Motion of a floating sphere pulled by a string and induced flows Open Access

In classical mechanics, a circular trajectory is associated with a fictitious force that tends to push objects away from the rotation center, in accordance with our sensitive experience during a turn in a ride. However, this intuition is challenged by other experiments in which the reverse effect appears. For instance, particles suspended in a rotating fluid are pushed toward the rotation center if their relative density is less than one. The centrifugal force acts in this case as an effective radial gravity which generates buoyancy directed toward the rotation center for light particles. More surprisingly, heavy particles placed at the bottom of a liquid stirred from the top by a rotating paddle also tend to aggregate at the rotation center. This effect is often referred to as “tea-leaf paradox,” as it can be observed with tea leaves when one mixes a tea cup with a spoon. Such peculiar particles transport finds its roots in the three-dimensional flows induced by the container edges¹ and has manifestation in sediments transport in rivers,^2–6 dissolution mediated by rotating paddles,⁷ aggregation control of nanoparticles,⁸ or blood-plasma separation.^9,10

However, the picture is completely different if the stirrer's trajectory can be affected in return by the flow it generates. In the experiments mentioned above, the circular trajectory of the stirrer is indeed imposed by an external operator, and the backaction of the flow simply consists in a friction force that the operator must overcome. The situation is completely different when the trajectory itself can be modified by the action of the flow, a commonly encountered example being the deflection experienced by a spinning sphere moving in a fluid due to lift (or Magnus) forces.^11–13 Similarly, the trajectory of magnetic disks set in rotation at a liquid interface were shown to be strongly impacted by the generated flow,^14,15 and several disks could interact and self-assemble under the generated flow's action.^16–18 On top of those translational degrees of freedom, object–flow interactions can also alter the shape of deformable bodies. For instance, a flexible plate placed in constant flow will tend to bend, which will modify the flow in return.^19–21 More generally, the interplay between deformable bodies and flow generation is at the heart of propulsion mechanisms for living organisms^22–25 or robots.^26–28 The extra degrees of freedom offered by the object's deformations and motion therefore offer a large panel of behaviors that are still to explore.

Inspired by this rich phenomenology, we propose to revisit the tea-leaf paradox when one replaces the rigid spoon by a deformable one, as depicted in Fig. 1(a). As an experimental model of such system, we will study the motion of a ball attached to a rotation arm by a deformable string as in Fig. 1(b). The motion at one end of the string is therefore imposed, but the ball attached to other end can move under the liquid's action. Although simple in principle, this system exhibits rich phenomenology that we briefly summarize here. At low rotation speed, the ball performs circular trajectories which radius decreases when the rotation rate increases, hence challenging the intuition on centrifugal force. At high rotation speed, a strong bifurcation is observed and the ball suddenly gets expelled or attracted toward the rotation center depending on the string length. In the latter case, the ball can moreover exhibit strong hysteretic behavior and gets robustly trap at the rotation center, showing that the system is sensitive to its past history. Moreover, those states are shown to drastically reduce the mixing efficiency of the liquid.

This work is organized as follows. We first present in Sec. II experimental results for different rotation rates and string length. Section III is dedicated to force analysis while Sec. IV focuses on the lift force exerted by the liquid, which must be taken into account to satisfactory explain the experimental results. Section V focuses on the impact of string deformation, and a full model of the system which shows good agreement with our data is proposed. We discuss in Sec. VI the appearance of hysteresis and discuss the existence of self-trapped states. Section VII finally discusses the emergence of tea-leaf paradox in our experimental case.

We describe in this section our experimental setup and results. All measurement are conducted in a rectangular tank of 26  $×$ 36 cm² filled with tap water with depth 6.5 cm. Our model deformable stirrer consists in a 3 D-printed plastic ball of radius $Rs=7$ mm, mass $m=1.3$ g, and relative density $ρ̂s=0.9$ attached to a rotation arm of length $W=4.4$ cm by a wool strand, as shown in Figs. 1(b) and 1(c). The arm is rotated at rate $Ω$ going from 0.2 to 3 round/s with a stepper motor. In all the experiments, the ball remains far (⁠ $>3$ cm) from any edge of the container to prevent boundary effect to occur. In order to recover the ball's trajectory, a camera (Basler) is placed below the tank and records 10 images per revolution of the rotation arm. The ball's position is retrieved on each image using a convolution algorithm. More details on the experimental procedure and analysis algorithm can be found in supplementary material.

In our experimental regimes, the strand is non-extensible, of negligible mass compared to the ball and free of static or plastic deformation after being constrained. Nevertheless, the way to attach it to the rotation arm is a sensitive parameter for experimental repeatability. After several attempts, the final design consists in knotting the strand to the rotation arm and guides it close to the surface with a tube as shown in Figs. 1(b) and 2. This avoids the introduction of constrains, as the strand is free to move in the tube, and confines the deformable part of the stirrer close to the horizontal interface. In this configuration, the camera placed below the tank can also be used to characterize the string's shape, as most of its deformation will occur in the horizontal plane. The results obtained with this setup were qualitatively similar to those obtained when the strand was directly tied to the rotation arm far from the surface. This shows that confining the strand in the horizontal plane did not affect significantly the underlying physics while simplifying experimental measurement and enforcing repeatability.

We performed experimental measurement of the ball's trajectory for progressively increased rotation speeds $Ω$ and different string lengths $Ls$⁠, measured from the tube to the ball. We plot in Figs. 3(a) and 3(b) the results for three different rotation speed $Ω=0.2,1.4$ and 2.0 round/s and two string length $Ls=4.6$ cm [Fig. 3(a)] and $Ls=5.3$ cm [Fig. 3(b)]. The trajectories are circular in all cases and their radius $Rt$ depends strongly on the rotation speed. For both string length, the radius of the trajectory decreases as $Ω$ goes from 0.2 to 1.4 round/s. This is opposite to what is expected from the tea-leaf paradox, in which floating particles are pushed toward the outside by secondary flows. When the rotation speed is increased to 2.0 round/s, two opposite behaviors were observed depending on the string length. For the shortest string, the ball was suddenly expelled from the rotation center in what we call a “diverged” state. Conversely, the ball was trapped at the rotation center in a “collapsed” state when the string was longer. Kinnograms obtained by stacking images from the side are shown in Fig. 3(c) and display the three “rotating,” “collapsed” and “diverged” regimes discussed above and shown respectively in Figs. 2(a)–,2(c) (multimedia available online). Experimental snapshots in Fig. 3(d) show that the string is straight in non-collapsed cases, but gains some curvature in the collapsed state. One can, therefore, qualitatively interpret such collapse as the inability of the system to maintain tension in the string during the ball's straddling.

More systematic measurement of the trajectory's radius $Rt$ for progressively increased rotation rate $Ω$ and string length $Ls$ are shown in Fig. 3(e). Two distinct behaviors emerge from those results. For the shortest string length (⁠ $Ls<5$ cm, red curves), the ball eventually reaches a diverged state at high rotation speed, where the radius no longer depends on the string length. This state is reached continuously for the shortest string length (⁠ $Ls=2.7$ cm and 3.5 cm) or by a brutal jump for the longest strings (⁠ $Ls=4$ cm and 4.6 cm). Before such diverged state is reached, there exists a range in which the trajectory radius decreases when the rotation speed increases. The associated range of rotation speeds increases as the string lengthens. Above a certain string length (⁠ $Ls≥5$ cm, green curves), the ball conversely reaches a collapsed state. The rotation speed at which the collapse occurs strongly decreases as the string becomes longer, going from 2.8 round/s for $Ls=5$ cm to almost zero for $Ls>6$ cm. Before such collapse occurs, we observe as for shorter string length that the radius globally decreases when the rotation speed increases.

In what follows, we aim to explain theoretically both why the radius decreases when the rotation speed increases and why the system collapses or diverges depending on the string length.

We propose in this section to perform a force analysis on the ball in the comoving frame. For now, we will consider the inertial force $F→in$⁠, the viscous force $F→v$⁠, and the string tension $T→$ as sketched in Fig. 4(a). We assume that the string is tensed and that the ball performs a circular motion of radius $Rt$ with rotation speed $Ω$⁠.The inertial force due to the acceleration writes $F→in=(m+madd)RtΩ2e→r$ where $madd$ is the added mass of fluid that needs to be pushed by the sphere. In the potential flow approximation, the added mass of a fully immersed sphere can be computed to be exactly $madd=m/2$⁠.²⁹ However, the flow is not irrotational [see Fig. 4(d)] and the presence of a free surface makes the computation of the added mass a very difficult task. In particular, the waves emitted during the ball's motion may significantly contribute to the added mass.^30,31 We have measured experimentally that the waves emitted had a maximum amplitude of typically $A∼0.2$ mm with wavevector $k∼2π/Rs$ (see supplementary material). A plane wave of such amplitude would apply on an object of size $Rs$ a force of typically $Fwave∼1/4(ρg+3γk2)A2Rs$⁠,³² and one can thus show that $Fwave/Fin<10−2$⁠. This justifies that we neglect the impact of the waves in what follows. Nevertheless, the free surface changes the boundary condition compared to infinite fluid, which impacts the exact value of the added mass. In what follows, we will thus write $madd=M̂addm$ and keep $M̂add∼1/2$ as an approximated value for discussion.

The Reynolds number Re typically ranges from 500 to 5000 in our experiments, so that a quadratic law can be used to estimate the friction force $F→v=1/2ρCDŜimπRs2(RtΩ)2e→θ$⁠, where $CD$ is the dimensionless drag parameter¹¹ and $Ŝim$ is the immersed cross-sectional surface of the object in the direction of motion normalized by the total normal surface $πRs2$⁠. For simplicity, we will assume that the immersed volume does not vary with the rotation speed. Experimental pictures taken from the side confirm this hypothesis except in the diverged regime, in which the immersed volume diminishes significantly (see supplementary material). The corresponding data will hence be discarded for the force analysis. We will furthermore only consider the cases were the string is tensed, and therefore discard all the collapsed states as well. An extensive discussion on the data selection is available in SI.

A good candidate of such force is the lift produced by the fluid. The latter arises from the complete solid rotation performed by the sphere during its revolution [see Fig. 4(a)]^13,29 and results from asymmetric flow between the two sides of the sphere. The flow can be visualized by seeding the liquid surface with small particles (pepper grains) while water is made opaque using a bit of powder milk to ensure good contrast. A camera placed above the tank recorded 150 frames/s during a full revolution of the ball, ensuring small displacement of the ball and the particles between two images. A stack of 15 consecutive images is shown in Fig. 4(d) and in Fig. 5 (multimedia available online). The same stack in the co-moving frame shows the streamlines bending by the ball. From this picture, one can qualitatively expect a lift force directed toward the inward of the trajectory as drawn in Fig. 4(d).

There exists exact formula to compute the lift force at low Reynolds numbers¹³ or in the case or irrotational flow through Kutta-Jukowski theorem.²⁹ In our case, the Reynolds number is large but not infinite (⁠ $Re∼103$⁠) and the flow is strongly rotational on an area comparable to the ball's surface, as shown by the large vortex structure of Figs. 4(d) and 5. An analytical computation of the lift force is thus difficult to perform, but we nevertheless estimate its magnitude through experimental measurement of the flow. We reconstruct the velocity field at the water surface by particle image velocimetry (PIV) with the free software PIVLab.^33,34 A typical result of velocity reconstruction in the comoving frame is shown in Fig. 4(e) with the norm of the velocity superimposed in the background. The fluid's velocity is higher in the region closer to the rotation center (inner part) compared to the external region. We estimate the mean inner and outer velocity $Vin$ and $Vout$ by averaging over the corresponding areas [see Fig. 4(e)]. Both values are very sensitive to the frame change that is performed numerically, but their difference $ΔV=Vin−Vout$ is insensitive to it. It is plotted as a function of the rotation speed $Ω$ in Fig. 4(f) and is compatible with a linear fit $ΔV=2ReffΩ$ with $Reff=0.45$ cm. The errorbars are obtained by computing the standard deviation over 20 consecutive PIV-frames and show low dispersion of the measurement along time.

Assuming as before that $madd=m/2$⁠, we get $ML=1.05×m$⁠, which is in fair agreement with the scaling law found above that gives $ML∼0.75×m$⁠. Our angle and flow measurement thus offer two independent and consistent estimate of the lift's magnitude. A full quantitative measurement of the lift force would require to obtain the full pressure and velocity field around the sphere, as well as to take into account possible viscous effects occurring near the ball in the boundary layer. Such study is beyond the scope of this article, but it is interesting to compare our result with other lift force measurement. For three-dimensional flows and high Reynolds numbers, the lift force acting on a rotating sphere of velocity V is often set in the form $FL=1/2CLρSimVb2$ with $CL$ the lift coefficient that depends on spin factor $RsΩ/Vb$⁠.³⁵ An approximate expression for the lift coefficient is $CL≈2CDRsΩ/Vball$⁠.³⁵ We can, thus, write $FL=CDρŜimπRs3RtΩ2$⁠, and we recover the same scaling law as in Eq. (3). The prefactor are, moreover, of the same order of magnitude since $CD∼0.5$⁠, but a quantitative comparison is not accessible despite our two independent measurement techniques. Indeed, the measurement from the angles requires an exact value of the added mass $madd$ to extract the lift mass $ML$⁠, while a flow measurement over the whole sphere is not accessible for direct measurement of the lift. Moreover, the presence of an interface may affect the expression of the lift force compared to the fully immersed case. It is, therefore, not obvious whether the previous expression would hold or not beyond the order of magnitude that we have performed.

For completeness of the force analysis, we would finally like to mention that Eq. (4) predicts the dependency of the proportionality constant with respect to the characteristics of the ball, which essentially scales as $α∼1/Rs$⁠. A complete discussion and additional experimental data for different ball's radius were performed to probe the found scaling law. The results are presented in supplementary material and are in fair agreement with the model.

So far, we have shown that our experimental measurement were consistent with the force analysis that includes the lift force induced by the fluid. Nevertheless, this is not enough to predict the trajectories radius $Rt$ shown in Fig. 3(e) as the force equilibrium admits an infinite number of solutions $(Rt,ϕ)$⁠. The geometry of the system will provide an extra relationship between those two quantities, which will finally fully determine the equilibrium state.

The geometrical constrain (5) is plotted in Fig. 6(d) for three horizontal length $Lh=5,5.4$ and 5.8 cm (plain colored lines). We plot on the same graph the force equilibrium from Eq. (4) (black dashed line). The intersections between the two curves graphically determines the solutions (⁠ $Rt,ϕ)$ that satisfy both force equilibrium and geometrical constrains for a given horizontal string length. Two solutions exist for $Lh=5$ cm and 5.4 cm, while no solution exists for $Lh=5.8$ cm. Among the two solutions, we show in supplementary material that only the largest one is linearly stable with respect to small radius perturbation. This solution is materialized by orange circles in Fig. 6(d), and corresponds to the radius effectively observed for a given horizontal string length. The latter increases with $Ω$⁠, which displaces the equilibrium radius $Rt$ as shown in Fig. 6(d) (orange arrow). In particular, one expects the radius to decrease as the rotation speed increases, which is consistent with our experimental results.

Such decreasing behavior is confirmed in Fig. 6(e), where we plot the expected radius $Rt$ a function of the horizontal length $Lh$⁠. When the string's horizontal length is larger than a critical value $Lhc≈5.5$ cm, no more stable solutions can be found [Figs. 6(c) and 6(d)]. This is in good agreement with our experimental results of Fig. 3(e), in which collapsed states occur for $Ls>5$ cm. In this case, the ball's equilibrium and the string's tension cannot be guaranteed simultaneously, which leads to a shape change of the strand. The critical horizontal length is reached for smaller rotation rates when the total length of the string $Ls$ is greater. The model, therefore, predicts that longer strings should collapse at lower rotation rates. Moreover, the smallest radius observed before collapse is approximately 2 cm and independent of $Lh$ [dashed line in Fig. 6(d)]. All those features are in quantitative agreement with our experimental results of Fig. 3(e).

Our model also sheds light on the existence of diverged states for shorter string length. The elongation eventually saturates at large rotation speed, so that a collapsed state cannot be reached if the string is too short $Lhmax<Lhc$⁠. At the bifurcation toward diverging state, we observed that the immersed volume of the ball suddenly decreases (see supplementary material). This is consistent with our force analysis and Eq. (1): a decrease in the immersed volume decreases the angle $ϕ$⁠, which corresponds to increase the rotation radius $Rt$ as the rope aligns with $e→r$ [see Fig. 4(a)]. Similarly, a decrease in the immersed volume tends to bring the dashed curve in Fig. 6(b) closer from zero, which tends to increase the equilibrium radius. The reason for such brutal variation of the immersed volume, at the heart of the sudden radius jump, is nevertheless still to be understood.

We have shown that the combination of force analysis on the ball and changing geometry of the string offers a consistent description of our experimental results. The apparent flexibility of the string in the horizontal plane, induced by its three-dimensional spatial reorganization, is at the heart of all the observed results. We, therefore, anticipate that the transition between rotating and collapsed trajectories described here is a rather general process, while the thresholds for their appearance depend on the chosen experimental configuration. We have already observed qualitatively the same three “rotating,” “collapsing,” and “diverging” regimes for different ball's size and rotation arm's length, but a more systematic study would be required to probe quantitative agreement with our model.

We now aim to discuss the stability of the collapsed state. Starting from the collapsed state obtained for $Ls=5$ cm and $Ω>Ωu=2.6$ round/s [blue circles in Fig. 7(a)], the rotation speed is progressively decreased (red squares). Interestingly, the ball does not reach its previous rotating state but rather remains trapped at the rotation center. Such bistability occurs on a large range of rotation speed and eventually breaks down for $Ω<Ωd=1.2$ round/s where circular trajectories appear again. The rotation range $[Ωd,Ωu]$ for which the hysteresis occurs strongly depends on the string length and rapidly decreases as the latter increases, as shown in Fig. 7(c). In comparison, no hysteresis was observed in the diverged cases as shown in Fig. 7(b).

The existence of hysteresis and memory effects is ubiquitous in fluid mechanics and occurs, for instance, in Basset forces on accelerating objects,³⁶ in the motion of bouncing droplets³⁷ or more generally in subcritical instabilities.^38,39 In the present case, the memory can be qualitatively interpreted in two ways. First, the tension lost in the string needs to be recovered to expel the ball from the flow's center. Second, the surrounding flow exhibits circular symmetry that needs to be broken to recover circular trajectories. Both features require a strong perturbation to be broken, leading to a robust hysteresis.

Contrary to the diverging/collapsing phenomena, which were observed in all the tested configurations, the occurrence and range of hysteresis strongly depend on the peculiar experimental details. For instance, no hysteresis was observed when the guiding tube was set 1.5 cm above the water surface. In this case, the collapsed state presents some intermittency instead of regular spinning. The ball alternates between no motion, during which the string wraps around it, and sudden fast rotation where of the tension accumulated in the string releases. Those “stop-and-go” trajectory provoke a lot of hysteresis, as the ball is sufficiently perturbed to reach back its rotating state. This effect is believed to come from the vertical component of the string's tension. It is consistent with the large hysteresis observed when the guiding tube is brought very close (3 mm) from the water surface to ensure horizontal pulling. We show in supplementary material that in this case, very large hysteresis cycles could be observed with $Ωu=2$ round/s and $Ωd<0.3$ round/s.

We discuss in this last section the secondary “tea-leaf” flows induced by the motion of the sphere. To get closer from the original version of the tea-leaf paradox occurring in a cup, we used a circular tank of 13 cm diameter filled with 4.5 cm of water. We introduce infused tea leaves in the water, which sink typically within a couple of seconds before settling at the bottom of the tank. A camera is placed above the tank while a LED panel is set below, so that the tea leaves appear as dark areas [see Figs. 8(a) and 8(b)]. This allows to monitor their spatial repartition by simple contrast analysis. In each experiment, the liquid is initially at rest and the tea leaves are uniformly spread at the bottom. We then start the ball's rotation at $Ω$ and record a movie from the top during one minute. The pictures in Figs. 8(a) and 8(b) show the initial and final state for $Ω=1.2$ round/s and $Ls=5.5$ cm. The leaves clearly gathered at the rotation's center, showing that the tea-leaf paradox also occurs with our “deformable spoon.” As mentioned in the introduction, the migration is caused by the secondary flow induced by the boundary layers between the rotating bulk and the nonmoving edges of the tank.² Our results show that adding some flexibility on the stirrer does not qualitatively impact the flow generation.

However, the deformation of the spoon can strongly impact the efficiency of those secondary flows and so particle's transport. In particular, the leave's transport is suppressed when the ball reaches a collapsed state. To prove this, we first compute the leave's spatial distribution at a given time along the horizontal direction by performing the average of the pixel's value over a small vertical window, as shown in Fig. 8(a). The resulting curves display either uniform [Fig. 8(a)] or plateau-like [Fig. 8(b)] distribution, corresponding respectively to uniformly spread or gathered tea leaves. Those two configurations can be easily distinguished by computing the average value p on the central area materialized by the green area in Figs. 8(a) and 8(b). Uniform spreading corresponds to $p∼100–150$ (equal mix of white and black pixels) while gathered leaves correspond to $p∼0$ (completely black pixels). The value of $p(t)$ along time is thus a good indicator of the leave's transport due to secondary flow. It is plotted in Fig. 8(c) for different rotation speeds. For the lower rotation speed $Ω=0.6$ round/s, $p(t)$ is approximately constant which indicates that no transport occurred during the acquisition. This suggests that the secondary flows are too weak to transport the leaves. For higher rotation speeds $Ω=0.9–1.8$ round/s, the ball performs circular trajectories and we clearly observe the gathering of the leaves after some delay. Interestingly, this delay seems to decrease as the rotation speed increases, which shows an increase in the mixing efficiency with the rotation speed. The same trend was observed for another dataset generated for $Ls=4.5$ cm, but more data would be needed to perform statistical analysis.

Interestingly, a further increase in the rotation speed to $Ω=2.1$ rad/s completely suppresses the tea-leaf transport. This loss of the tea-leaf paradox is concomitant with the appearance of a collapsed state, leading to sudden decrease in the mixing area. We have let the system evolve in this state for tens of minutes without observing any noticeable migration of the leaves. This suggests that the secondary flow is, as for the lowest rotation case, too weak to perform transport. Those results clearly show that in the presence of a flexible stirrer, increasing the rotation speed does not always lead to higher mixing efficiency. A sudden drop of the mixing area can indeed occur when the stirrer reaches a collapsed state, which significantly reduces the momentum transfer between the fluid and the sphere.

We have discussed in this work the circular motion of a ball pulled by a string and the induced flows. The unexpected radius shrinking when the rotation speed increases results from an equilibrium between the string's deformation and flow's action. Our deformable system allowed to measure the ratio between the lift and the added mass for a sphere moving circularly at the water surface, and the obtained results were consistent with direct flow measurement. At higher rotation speed, we have shown the existence of bifurcation and robust self-trapped states. Interestingly, those trapped states were shown to strongly decrease the mixing efficiency compared to rotating state at lower rotation speed.

The results presented in this work open several interesting avenues for the future. It would be interesting to perform the same experiment for other size or body shapes, as well as to study analytically the impact of the free surface on the lift force expression. It also strongly suggests to perform the same experiments with several objects to introduce pairwise interactions mediated by the flow, as it was done for rotating magnetic disks.¹⁶ Last, one may replace water with a non-Newtonian fluid, which were, for instance, shown to modify swimmers efficiency,⁴⁰ or replace the inextensible rope by a flexible one to introduce flow-elasticity coupling. The impact of those modifications on the object's trajectories and induced mixing is expected to exhibit rich phenomenological features.

See the supplementary material for this article include details on the experimental setup, details on data selection for the force analysis, detailed computation on the stability of equilibrium position, and additional experimental data with different balls.

The authors acknowledge J.B. Gorce for fruitful discussions, M. Mallejac for his help with Blender, and G. Noetinger for precious feedbacks.

The authors have no conflicts to disclose.

Benjamin Apffel: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Writing – original draft (lead). Romain Fleury: Funding acquisition (lead); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.