d}Re/\text{d}t=0.0061~\text{s}^{-1}$ ( $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=0.013$ ) to $Re=210$ . The flow structures during this $5.5$ hour experiment were photographed using the low speed camera at an interval of 50 s between frames. At regular intervals, photographs for short durations ( $t=2.2~\text{s}$ ) at a rate of $60~\text{frames}~\text{s}^{-1}$ were recorded using the high speed camera.

Figure 4.

(a) Photographs of the flow structures, and (b) space–time diagrams of CCF, partial CCF, TVF and WTV for pure fluid at $Re=90,113,125$ and $152$ , respectively, during an increasing- $Re$ ramp with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=0.013$ . The space–time diagrams are constructed from images taken at $60~\text{frames}~\text{s}^{-1}$ . Note that at each $Re$ in (b), the duration is $\unicode[STIX]{x0394}t=2.2~\text{s}$ .

Figure 5.

Space–time diagrams constructed from images taken at intervals of 50 s between frames over a $5.5$ hour period for pure fluid: (a) during increasing- $Re$ ramp from $Re=90$ to $Re=210$ with $\text{d}Re/\text{d}t=0.0061~\text{s}^{-1}$ ( $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=0.013$ ), and (b) during decreasing- $Re$ ramp from $Re=210$ to $Re=90$ with $\text{d}Re/\text{d}t=-0.0061~\text{s}^{-1}$ ( $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.013$ ). Flow transitioned from CCF to TVF to WTV during the increasing- $Re$ ramp and from WTV to TVF to CCF during the decreasing- $Re$ ramp.

Figure 4(a) shows four photographs of different flow structures at $Re=90$ , 113, 125 and 152. Space–time diagrams, which show the time modulation of the flow structures, are constructed from sequences of such photographs by extracting the light intensity from an area of $1$ pixel width (at the centre of the image in the horizontal direction) over the full length of the annulus and plotting it as a function of $t$ . Figure 4(b) shows the space–time diagrams at various $Re$ , corresponding to the flow structures shown in figure 4(a), constructed from photographs recorded at a rate of $60~\text{frames}~\text{s}^{-1}$ for a duration of 2.2 s. Since the slowly ramped $Re$ changes negligibly over this time, each space–time diagram constructed from such data represents a particular $Re$ . Figure 5(a) shows the space–time diagram constructed from photographs recorded at intervals of 50 s between frames for $90\leqslant Re\leqslant 210$ . As the inner cylinder rotation rate at any given time is known, time on the horizontal axis can be replaced by $Re$ . At $Re=90$ , the flow was observed to be circular Couette flow (CCF) with Ekman rolls near top and bottom boundaries. This flow state was observed to be stable with increase in $Re$ . Close to $Re=120$ , counter-rotating pairs of Taylor vortices appeared near top and bottom boundaries with a stable CCF in the middle section. These rings eventually filled the whole length of the annulus, transitioning into Taylor vortex flow (TVF) at $Re=120$ . This is in agreement with experiments and linear stability theory for the radius ratio of $\unicode[STIX]{x1D702}=0.877$ (Andereck et al. Reference Andereck, Liu and Swinney1986; Recktenwald, Lücke & Müller Reference Recktenwald, Lücke and Müller1993; Ali et al. Reference Ali, Mitra, Schwille and Lueptow2002). The TVF state was observed to be stable for $120\leqslant Re\leqslant 151$ . At $Re=151$ , TVF transitioned into wavy Taylor vortices (WTV) in which the vortices oscillate in sinusoidal fashion in the azimuthal direction. With increase in $Re$ , the WTV state was observed to be stable up to $Re=210$ . The time modulation of CCF, partial CCF, TVF and WTV flow states can be seen in the space–time diagrams at $Re=90,113,125$ and $152$ , respectively, in figure 4(b).

Figure 5(b) shows the space–time diagram when $Re$ was reduced from $Re=210$ to $Re=90$ at a rate of $\text{d}Re/\text{d}t=-0.0061~\text{s}^{-1}$ ( $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.013$ ). The flow in this decreasing- $Re$ ramp experiment was observed to follow the sequence of WTV $\rightarrow$ TVF $\rightarrow$ CCF. These flow structures at various $Re$ and the transition $Re$ corresponding to WTV–TVF and TVF–CCF transitions match closely to those observed in the increasing- $Re$ ramp, showing no hysteresis in the pure fluid experiments.

3.2 Procedure for experiments with suspension

The objective of the present work is to understand deviations in flow transitions of fluid–particle suspensions compared to those of the pure fluid. The particle concentrations probed in the present work, including the pure fluid ( $\unicode[STIX]{x1D719}=0$ ), are $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ for $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D719}=0.10$ for $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ . We have applied a protocol in which $Re$ was ramped down rather than up for the study of the suspensions. In this protocol, the inner cylinder rotation rate was increased as rapidly as possible from rest to a value of $Re$ yielding WTV (typically $Re>160$ ). Subsequently, $Re$ was reduced in a quasi-steady manner with $|\text{d}Re/\text{d}\unicode[STIX]{x1D70F}|\ll 1$ until CCF was established in the full length of the annulus. The flow structures were photographed with two cameras as described in § 2. The rationale for not using the standard increasing- $Re$ protocol came from a set of experiments we performed with very dilute suspensions ( $\unicode[STIX]{x1D719}<0.01$ ), which showed that an increasing- $Re$ ramp could lead to concentration inhomogeneity along the annulus. In these experiments, we observed CCF at low $Re$ . As the $Re$ was increased, Taylor vortices appeared near top and bottom boundaries with circular Couette flow in the middle section. Associated with this flow variation along the annulus length, we observed particle depletion in the Couette region and particle accumulation in Taylor vortex rings; while the details of the mechanism leading to this concentration inhomogeneity are not clear, it is apparently due to a secondary flow in the Couette region driven by the requirement of continuity with the azimuthally directed vorticity in the Taylor rings near top and bottom boundaries. Hence, the increasing- $Re$ protocol does not give the transition $Re$ corresponding to uniform suspension concentration. To avoid particle segregation, for experiments with suspension we used the decreasing- $Re$ protocol starting with WTV and reaching CCF at the end. Because all the flow states prior to reaching CCF have axial and radial velocity components, this method maintains a quite uniform distribution of the particles. Thus the flow states are expected to correspond closely to those with fully uniform suspension concentration.

A detailed discussion of experiments with $\unicode[STIX]{x1D719}=0.10$ suspension is presented in § 3.3 including the effect of particle size on the flow transitions. Suspension experiments with $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ to study the effect of particle loading on the flow transitions are presented in § 3.5, following a discussion of spiral and ribbon structures in § 3.4.

3.3 Detailed example: $\unicode[STIX]{x1D719}=0.10$

3.3.1 Quasi-steady experiments

For $\unicode[STIX]{x1D719}=0.10$ suspensions of both particle sizes ( $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ and $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ ), the effective viscosity is $\unicode[STIX]{x1D707}_{s}\approx 1.44\unicode[STIX]{x1D707}_{l}$ and the diffusion time scale $t_{d}\approx 1.5~\text{s}$ . The flow transition values of $Re$ and the flow structures observed differ substantially from those observed with pure fluid. Note that for all suspensions we compute $Re$ using the effective viscosity of the suspension, $Re=\unicode[STIX]{x1D6FF}d_{i}\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D70C}/2\unicode[STIX]{x1D707}_{s}(\unicode[STIX]{x1D719})$ . We recall that only the inner cylinder was rotated while the outer cylinder was kept stationary. In these experiments, the inner cylinder $Re$ was increased very rapidly from rest to a $Re$ corresponding to wavy vortex flow and held at this condition for $20$ minutes to ensure fully established WTV flow. The inner cylinder rotation rate was then reduced at $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ until the full annulus was observed to be in CCF.

Figure 6.

Space–time diagrams for (a) pure fluid, (b) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles, and (c) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles, each constructed from images taken at an interval of 50 s between frames from decreasing- $Re$ experiments with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The $Re$ values corresponding to flow transitions are labelled. The flow transition sequence is from WTV to TVF to CCF for the pure fluid while for $\unicode[STIX]{x1D719}=0.10$ suspensions, the sequence is from WTV to TVF to non-axisymmetric states (spirals and ribbons) to CCF. Flow transitions took place at lower $Re$ for the suspensions compared to the pure fluid, while reduction of particle size reduced the range of $Re$ over which the non-axisymmetric flow states are seen.

Figure 7.

Space–time diagrams at various $Re$ for (a) pure fluid, (b) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles, and (c) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles, each constructed from images taken at $60~\text{frames}~\text{s}^{-1}$ from a decreasing- $Re$ experiment with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The diagram for each $Re$ represents a time duration of 3.8 s for pure fluid and 4 s for the suspensions. The flow transitions were from WTV to TVF to CCF for the pure fluid. For $\unicode[STIX]{x1D719}=0.10$ suspension of $230~\unicode[STIX]{x03BC}\text{m}$ particles, additional spiral vortex flow and ribbons appeared between TVF and CCF. For $\unicode[STIX]{x1D719}=0.10$ suspension, when the particle size is reduced to $d_{p}=70~\unicode[STIX]{x03BC}\text{m}$ , only the ribbon state appeared between TVF and CCF.

Figure 8.

Photographs of flow structures for $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles observed during a decreasing- $Re$ ramp with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The red rectangle in the top left section shows the area photographed for this figure. WTV and TVF states can be seen at $Re=134.5$ and $Re=118$ , respectively. Spirals merging into the bottom Ekman layer can be seen in the SVF at $Re=107.6$ , as in the space–time diagrams for $112\geqslant Re\geqslant 109$ in figure 7(b). The RIB state at $Re=110$ shows the azimuthal location where upward rotating vortices change into downward rotating vortices and vice versa at the same axial height (also seen in the space–time diagram for $Re=108$ in figure 7(b)).

Figure 9.

Compilation of $Re$ corresponding to various flow structures and transitions for (a) pure fluid, and suspensions at $\unicode[STIX]{x1D719}=0.10$ for particles of size (b) $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles, and (c) $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles. Each case displays the average of seven decreasing- $Re$ ramp experiments at $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . Error bars show the standard deviation from the average transition $Re$ . Non-axisymmetric flow structures (SVF, RIB) appear for the suspensions which do not appear for pure fluid. SVF and RIB appeared for the suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ , while only RIB appeared for $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ .

Figure 6 shows the space–time diagrams constructed from photographs recorded at intervals of 50 s for three cases: (a) pure fluid for $209\geqslant Re\geqslant 89$ , (b) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles for $145\geqslant Re\geqslant 93$ , and (c) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles for $145\geqslant Re\geqslant 93$ . In the range of $Re$ studied, the flow transition sequence was from WTV to TVF to CCF for the pure fluid. By contrast, for the $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles, the flow transitions were from WTV to TVF to non-axisymmetric flow states (spirals and ribbons, as described below) to CCF. When the particle size was reduced to $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ , the suspension followed the same sequence of flow transitions as those for the $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particle suspension, but the $Re$ range corresponding to the non-axisymmetric flow states was reduced.

Figure 7 shows space–time diagrams constructed from 3–4 s durations of $60~\text{frames}~\text{s}^{-1}$ images for (a) pure fluid, (b) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles, and (c) $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles. For the $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles, space–time diagrams at $Re=145,131,130$ and $125$ show counter-rotating pairs of wavy Taylor vortices (WTV) with larger azimuthal wavelength at smaller $Re$ . Between $Re=125$ and $Re=124$ , a transition from WTV into Taylor vortex flow (TVF) is observed. Space–time diagrams at $Re=124$ and $113$ show axisymmetric counter-rotating pairs of Taylor vortices filling the full annulus length; note that non-CCF states persist to lower $Re$ than seen for the pure fluid.

In addition to the reduced $Re$ for the transition back to CCF, between $Re=113$ and $Re=112$ , axisymmetry is lost as the TVF breaks down to form spiral vortex flow (SVF). The space–time diagram at $Re=112$ shows spiral structure in the middle section of the annulus with a pair of counter-rotating vortices spiralling around the inner cylinder merging into Taylor vortices at top and bottom. The merging location of spirals and Taylor vortices creates defects and disturbs the axisymmetry of these Taylor vortices, resulting in their azimuthal waviness. Between $Re=109$ and $Re=108$ , the spirals give way to another non-axisymmetric flow structure called ribbons (RIB), in which pairs of time-modulated counter-rotating vortices fill the annular length. The two counter-rotating vortices in each pair alternate in time between upward and downward rotation (as seen at the outer wall). The space–time diagram for the RIB state can be seen in figure 7(b). Between $Re=108$ and $Re=106$ , the flow transitioned from ribbons into CCF.

Figure 8 shows photographs of flow structures for $\unicode[STIX]{x1D719}=0.10$ and $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ : the flow states observed are WTV at $Re=134.5$ , TVF at $Re=118$ , SVF at $Re=107.6$ , RIB at $Re=106.6$ and CCF at $Re=82.8$ . The spirals shown in figures 7 and 8 appear to translate upwards in the laboratory frame of reference, but in other experiments with the same parameters, spirals translating downwards were observed indicating symmetry with respect to this flow state. For the smaller particles of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ , the space–time diagrams in figure 7 show the flow transition with decreasing $Re$ from WTV to TVF to RIB to CCF without appearance of SVF.

The $Re$ ramp down experiments for pure fluid, as well as suspensions at $\unicode[STIX]{x1D719}=0.10$ for both $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ and $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles, described above were repeated seven times for each material. In two of the seven experiments, we started at $Re$ corresponding to WVF and then ramped down the rotation rate until CCF was observed. In the other five experiments we started at $Re$ corresponding to TVF and ramped down to CCF. In figure 9, $Re$ corresponding to various flow transitions are compiled with error bars showing the standard deviation from the average value of transition $Re$ ; figure 7 shows space–time diagrams of the flow structures. The first primary difference between the pure fluid and suspension is that the boundary between WTV and TVF for $\unicode[STIX]{x1D719}=0.10$ and $d_{p}=230~\unicode[STIX]{x03BC}\text{m}$ was observed at smaller $Re$ than for pure fluid. The pure fluid went through a transition from axisymmetric TVF to axisymmetric CCF. By contrast, an even more striking deviation from the pure fluid behaviour was seen for the suspension in an interval of $113\geqslant Re\geqslant 107$ , as stable non-axisymmetric flow structures (SVF and RIB) appeared between the axisymmetric TVF and CCF. The transition into CCF for $\unicode[STIX]{x1D719}=0.10$ and $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ is at $Re=107$ , well below the $Re=120$ found for this transition in the pure fluid. For the smaller particle size of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ , the non-axisymmetric region consisted only of RIB, as SVF did not appear. As a final point, the range of $Re$ for this region reduced to $107.8\geqslant Re\geqslant 106.8$ , but reduction of the particle size did not noticeably change the $Re$ corresponding to transitions of WVF to TVF, into RIB or from RIB to CCF.

Figure 10.

Compilation of $Re$ corresponding to various flow structures and transitions for $\unicode[STIX]{x1D719}=0.10$ suspension with particles of size $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ during (a) decreasing- $Re$ ramp, and (b) increasing- $Re$ ramp. Various flow transitions were found to occur at higher $Re$ during the increasing- $Re$ ramp compared to during the decreasing- $Re$ ramp showing significant hysteresis. Note that in figure (b) the CCF–RIB transition is not captured as in these experiments we started with the $Re$ in the RIB region and increased  $Re$ .

Figure 11.

Space–time diagram constructed from images recorded at $60~\text{frames}~\text{s}^{-1}$ during decreasing- $Re$ ramp with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The TVF–SVF transition for $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles occurs in the range $110.57\geqslant Re\geqslant 110.43$ .

Figure 12.

Space–time diagram constructed from images recorded at $60~\text{frames}~\text{s}^{-1}$ during decreasing- $Re$ ramp with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The SVF–RIB transition for $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles occurs gradually in the range $106.57\geqslant Re\geqslant 100.09$ .

Figure 13.

Space–time diagram constructed from images recorded at $60~\text{frames}~\text{s}^{-1}$ during decreasing- $Re$ ramp with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The TVF–RIB transition for $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles occurs in the range $107.88\geqslant Re\geqslant 107.54$ .

Figure 14.

Space–time diagram constructed from images recorded at $60~\text{frames}~\text{s}^{-1}$ during decreasing- $Re$ ramp with $\text{d}Re/\text{d}\unicode[STIX]{x1D70F}=-0.0031$ . The RIB–CCF transition for $\unicode[STIX]{x1D719}=0.10$ suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles occurs in the range $106.99\geqslant Re\geqslant 106.78$ .

Additional experiments were performed with $\unicode[STIX]{x1D719}=0.1$ suspension with particles of size $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ to study the hysteresis in flow transitions. Starting with a value of $Re$ in the TVF regime, $Re$ is reduced slowly until RIB is reached and ramped up slowly to a $Re$ corresponding to WVF. The flow transitions and flow structures during a decreasing- $Re$ ramp experiment are compared against those during an increasing- $Re$ ramp in figure 10. Note that the CCF–RIB transition for the increasing- $Re$ was not captured as we started from a $Re$ that is above this transition. The flow transitions in both cases are observed to be qualitatively the same but quantitatively different. All the flow transitions during the increasing- $Re$ ramp were observed to occur at higher $Re$ compared to during decreasing- $Re$ ramp experiments showing significant hysteresis. Recall that the pure fluid showed no hysteresis with respect to the direction of ramp rate.

3.3.3 Transient evolution

Experiments to examine the transient evolution of suspension flow at fixed $Re$ over extended periods of time (rather than slowly ramped as described previously) were performed. These experiments only considered the single concentration of $\unicode[STIX]{x1D719}=0.10$ with both particle sizes, $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ and $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ . This provides information on system dynamics distinct from and complementary to the quasi-steady evolution probed with a slowly ramped rotation rate. First, the inner cylinder was rotated at $Re=165.6$ to establish WTV and held for 120 s for particle mixing. The inner cylinder was then stopped and held stationary for 120 s to ensure no residual flow. The inner cylinder was then rotated at $Re=99.3$ for 120 s to establish CCF, after which $Re$ was raised as rapidly as possible to a given fixed $Re$ . The flow evolution with time at this latter $Re$ was recorded over $30$ min, during which time the flow was photographed at regular intervals for a duration of 3 s at a rate of $60~\text{frames}~\text{s}^{-1}$ . The resulting images were used to construct space–time diagrams. The procedure was repeated for several $Re$ in the ranges $105.5\leqslant Re\leqslant 110.7$ for $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ and $101.4\leqslant Re\leqslant 119.0$ for $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ . Note that the diffusion time scale here is $t_{d}\approx 1.5~\text{s}$ , far shorter than the 1800 s duration of each of these experiments.

Figure 15.

Transient evolution experiments. Space–time diagrams for a suspension of $\unicode[STIX]{x1D719}=0.10$ and $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ when the $Re$ is rapidly increased from $Re=99.3$ (CCF) to a higher value (shown on the right) and held constant for 1800 s (30.0 min). The diagrams are constructed from images taken at a rate of $60~\text{frames}~\text{s}^{-1}$ . The diagram at each $Re$ in the range $101.4\leqslant Re\leqslant 119.0$ is a compilation of space–time slices each 3 s in duration separated by 60 s. The flow structures are CCF at $Re=101.4$ , partial RIB at $Re=103.5$ , RIB at $Re=105.5$ , SVF at $Re=107.6$ and $Re=113.8$ , and TVF at $Re=116.9$ and $Re=119.0$ .

The space–time diagrams at various $Re$ for the suspension of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ are shown in figure 15. At each $Re$ , the space–time diagrams of 3 s duration are separated by $1$ min. The space–time diagrams for $0\leqslant t\leqslant 14$ min show the evolution of the flow structures with time and the diagrams at $t=29$ and $30$  min show the fully developed flow structures at the end of the experiment. At $Re=101.4$ , which is below the TVF–CCF transition $Re$ , the flow rapidly formed CCF in the full length of the annulus with the exception of top and bottom Ekman layers, and this structure was retained for the remainder of the experiment. At $Re=105.5$ , the initial CCF slowly became unstable and took $13$ min to form ribbon structures (RIB) in the full annular length. The fully developed RIB was seen at the end of the $30$ min duration of the experiment. At $Re=103.5$ , the flow was CCF for most of the time, but towards the end RIB structures appeared in the bottom half of the annular section. At $Re=107.6$ and $113.8$ , the flow became unstable in short time and evolved from CCF to RIB and finally into a combination of spirals and Taylor vortices with azimuthal waviness that remained to the end of the run. At $Re=116.9$ and $119$ , the flow in the annulus became unstable immediately to form a combination of spirals and Taylor vortices in the full length, eventually evolving to the final state of TVF.

Space–time diagrams showing the time evolution of flow structures for the suspension of $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ are shown in figure 16. The diagram at each $Re$ between $Re=105.5$ and $Re=110.7$ is a compilation of space–time slices of 3 s, separated by 30 s. The space–time diagrams between $0\leqslant t\leqslant 5.0$  min show the evolution of the flow structures with time and the diagrams at $t=29.5$ and $30.0$  min show the fully developed flow structures. At $Re=105.5$ and $Re=106.6$ the flow quickly established CCF with Ekman layers at top and bottom. At $Re=107.6$ , the flow slowly became unstable to form a regular RIB structure after $5$ min and this remained unchanged to 30 min. At $Re=108.6$ , TVF was established quickly in the bottom part and the fluid between this region and top Ekman layer became unstable to form RIB after $1.5$  min. This partial TVF–RIB structure persisted to 30 min. Finally, at $Re=109.7$ and $Re=110.7$ , the flow quickly established TVF, which persisted until $30$  min.

Figure 16.

Transient evolution experiments. Space–time diagrams for a suspension of $\unicode[STIX]{x1D719}=0.10$ and $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ when $Re$ is rapidly increased from $Re=99.3$ (CCF) to a specific value and kept constant for 1800 s (30 min). The diagrams are constructed from images taken at a rate of $60~\text{frames}~\text{s}^{-1}$ . The diagram at each $Re$ in the range $105.5\leqslant Re\leqslant 110.7$ is a compilation of space–time slices each 3 s in duration separated by 30 s. The flow structures are CCF at $Re=105.5$ and $Re=106.6$ , RIB at $Re=107.6$ and $Re=108.6$ , and TVF at $Re=109.7$ and $Re=110.7$ .

Figure 17.

Photographs of (a) spiral vortex flow (SVF) at $Re=107.6$ , and (b) ribbons (RIB) at $Re=106.6$ for a suspension of $\unicode[STIX]{x1D719}=0.10$ and $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles at various times ( $t$ ) during two rotations of the inner cylinder. Here $T$ is the time for the inner cylinder to complete one full rotation. Both SVF and RIB complete one full cycle in two inner cylinder rotations indicating an azimuthal wave speed that is half of the inner cylinder rotation speed. One pair of vortices in SVF originating at the bottom shift axially by one pair width (2 $\unicode[STIX]{x1D6FF}$ ) in one rotation around the inner cylinder ( $\unicode[STIX]{x1D703}=2\unicode[STIX]{x03C0}$ ). The RIB cycle here has half-downward rotating (dark region marked with dashed yellow rectangles) and half-upward rotating (bright region) vortices with rotation direction described as seen at the outer wall.

Figure 18.

Space–time diagrams for (a) spiral vortex flow (SVF) at $Re=107.6$ , and (b) ribbons (RIB) at $Re=106.6$ for a suspension of $\unicode[STIX]{x1D719}=0.10$ and $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles during eight rotations of the inner cylinder. Here, $T$ is the time for the inner cylinder to complete one full rotation. SVF and RIB completed one full cycle in two inner cylinder rotations indicating an azimuthal wave speed that is half of the inner cylinder rotation rate. One pair of vortices in SVF originating at the bottom shift one vortex pair width (2 $\unicode[STIX]{x1D6FF}$ ) in the axial direction during one rotation around the inner cylinder. The RIB cycle here has half-downward rotating (dark region) and half-upward rotating (bright region) vortices around the inner cylinder.

Figure 19.

Marginal stability curves from linear stability analysis for the pure fluid sheared between the rotating inner cylinder and stationary outer cylinder. Here, $k_{\unicode[STIX]{x1D703}}=0,1$ and $2$ represent Taylor vortex flow (TVF), mode 1 spiral vortex flow (SVF) and mode 2 SVF structures. Circles in (a) indicate the minimum $Re$ on each marginal stability curve. The lowest $Re$ on the curves is $Re=120.5$ at $k_{\unicode[STIX]{x1D703}}=0$ and $k_{z}/2\unicode[STIX]{x03C0}=0.996$ which corresponds to TVF. The vertical axis in (b) represents the ratio of azimuthal wave speed for the flow structure to the inner cylinder rotation rate, e.g. the mode 1 spiral has an azimuthal wave speed that is half that of the inner cylinder rotation. The circles correspond to the minimum $Re$ on each curve.

Figure 20.

Space–time diagrams of (a) L-spiral (mode -1), (b) R-spiral (mode 1), and (c) ribbons constructed from velocity fields computed using linear stability analysis for the pure fluid with rotating inner cylinder and stationary outer cylinder. The ribbon structure is created by superimposing an L- and an R-spiral. Shown are the velocity vectors in the $z$ – $t$ plane at $\unicode[STIX]{x1D703}=0$ and $r=d_{i}/2+0.8\unicode[STIX]{x1D6FF}$ , where $d_{i}$ is the inner cylinder diameter and $\unicode[STIX]{x1D6FF}$ is the annular gap; $T$ is the time for the inner cylinder to complete one full rotation. The space–time diagrams for the R-spiral and ribbon structures here are the same as those for the SVF and RIB in figure 18 in terms of wavelengths and wave speeds.

The final flow structures at fixed $Re$ observed at the end of the $30$ min of these experiments were the same as those determined from the quasi-steady experiments with slow ramping of $Re$ as shown in figure 9. The conclusion we draw is that the ultimate states are relatively independent of the path to them for all of these conditions. The flow for the suspension with $d_{p}\approx 70~\unicode[STIX]{x03BC}\text{m}$ particles took less time to evolve into fully developed structures than for the suspension with $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ particles. This leads to the same conclusion drawn from the quasi-steady experiments, namely that the reduction of the particle size leads to a behaviour closer to that of the pure fluid.

3.4 Characterization of spiral and ribbon structures

Photographs and space–time diagrams of spiral and ribbon structures during two rotations of the inner cylinder are displayed in figures 17 and 18. In the case of spirals at $Re=107.6$ , an axial length of two times the annular gap ( $2\unicode[STIX]{x1D6FF}$ ) was found for a pair of counter-rotating vortices (downward rotating and upward rotating at the outer wall are seen as darker and brighter regions, respectively, in these figures). The vortex pair was observed to shift by $2\unicode[STIX]{x1D6FF}$ in the axial direction while it completed one rotation, or $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=2\unicode[STIX]{x03C0}$ , around the inner cylinder, indicating mode 1 spiral structure (various modes of the spiral structures are discussed in the later part of this section in connection with linear stability theory). The spiral completed one cycle in the azimuthal direction in the time the inner cylinder took to complete two rotations, i.e. the azimuthal spiral wave speed is half that of the inner cylinder.

For ribbons at $Re=106.6$ , the axial width of each vortex structure was observed to be the same as the annular gap ( $\unicode[STIX]{x1D6FF}$ ). The structures in the azimuthal direction consist of half ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ ) downward rotating vortex (marked with dashed yellow rectangle) and half ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ ) upward rotating vortex (with rotation directions seen at the outer wall). At the two azimuthal locations where these two vortices meet, cancelation of vorticities resulted in circular Couette flow. This vortex around the inner cylinder completed one cycle in two inner cylinder rotations indicating an azimuthal wave speed that is half of the inner cylinder rotation speed. This ribbon structure can be obtained by superposition of two mode 1 spirals, one spiralling clockwise and the other in the counter-clockwise sense when seen from the top view.

We sought a quantitative comparison of the spiral and ribbon structures described above for the suspension with those obtained for pure fluid. For this purpose, we have used linear stability analysis (LSA). The primary question to be answered is whether the observed wavelengths and wave speeds of the structures in the suspension agree with those predicted by LSA for the pure fluid with only the inner cylinder rotating; note these are not structures accessed by the pure fluid, and differ from those seen with both inner and outer cylinder moving with counter-rotation. The circular Couette base flow has only azimuthal component $U_{\unicode[STIX]{x1D703}}(r)$ which for a Newtonian fluid with rotating inner cylinder and stationary outer cylinder in cylindrical coordinates ( $r,\unicode[STIX]{x1D703}$ and $z$ denoting radial, azimuthal and axial coordinates, respectively) is given by

(3.1a-c ) $$\begin{eqnarray}U_{\unicode[STIX]{x1D703}}(r)=Ar+Br^{-1};\quad A=\frac{-d_{i}^{2}\unicode[STIX]{x1D6FA}}{d_{o}^{2}-d_{i}^{2}};\quad B=\frac{d_{o}^{2}d_{i}^{2}\unicode[STIX]{x1D6FA}}{d_{o}^{2}-d_{i}^{2}}.\end{eqnarray}$$

This flow generates a centrifugal acceleration which is balanced by the pressure field. In the standard fashion, a perturbation of the base flow and pressure fields by $\boldsymbol{u}^{\prime }=(u_{r}^{\prime },u_{\unicode[STIX]{x1D703}}^{\prime },u_{z}^{\prime })$ , $p^{\prime }$ is summed with the base flow and the resulting velocity and pressure fields are inserted to the Navier–Stokes equations, with neglect of nonlinear terms in perturbed quantities. It should be noted that values of the perturbed field are zero at the inner and outer cylinder walls. Details are shown in appendix A. We provide only results of primary interest to the current study.

The marginal stability curves determined from the analysis shown in figure 19 indicate the relations between ${k_{t}}_{i},k_{\unicode[STIX]{x1D703}}$ and $k_{z}$ which are the wavenumbers in time, azimuthal and axial directions. Here $k_{\unicode[STIX]{x1D703}}=0,1$ and $2$ represent TVF, mode 1 spiral and mode 2 spiral, respectively. For pure fluid, the first non-CCF state which is physically realized as $Re$ increases corresponds to the minimum $Re$ point on the lowest marginal stability curve, which is at the parameter values $k_{\unicode[STIX]{x1D703}}=0$ , $Re=120.5$ , $k_{z}/\unicode[STIX]{x03C0}=0.996$ and $k_{ti}=0$ : this is Taylor vortex flow. For flow driven by a rotating inner cylinder and stationary outer cylinder, the spiral vortex flow solution is not realized for the pure fluid. However, we can compare the analytical predictions of the two (left L, and right R) spiral structures to our observations for the suspension. The L-spiral has parameters $k_{\unicode[STIX]{x1D703}}=-1$ , $Re=121.303$ , $k_{z}/\unicode[STIX]{x03C0}=1.0008$ and $k_{ti}=-8.1353$ and the R-spiral has $k_{\unicode[STIX]{x1D703}}=1$ , $Re=121.303$ , $k_{z}/\unicode[STIX]{x03C0}=1.0008$ and $k_{ti}=8.1353$ . The variation in time of the velocity vectors of the L- and R-spirals along a line ( $r=d_{i}/2+0.8\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D703}=0$ , $z$ ) is shown in the space–time diagrams of figure 20(a,b). The space–time diagram for ribbons in figure 20(c) is constructed from superimposing R- and L-spirals. The space–time diagrams for the R-spiral and ribbon structures reveal that the azimuthal wave speed is half of the inner cylinder rotation speed and the axial wavelength is the same as the annular gap, $\unicode[STIX]{x1D6FF}$ . These results agree with the wave speed and wavelength of spirals and ribbons observed in the suspension flow (see figure 18). From this, we conclude that the suspension is apparently accessing the spiral and ribbon structures that satisfy the pure fluid problem, rather than distinct flow structures.

The experiments with the suspensions showed that the flow structures at a given $Re$ are the same for both cases: the quasi-steady ramp rate (§ 3.3.1) and instant rise of $Re$ from CCF (§ 3.3.3). This indicates that the inertially migrated concentration profiles are not the reason for accessing spirals and ribbons. Note that the radial concentration profiles in the spirals and ribbons are uniform as the time scale of the transient modulation of these structures is much smaller compared to the time scale of inertial migration. Furthermore, the linear stability analysis of circular Couette flow of a fluid with inertially migrated concentration profiles showed that these radially varying profiles neither produce spirals and ribbons after instability nor the instability at lower $Re$ compared to the pure fluid. The details of the analysis are described in the supplementary material available at https://doi.org/10.1017/jfm.2017.754 attached to this paper, and the key points are given below.

An additional linear stability analysis of the circular Couette flow is performed for a fluid with radially varying viscosity profile in the annular region to take into account the concentration variation due to inertial migration. Three profiles of radially varying relative viscosity, $f(r)$ , corresponding to the three concentration profiles, are considered. Two centre peaking concentration profiles for $\unicode[STIX]{x1D719}_{avg}=0.025$ and 0.1, to simulate the steady concentration profiles taking into account the inertial migration, and one uniform concentration profile with zero concentration near the walls for $\unicode[STIX]{x1D719}_{avg}=0.1$ , to simulate profile corresponding to instant rise of $Re$ from rest to a given value, were considered. The neutral stability curves for the three cases indicate that the transition $Re$ for the suspension is higher than that of the pure fluid whereas the experiments showed otherwise. The predicted transition $Re$ increased with increase in $\unicode[STIX]{x1D719}_{avg}$ for the centre peaked profiles and reduced with increase in uniformity of the concentration profile for $\unicode[STIX]{x1D719}_{avg}=0.1$ . In all three cases the mode of instability is that of $k_{\unicode[STIX]{x1D703}}=$ 0 which is Taylor vortex flow, but it is interesting and potentially significant that the wavenumber for time modulation $k_{ti}$ is non-zero. Note that, for a pure fluid with uniform viscosity across the annular gap, the wave speeds for $k_{\unicode[STIX]{x1D703}}=0$ (TVF) and $+/-1$ (L-SVF/R-SVF) are 0 and $+/-0.5$ , respectively. For the central peak $\unicode[STIX]{x1D719}$ profiles, the absolute value of the wave speed was observed to increase with increase in $\unicode[STIX]{x1D719}_{avg}$ . For $\unicode[STIX]{x1D719}_{avg}=0.1$ , the absolute value of the wave speed reduced towards zero with increasing uniformity of the concentration profile.

Figure 21.

Flow state map for suspension of concentrations $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ , with $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ . (a) Flow structures observed for suspension of $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ . Lines delineating various flow transitions: ——, CCF-RIB; - - - - - -, RIB-SVF; - - -, SVF-TVF; – - – - –, TVF-WTV. (b) Close-up of (a) to show the ribbon and spiral vortex regions clearly.

The results of the linear stability analysis show that a simple effective model with radially varying viscosity profiles failed to explain the experimentally observed deviations of the suspension from pure fluid in terms of the flow transition $Re$ and the flow structures. The analysis predicted that the modified viscosity profiles due to inertial migration stabilizes the flow whereas the experiments showed that the addition of particles destabilized the flow. We feel that a most likely candidate for the role of suspended particles is the finite-size perturbations they create in the background fluid.

3.5 Suspension experiments for $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$

The quasi-steady experiments, in which the $Re$ was slowly ramped as described in § 3.3.1, were performed for several suspension concentrations in the range $0\leqslant \unicode[STIX]{x1D719}\leqslant 0.30$ all for the larger particles of $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ . A flow state diagram shown in figure 21 consolidates the various flow transitions and flow structures observed on the $\unicode[STIX]{x1D719}$ – $Re$ plane. Figures 22 and 23 show the space–time diagrams at different concentrations during decreasing- $Re$ ramp experiments, constructed from images taken at $60~\text{frames}~\text{s}^{-1}$ .

Figure 22.

Space–time diagrams for suspensions of (a) $\unicode[STIX]{x1D719}=0$ , (b) $\unicode[STIX]{x1D719}=0.05$ , (c) $\unicode[STIX]{x1D719}=0.10$ , and (d) $\unicode[STIX]{x1D719}=0.15$ , all with $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ , constructed from images taken at $60~\text{frames}~\text{s}^{-1}$ during decreasing- $Re$ experiments. The time duration for each diagram is 3.8, 3.7, 3.0 and 3.0 s for $\unicode[STIX]{x1D719}=0$ , $\unicode[STIX]{x1D719}=0.05$ , $\unicode[STIX]{x1D719}=0.10$ and $\unicode[STIX]{x1D719}=0.15$ , respectively.

Figure 23.

Space–time diagrams for (a) $\unicode[STIX]{x1D719}=0.20$ , and (b) $\unicode[STIX]{x1D719}=0.30$ suspensions with $d_{p}\approx 230~\unicode[STIX]{x03BC}\text{m}$ , constructed from images taken at $60~\text{frames}~\text{s}^{-1}$ during decreasing- $Re$ experiments. The time duration for each space–time diagram is 3.7 s and 1.8 s for $\unicode[STIX]{x1D719}=0.20$ and $\unicode[STIX]{x1D719}=0.30$ , respectively.

As $Re$ decreased, the pure fluid transitioned from WTV to TVF to CCF as shown for $\unicode[STIX]{x1D719}=0$ in figure 22. For dilute suspensions with $0<\unicode[STIX]{x1D719}\leqslant 0.0375$ , TVF transitioned into CCF and the transition $Re$ was found to be decreasing with $\unicode[STIX]{x1D719}$ . By contrast, for $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.15$ , we found that the flow transitions all follow the sequence described previously in § 3.3.1 for the $\unicode[STIX]{x1D719}=0.10$ suspension: as $Re$ was reduced, the sequence was WTV to TVF to SVF to RIB to CCF (refer to figure 22). For $\unicode[STIX]{x1D719}=0.20$ , an additional stable state of wavy spiral vortices (WSV) was observed at higher $Re$ . With reduction in $Re$ , the observed flows for the $\unicode[STIX]{x1D719}=0.20$ suspension were WSV ( $164\geqslant Re\geqslant 141$ ), WTV ( $141\geqslant Re\geqslant 125$ ), WSV ( $125\geqslant Re\geqslant 107$ ), TVF ( $107\geqslant Re\geqslant 102$ ), SVF ( $102\geqslant Re\geqslant 96$ ), RIB ( $96\geqslant Re\geqslant 94$ ) and finally CCF ( $Re\leqslant 94$ ). The WSV consists of a pair of counter-rotating vortices spiralling around the inner cylinder, similar to SVF but with waviness along the axis of the vortices. This is shown for $\unicode[STIX]{x1D719}=0.20$ in figure 23 for $Re>120$ .

With increase in the solid concentration to $\unicode[STIX]{x1D719}=0.30$ , WTV, TVF and RIB did not appear in the $Re$ range studied. As $Re$ decreased, the suspension at $\unicode[STIX]{x1D719}=0.30$ exhibited only a sequence of non-axisymmetric flow states: from WSV ( $142\geqslant Re>82$ ) to SVF ( $82\geqslant Re>75$ ) before transitioning into CCF ( $Re\leqslant 75$ ) as shown in figure 23.

The first dominant observation from the results displayed in figures 21–23 is that CCF became unstable at lower $Re$ with each increase in particle loading $\unicode[STIX]{x1D719}$ . Recall that $Re$ is here defined in terms of the measured effective viscosity of the bulk suspension. The second observation is that the presence of particles led to a transition in which CCF and TVF are separated by non-axisymmetric flows, including ribbons, spirals and wavy spirals. For the pure fluid with rotating inner cylinder and stationary outer cylinder, the CCF becomes unstable first to the axisymmetric TVF, and counter-rotation of the outer cylinder at $Re$ is required to observe non-axisymmetric SVF, as shown in figure 1.