2.1. Particles and fluids

This study considers oblate particles of various shapes, some of which are displayed in figure 1. These include axisymmetric bodies such as spheroids, disks and rings, classified based on their particle aspect ratio $r$ , defined as the ratio between their length $2 \ell$ and their diameter $2a$ . A variety of rings with differing cross-sectional shapes are manufactured, including those with circular, triangular, L-shaped and T-shaped cross-sections. The shape of these rings is contingent upon a third parameter, namely the radius of the inner hole, which is denoted as $b$ . Images of the particles are captured with a Hirox RH-2000 digital microscope, which offers a resolution of more than 200 pixels per mm. Ten digital measurements of the particle sizes are then performed using the software ImageJ, and the statistical analysis of the results is reported in table 1.

Figure 1. Typical particles used in the experiments: (a) side and (c) top views of the circular ring R02; (b) side and (d) top views of the triangular ring TR012. (e) Illustration of the four different ring sections considered in this study. The particle symmetry axis is drawn as a dash-dotted line for each case.

Particles are digitally designed using Blender Community (2018) and fabricated by three different rapid prototyping methods. The first batch (batch I) is constituted by disks (D002, D003, D004) that have been laser cut from rigid Plexiglas sheets with a density of $1160$ kg m^– $^3$ . Stereolithography is employed to manufacture the particles constituting the second batch (batch II), which includes one ellipsoid (ELL06), all circular rings, the less flat triangular rings (TR01, TR012, TR04), and the special L-shaped (RL01) and T-shaped (RT01) rings. This method is based on the polymerisation of a UV-sensitive resin over three-dimensional layers, resulting in the production of high-resolution objects (with a resolution of $25\rm \,\unicode{x03BC}$ m) with an estimated density of $1200$ kg m^– $^3$ . The third and final batch (batch III) comprises the most slender triangular rings (TR003, TR005, TR008), which were fabricated using a computer numerical control (CNC) three-dimensional (3-D) milling machine on thin sheets of Plexiglas. All the selected materials possess a Young modulus of a few gigaPascals, which is sufficient to resist deformation within the viscous shear flow, even for the thinnest sections (TR003).

In our experiments, it is necessary to suspend a single particle at a time in a viscous shear flow under neutrally buoyant conditions. To achieve this, citric acid and Ucon oil are mixed with pure water to prepare the necessary density-matched fluid. A variable quantity of Ucon oil is added to the solution to adjust its viscosity within the range of $0.02 \leqslant \mu \leqslant 0.8$ Pa s. Thereafter, the concentration of citric acid is incrementally increased until the density of the fluid $\rho _f$ matches that of the considered particle $\rho _p$ . The final density of the fluid is determined by means of an Anton Paar densimeter, with an estimated uncertainty of $4$ kg m^– $^3$ . Rheological measurements are employed to ascertain the viscosity of the solution, with an uncertainty below 0.01 Pa s.

Table 1. Characteristics of all the particles used in the experiments. The columns from left to right provide the following information: code name, shape, mean aspect ratio $r$ , radius $a$ , half-length $\ell$ , hole radius $b$ , confinement ratio $\kappa$ and identification of the production method.

2.2. Shearing cell

A linear shearing cell apparatus is employed in the experiments, as illustrated in figure 2 and described in detail by Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024). The apparatus comprises a small tank (500 mm long, 40 mm wide and 90 mm deep) with transparent walls. A transparent and flexible Mylar film is driven by an electric motor and pulled in tension by two cylinders hanging at the extremities of the cell lid along its long side. During each experiment, approximately 1.2 l of viscous fluid are poured into the tank at a controlled room temperature of $23^{\circ } \pm 1^{\circ }$ . As the film continuously rotates at a constant velocity, it shears the fluid confined between its inner surfaces, held at a distance $L_y$ of 27 mm. The sheared particles are inevitably confined between the two sides of the transparent belt, with a confinement ratio $\kappa = 2 a / L_y$ . Mean values of the confinement ratio are reported in table 1.

Figure 2. Sketch of the experimental apparatus. The tank (1) is shown with its perforated lid (2), through which the two cylinders are hanging on both ends. The first cylinder is free to rotate (3), being coupled to a transmission shaft (4) through a rolling bearing. The second cylinder is fixed (5). Between them, a transparent plastic belt made of Mylar is kept under tension (6). The two cameras are also depicted: one is oriented to observe the flow-gradient (x, y) plane (7), while the other is focused on the flow-vorticity (x, z) plane (8). The operative volume where the experiments are performed is also shown in blue (9).

The operative volume of the experiments extends for 140 mm in the flow direction $x$ , 30 mm in the vorticity/gravity direction $z$ and 27 mm in the gradient direction $y$ , with the boundaries defined by the two inner surfaces of the transparent film. Two Allied Prosilica GX1910 cameras are positioned to capture this region of the shear cell, as illustrated in figure 2. Both cameras are equipped with a Nikon Micro-Nikkor 55 mm f2.8 objective, which enables the imaging of the shear cell with $1920 \times 1080$ pixels from a distance of approximately half a metre. This produces undistorted images with a sufficient focal depth and a resolution of approximately 20 pixels per mm over the considered particles. The top camera is aligned with the vorticity axis z and images the flow-gradient (x, y) plane, while the side camera is aligned with the gradient axis y and views the flow-vorticity (x,z) plane. Planar $(x,y)$ particle image velocimetry (PIV) measurements are recorded with the top camera by shedding a laser beam at three different locations along the vorticity axis z, corresponding to the upper, middle and lower sections of the operative volume. This results in the observation of a constant shear rate $\dot {\gamma }$ across the depth of the control volume. A total of 40 PIV measurements is recorded using various viscous fluids, with shear rates linearly varying between $\dot {\gamma } = 3.18 \pm 0.1$ s $^{-1}$ at $\mu = 0.02$ Pa s and $\dot {\gamma } = 3.63 \pm 0.01$ s $^{-1}$ at $\mu = 0.8$ Pa s.

As fluid and particle densities are always matched to prevent sedimentation effects ( $\rho _p\,\simeq \,\rho _f$ ), and the fluid is sheared at the highest rate $\dot {\gamma }$ possible, the particle Reynolds number $Re_p=\rho _f a^2 \dot {\gamma } / \mu$ is controlled through the fluid viscosity $\mu$ . Hence, inertial effects are produced by reducing the Ucon oil concentration during the preparation of the fluid as it reduces the viscosity $\mu$ . This implies that fluid and particle inertia effects are indiscernible ( $Re_p = St$ , with the Stokes number $St=\rho _p a^2 \dot {\gamma }/ \mu$ ).

2.3. Measurements

2.3.1. Particle tracking

The experiments are prepared by filling the cell with the selected fluid and initiating the shearing motion of the transparent plastic belt. Subsequently, one particle from those presented in table 1 is selected and positioned at the centre of the operative volume, as close as possible to the zero velocity streamline. In this procedure, the initial orientation of the particle is not precisely controllable and the result is essentially random. At this moment, the video recording by the two cameras is activated and synchronised by a luminous signal, signifying the initiation of the experiment. The frame acquisition is conducted at a rate of 7.5 fps until the particle exits the camera field of view. Experiments are typically repeated between 5 and 10 times for a given particle at each particle Reynolds number, as reported in table 2. This methodology is adequate to measure multiple particle rotations, but does not permit the observation of transverse migration over extended time scales. For this reason, inertial lift measurements are not attempted in this work.

Table 2. For each particle type used in the experiments (first column), the particle Reynolds number, $Re_p$ , the number of runs, $n_{runs}$ , and the mean duration of the shearing normalised by the (experimentally measured) mean period of rotation, $\overline {\Delta t_{run}} / T$ , are provided. In the case of aligning particles, the mean duration of the shearing is not calculated.

A Python script was developed in-house to track the particle across both top and side camera fields during the recordings of the experiments. The script implements a standard version of the Watershed method calling the high-level functions provided in the OpenCV library (Bradski Reference Bradski2000). In detail, the image is binarised by thresholding to separate between the background and the particle by simple morphological operations. Then, the watershed algorithm is applied to find an accurate estimation of the particle centre in the image plane. This operation is particularly challenging for slender rings, since they offer the lowest surface to the camera field of view. Consequently, each frame is cropped to a $4a \times 4a$ image, which displays the particle under consideration.

2.3.2. Orientation measurements with convolutional neural network

The measurement of the orientation of both axisymmetric and asymmetrical particles represents a significant challenge. Cylindrical and ellipsoidal shapes present simple geometrical relations that can be exploited to regress their orientation from their axes-aligned bounding boxes (Di Giusto et al. Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024). However, this method is not readily generalisable to asymmetric particle shapes. Consequently, we have developed a new data-driven approach, which is sketched in figure 3.

Figure 3. Sketch of the convolutional neural network. The operative process is as follows: a given particle geometry, represented by a ‘.stl’ file in panel (a), is used as the basis for the generation of a synthetic data set in Blender in panel (b). This data set is then used to train a deep learning model, with the objective of estimating the particle orientation given two perpendicular projections in panel (c). A physical particle corresponding to the ‘.stl’ file is also created through rapid prototyping in panel (d) and employed in the experiments in panel (e). Subsequently, the Watershed method in panel (f) is applied to the recorded data from the experiments prior to the deep learning model inference operation in panel (g), which estimates the time-evolution of the three-dimensional particle orientation vector $\boldsymbol{n}$ in the given experiment.

As previously outlined in § 2.1, the particle fabrication methods necessitate the generation of a digital model of each object in the format of a ‘.stl’ file, see panel (a) of figure 3. It is thus possible to create a synthetic dataset of virtual images of oriented particles that closely resemble the frames recorded during the experiments, as illustrated in panel (b) of figure 3. Further details regarding the dataset preparation can be found in Appendix A. The labelled data can then be used to train a deep learning model, see panel (c) of figure 3, and to measure the particle orientation from pairs of perpendicular views, as detailed in Appendix B. Concurrently, the ‘.stl’ file is used to reproduce a physical copy of the considered particles through rapid prototyping, see panel (d) of figure 3, as described in § 2.1. The rotational dynamics of the aforementioned object are then recorded during the experiments, see panel (e) of figure 3, using the set-up described in § 2.2.

Two additional operations are conducted during the post-processing of the experimental recordings. First, the Watershed algorithm is applied to track each particle across the camera frames, as described in § 2.3.1. Subsequently, the deep learning model can infer the three-dimensional particle orientation from each couple of perpendicular frames and estimate the rotational dynamics of the considered particles, see panels (f) and (g) of figure 3.

Among all deep learning architectures, convolutional neural networks (CNNs) are particularly adept at extracting features from visual inputs (Szegedy et al. Reference Szegedy, Liu, Jia, Sermanet, Reed, Anguelov, Erhan, Vanhoucke and Rabinovich2015), as evidenced by their successful application to a range of tasks, including face recognition (Taigman et al. Reference Taigman, Yang, Ranzato and Wolf2014), real-time object detection (Redmon et al. Reference Redmon, Divvala, Girshick and Farhadi2016) and autonomous driving (Bojarski et al. Reference Bojarski2016). CNNs were directly inspired by the hierarchical functioning of neurons in the visual cortex of simple mammals (Hubel & Wiesel Reference Hubel and Wiesel1959, Reference Hubel and Wiesel1968). This resulted in the implementation of a simple convolution down sampling architecture into a first neural network for visual pattern recognition (Fukushima Reference Fukushima1980). Subsequently, LeCun et al. (Reference LeCun, Bottou, Bengio and Haffner1998) stacked convolutional and pooling layers with a fully connected neural network to build the LeNet-5 architecture, one of the first CNNs, and successfully applied it to handwritten digit recognition.

In this study, we adapt the LeNet-5 architecture to quantify the three-dimensional orientation of axisymmetric and asymmetric particles from each pair of perpendicular frames of the video recording of the experiments, as sketched in figure 3. The image pairs are initially processed separately, with each frame of a pair allocated to a dedicated branch comprising three consecutive two-dimensional convolutional layers. These layers have kernel sizes of $5\times 5$ pixels, the first layer containing 8 feature maps and the second 16. Each convolutional layer is followed by a two-dimensional max-pooling operation with a pool size of $2\times 2$ pixels. Subsequently, the two streams are concatenated, and three fully connected layers with 256 features each are applied to ascertain the orientation vector of the particle, represented by $\boldsymbol{n}$ . In accordance with Krizhevsky, Sutskever & Hinton (Reference Krizhevsky, Sutskever and Hinton2012), the Rectified Linear Unit (ReLU) was selected as the activation function for all layers except the final one. In this case, a linear mapping was employed to determine the three components of $\boldsymbol{n}$ , which were then normalised by their Euclidean norm to ensure the consistency of the orientation. The parameters of the CNN are trained over synthetic images of axisymmetrical and asymmetrical particles generated following the procedure described in Appendix A. The network is trained to infer the particle orientation from its two perpendicular views by minimising a loss function given by the Euclidean norm of the difference between the true particle orientation vector, $\boldsymbol{n}_{true}$ , and the predicted particle orientation vector, $\boldsymbol{n}_{pred}$ . An additional penalisation is determined whenever a negative value of the $n_3$ component is observed. The optimisation is performed through the Adam algorithm. Further details about the training procedure can be found in Appendix B.

CNNs have been successfully applied to estimate fluid motion (Cai et al. Reference Cai, Zhou, Xu and Gao2019) or to detect submicron particles (Newby et al. Reference Newby, Schaefer, Lee, Forest and Lai2018). However, to the best of our knowledge, this is the first time that such a method has been proposed to measure the orientation of macroscopic axisymmetric and asymmetric particles. Convolutional neural networks (CNNs) are found to provide a satisfactory compromise between simplicity and effectiveness. However, other neural architectures may prove to be more effective. Consequently, a benchmark consisting of a comprehensive set of experimental videos is provided, accompanied by the spheroid ELL06 ‘.stl’ file and training data.

Figure 4. Evolution of the components of the orientation vector $\boldsymbol{n}$ , displayed as vertically aligned panels, against the dimensionless time $t\dot {\gamma }$ for the circular ring R02 with aspect ratio r = 0.23 at a small $Re_p$ = 0.06. See Supplementary movie 1 for animations and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_4/Figure_4.ipynb) of the figure including the data.

2.3.3. Data analysis

Once trained, the neural network is capable of reconstructing the time evolution of the particle orientation vector $\boldsymbol{n}$ during each experiment in a few seconds. A typical result with the circular ring R02 is displayed in figure 4. Despite the presence of occasional noise, the periodic nature of the three components, namely $n_1$ , $n_2$ and $n_3$ , can be discerned against the dimensionless time $t\dot {\gamma }$ . Subsequently, other significant quantities can be determined. The Fourier transform of the signals can be calculated, thereby providing the period, $T$ . Furthermore, a minimum difference threshold criterion is established on the two highest peaks of the power spectrum of the particle polar angles, with the objective of distinguishing between rotational and aligned dynamical states. The components of the reconstructed vector $\boldsymbol{n}$ can be used to determine the azimuthal angle, $\phi$ , and polar angle, $\theta$ , of the orientation. This is achieved by calculating the following expressions:

(2.1) \begin{align} n_1 = \sin (\phi )\sin (\theta ),\end{align}

(2.2) \begin{align} n_2 = \cos (\phi )\sin (\theta ),\end{align}

(2.3) \begin{align} n_3 = \cos (\theta ).\end{align}

The distinction between rotational and aligned dynamical states based on the power spectrum is always visually verified by examining the components of the vector $\boldsymbol{n}$ . It is acknowledged that the data may contain some errant points due to minor imprecisions in particle tracking or 3-D orientation estimation. However, the methods used to calculate the period of rotation and to determine the aligned dynamical states are based on Fourier analysis, which results in robustness to these small inaccuracies. Further details can be found in Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024).

3.1. Period of rotation

Figure 5 depicts the dimensionless period of rotation, $T\dot {\gamma }/2\pi$ , of the flat bodies of revolution as a function of particle aspect ratio, $r$ . The data are obtained by averaging over all available experiments for small particle Reynolds number, $Re_p \lesssim 0.5$ . The measured period of these particles consistently falls below the Jeffery curve, indicating that the period is consistently smaller than that of the corresponding spheroid at the same $r$ . As previously stated in § 1, an equivalent aspect ratio, $r_{eq}$ , can be determined for any body of revolution to recover the Jeffery period. To be more precise, $r_{eq}$ can be calculated from the equation $T \dot {\gamma } = 2\pi (r_{eq} + 1/r_{eq})$ , where $T$ represents the measured period of rotation. When the ring has a circular-shaped cross-section, i.e. when the ring is a circular torus, Singh et al. (Reference Singh, Koch and Stroock2013) proposed that the effective aspect ratio at the leading order is given by $r_{eq}=0.903 r \sqrt {\ln 1/r}$ . This theoretical prediction is in excellent agreement with the present measurements for rings with circular-shaped cross-section, and even for rings with triangular-shaped and L-shaped cross-section. The period of the other bodies (rings with T-shaped cross-section, oblate ellipsoids and disks) appears to be more closely in agreement with the empirical correlation proposed by Harris & Pittman (Reference Harris and Pittman1975), namely $r_{eq} = 1.14\,r / r^{0.156}$ . It is important to highlight that all these particles have always been observed to rotate for $Re_p \lesssim 0.5$ . It is possible that the rings may lack the requisite flatness to permit long-term alignment at low Reynolds numbers, as postulated by Borker et al. (Reference Borker, Stroock and Koch2018).

The influence of inertia on the period of rotation is addressed in figure 6, where the experimentally measured period of rotation, $T$ , is plotted against the particle Reynolds number, $Re_p$ . The period is normalised by the Jeffery period, $T_J=2 \pi (r_{eq}+1/r_{eq} )/\dot {\gamma }$ , using the equivalent aspect ratio $r_{eq}$ inferred from the data of figure 5 at low $Re_p$ . For $Re_p \lt 0.8$ , inertia has a negligible effect on the period of rotation, which remains equal to the Jeffery period, $T_J$ . As $Re_p \gtrsim 1$ , the period exhibits a noticeable increase and, in the case of flat disks and rings with triangular cross-sections, may even reach infinity.

Figure 5. Period of rotation, $T$ , of the flat bodies of revolution against the particle aspect ratio $r$ . The period is made dimensionless using the shear rate $\dot {\gamma }$ and normalised by a factor $2 \pi$ . The experimental values are displayed as coloured circles (rings with circular-shaped cross-section), triangles (rings with triangular-shaped cross-section) and squares (disks). The special rings with L- and T-shaped cross-section as well as the oblate ellipsoid are displayed with the letters $\boldsymbol{ {L}}$ , $\boldsymbol{ {T}}$ and $\boldsymbol{ {O}}$ . Each point is the average over all the available experiments for small particle Reynolds numbers ( $Re_p \lesssim 0.5$ ). The theory of Jeffery (Reference Jeffery1922), the semi-empirical correlation of Singh et al. (Reference Singh, Koch and Stroock2013) and the empirical expression of Harris & Pittman (Reference Harris and Pittman1975) are displayed as a solid black line, a dashed blue line and a dash-dotted pink line, respectively. See the Supplementary material for the directory of the figure including the data and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_5/Figure_5.ipynb).

Figure 6. Inverse of the experimentally measured period of rotation $T$ normalised by the Jeffery period, $T_J=2 \pi (r_{eq}+1/r_{eq})/\dot {\gamma }$ , against the particle Reynolds number, $Re_p$ . The data symbols are identical to those in figure 5. Each point is the average over at least three experiments. The Jeffery period is calculated at the lowest particle Reynolds number for each particle, corresponding to the dotted black line within this normalisation. See the Supplementary material for the directory of the figure including the data and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_6/Figure_6.ipynb).

3.2. Dynamics

Having previously examined the influence of inertia on the period in figure 6, we now turn our attention to the effect of inertia on the global dynamics of the flat bodies. Figures 7, 8 and 9 display the three components of $\boldsymbol{n}$ in the flow ( $n_1$ ), gradient ( $n_2$ ) and vorticity ( $n_3$ ) directions against the dimensionless time, $t\dot {\gamma }$ . To facilitate clarity of presentation, we have chosen to focus the discussion on a subset of four runs for a typical circular ring (i.e. a ring with circular-shaped cross-section), triangular ring (i.e. a ring with triangular-shaped cross-section) and disk at small $Re_p \lesssim 1$ and at larger $Re_p \gtrsim 1$ . We also compare our results with the theory of Jeffery (Reference Jeffery1922) and the asymptotic theories of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) and Dabade et al. (Reference Dabade, Marath and Subramanian2016). It is important to stress that, although the theories consider an unbounded system, there is some degree of confinement in the experiments ( $\kappa \approx 0.2{-}0.7$ ) which may influence the dynamical behaviours as outlined by Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ). Moreover, the asymptotic theories of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) and Dabade et al. (Reference Dabade, Marath and Subramanian2016) are strictly valid only for $Re_p\ll 1$ . It is therefore not possible to expect these theories to accurately describe the dynamics at $Re_p \gtrsim 1$ , as also discussed by Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ).

Figure 7. Evolution of the components of the orientation vector $\boldsymbol{n}$ , displayed as vertically aligned panels, against the dimensionless time $t\dot {\gamma }$ for the circular ring R05 with aspect ratio $r=0.45$ at $Re_p=0.06$ , $Re_p=0.06$ , $Re_p=0.15$ and $Re_p=1.19$ (from left to right). The two cases at $Re_p=0.06$ correspond to different initial orientations, leading to different Jeffery orbits. Comparison with the model of Jeffery (Reference Jeffery1922) and of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) are also given as black solid lines and black dashed lines, respectively. See Supplementary movies 2–4 for animations and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_7/Figure_7.ipynb) of the figure including the data.

Figure 8. Evolution of the components of the orientation vector $\boldsymbol{n}$ , displayed as vertically aligned panels, against the dimensionless time $t\dot {\gamma }$ for the triangular ring TR008 with aspect ratio $r = 0.09$ at $Re_p$ = 0.15, $Re_p$ = 1.07, $Re_p$ = 4.90 and $Re_p$ = 4.90 (from left to right). The two cases at $Re_p = 4.90$ demonstrate the systematic alignment of the particle in the plane of shear. Comparison with the model of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) is also given as black dashed lines. See Supplementary movies 5–7 for animations and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_8/Figure_8.ipynb) of the figure including the data.

Figure 9. Evolution of the components of the orientation vector $\boldsymbol{n}$ , displayed as vertically aligned panels, against the dimensionless time $t\dot {\gamma }$ for the disk D003 with aspect ratio $r = 0.03$ at $Re_p$ = 0.52, $Re_p$ = 0.52, $Re_p$ = 2.06 and $Re_p$ = 2.06 (from left to right). Comparison with the model of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) is also given as black dashed lines. The two cases at $Re_p=0.52$ correspond to different initial orientations, resulting in an inertial drift towards the spinning (first column) and tumbling (second column) limiting cycles. The two cases at $Re_p=2.06$ correspond to different initial orientations, determining an alignment of the particle in the plane of shear (third column) or in the vorticity direction (fourth column). See Supplementary movies 8–9 for animations and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_9/Figure_9.ipynb) of the figure including the data.

We start by examining the influence of inertia on the dynamics of the circular ring with aspect ratio $r=0.45$ . At very small $Re_p=0.06$ , the ring approximately follows a Jeffery orbit with no significant drift between the successive rotations. At larger $Re_p=0.15$ and $Re_p=1.19$ , the ring appears to remain in the tumbling orbit in the plane of shear during several periods of rotation. At $Re_p = 0.15$ , the ring dynamics are well described by the asymptotic theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) and a slow drift toward the spinning orbit may be discernable. At $Re_p = 1.19$ , the ring is predicted to drift into a spinning orbit, but in the experiments, the ring remains in a tumbling orbit. As observed in experiments on oblate spheroids and disks (Di Giusto et al. Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024), the bifurcation towards a single stable spinning orbit above a critical aspect ratio of approximately 0.14, predicted by asymptotic theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) as well as by the numerical simulations of Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ) at $Re_p=1{-}5$ , does not appear to be clearly observed for a circular ring. However, since inertial effects also depend on the detailed particle geometry, there is no fundamental reason why the critical aspect ratio for rings should be the same as that of spheroids, even with the same equivalent aspect ratio. As previously noted, these theories are strictly valid for $Re_p\ll 1$ . It may be that discerning a clear bifurcation towards the spinning orbit in the experiments may necessitate a considerably longer observational timeframe, i.e. over more than 10 rotational periods, which is challenging to achieve experimentally.

We then consider the triangular ring with aspect ratio $r = 0.09$ in figure 8. At small $Re_p=0.15$ , a drifting behaviour is observed in agreement with the asymptotic theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ). For larger $Re_p=1.07$ , the asymptotic theories predict a slightly faster drift, which is expected since these theories are strictly valid only for $Re_p\ll 1$ . It is noteworthy that the period of rotation is observed to increase from $Re_p=0.15$ to $Re_p=1.07$ . For even larger $Re_p=4.09$ , a permanent alignment in the plane of shear is observed.

Finally, we turn to the discussion of the disk D003 with aspect ratio $r = 0.03$ in figure 9. For the smallest $Re_p=0.52$ , the particle experiences a drift towards either a spinning or a tumbling orbit, contingent upon its initial orientation. The theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a ) successfully predicts the existence of these two limiting orbits. For larger $Re_p=2.06$ , alignments with orientation either in the plane of shear or in the vorticity direction are observed.

3.3. Bifurcations

A summary of the dynamical behaviours exhibited by the circular and triangular rings as well as the disks can be found in the bifurcation map displayed in figure 10 which follows that proposed by Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ). The $Re_p$ versus $r_{eq}$ map shows whether the observed dynamics is rotational or aligning. Empty data symbols represent rotational dynamics and full data symbols represent aligning dynamics. The map depicts various bifurcation limits. It should be noted that the map provides a comprehensive representation of both oblate and prolate sides for the sake of completeness.

Figure 10. Particle Reynolds number, $Re_p$ , against the equivalent particle aspect ratio, $r_{eq}$ . The experiments are displayed as coloured circles (rings with circular-shaped cross-section), triangles (rings with triangular-shaped cross-section) and squares (disks), which are empty (or full) when rotational (or aligning) dynamics is observed. Each point is the average over at least three experiments. The equivalent particle aspect ratio is calculated at the lowest particle Reynolds number for each particle. The bifurcations between stable tumbling and stable fixed orientation in the slender oblate and prolate limits derived by Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ) are shown as black solid lines, while the bifurcation between stable and unstable tumbling given by Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) and Dabade et al. (Reference Dabade, Marath and Subramanian2016), and extended to the small inertial regime by Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ), is drawn as a brown dashed line. The limits between tumbling and stable orientation given by the lattice Boltzmann as well as the steady-state simulations of Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ) are also displayed as solid cyan octagons. See the Supplementary material for the directory of the figure including the data and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_10/Figure_10.ipynb).

The predicted bifurcation from stable to unstable tumbling is indicated by the brown dashed line. This bifurcation was predicted to occur at a critical aspect ratio of 0.137 in the asymptotic theories (Einarsson et al. Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ; Dabade et al. Reference Dabade, Marath and Subramanian2016). This bifurcation limit was extended in the time-resolved lattice Boltzmann simulations of Rosén et al. (Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ) and shown to survive up to $Re_p=5$ , even with a confinement of $\kappa =0.2$ . This bifurcation was unexpectedly not observed in the previous experiments of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024) having similar confinement and is still not clearly observed in the present experiments. As previously mentioned in § 3.2, it is possible that the transition from a tumbling to a spinning orbit may require a significantly longer observational timeframe.

The bifurcation from tumbling in the flow-shear plane to a stable fixed orientation is given by the solid lines in the oblate and prolate asymptotic limits as predicted by the asymptotic theory (Rosén et al. Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ). Bifurcations to a fixed point, as observed in the lattice Boltzmann as well as steady-state simulations, are also indicated by solid cyan octagons. The agreement between the predictions of the asymptotic theory and the simulations is only qualitative, as the theory is only valid for $Re_p\ll 1$ . The present experimental bifurcation limit for disks and rings appears to be in good agreement with the numerical simulations carried out for spheroids. This further suggests that the $Re_p$ versus $r_{eq}$ bifurcation map is the most appropriate description for these systems beyond spheroids.

We now examine the critical behaviour of the period below the bifurcation, as illustrated in figure 11. The period, which has been normalised by the shear rate, $T\dot {\gamma }$ , is plotted against the distance to the bifurcation towards a fixed orientation, $Re_{p,c}-Re_p$ . It tends to infinity as $(Re_{p,c}-Re_p)^{-1/2}$ using the critical particle Reynolds number predicted by the asymptotic theory (Rosén et al. Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ), $Re_{p,c}=15 \,r_{eq}$ (as $r_{eq}\rightarrow 0$ ), which is indicated in figure 10. This result is in agreement with the prediction of Ding & Aidun (Reference Ding and Aidun2000) and the experiments of Zettner & Yoda (Reference Zettner and Yoda2001). The fitting of the data yields a prefactor of $67\pm 13$ . The experimental data deviate from this scaling law when the distance from the threshold becomes too large, i.e. $Re_{p,c}-Re_p \gtrsim 3$ , as can be expected from a critical behaviour.

Figure 12 compares the behaviour of the alignment angle in the flow-shear plane, $\phi _a$ , with the aspect ratio, $r_{eq}$ , above the bifurcation with the prediction of the asymptotic theory (Rosén et al. Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a). This theory predicts that the flat body aligns in the flow-shear plane at the azimuthal angle $\phi _a= \pi /2 + r_{eq} + (30\,r_{eq})^{1/2} (Re_{p,c}-Re_p)^{1/2} / 15$ with $Re_{p,c}=15 \,r_{eq}$ as $r_{eq}\rightarrow 0$ . The theoretical predictions have been calculated using the experimental, $r_{eq}$ , and experimental distance to the bifurcation, $Re_{p,c}-Re_p$ . The experimental data qualitatively follow the predictions. It is clear that as the flatness increases, the orientation angle becomes more perpendicular to the flow direction.

Figure 11. Dimensionless period, $T\dot {\gamma }$ , versus the distance to the bifurcation to a fixed orientation, $Re_{p,c}-Re_p$ . The dashed line is the best fit of the experimental data to the scaling law $(Re_{p,c}-Re_p)^{-1/2}$ using the critical particle Reynolds number predicted by the asymptotic theory, $Re_{p,c}=15 \,r_{eq}$ (Rosén et al. Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ). The data symbols are identical to those in figure 5. See the Supplementary material for the directory of the figure including the data and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_11/Figure_11.ipynb).

Figure 12. Alignment angle in the flow shear plane, $\phi _a$ , versus the equivalent aspect ratio, $r_{eq}$ . Comparison with the prediction of the asymptotic theory, $\phi _a= \pi /2 + r_{eq} + (30\,r_{eq})^{1/2} (Re_{p,c}-Re_p)^{1/2} / 15$ with $Re_{p,c}=15 \,r_{eq}$ as $r_{eq}\rightarrow 0$ (Rosén et al. Reference Rosén, Einarsson, Nordmark, Aidun, Lundell and Mehlig2015a ). See the Supplementary material for the directory of the figure including the data and the Jupyter Notebook (https://www.cambridge.org/S0022112025104217/JFM-Notebooks/files/figure_12/Figure_12.ipynb).