2 Equilibrium position of a sphere in a rotating viscous flow

Figure 1. Schematic of an end view of the flow domain. A cylinder of fluid (outer circle) rotates about its axis $O$ at angular velocity $\unicode[STIX]{x1D6FA}$ , and a buoyant particle of radius $a$ (inner filled circle) is held at an equilibrium position by a balance of Stokes drag $\boldsymbol{F}_{D}$ , centripetal forces $\boldsymbol{F}_{C}$ , buoyancy $\boldsymbol{F}_{B}$ and lift $\boldsymbol{F}_{L}$ .

In this section, a model is described for the motion of a buoyant sphere within a cylinder rotating about its axis, which is perpendicular to gravity. This model follows the results of Candelier (Reference Candelier2008), who derived the history-dependent forces on a sphere in an unbounded rotating flow at low Reynolds number, and showed that these forces correspond to the lift and drag corrections predicted by Gotoh (Reference Gotoh1990) when the sphere is in equilibrium.

The cylinder rotates with angular velocity $\unicode[STIX]{x1D6FA}$ , and is filled with viscous fluid of density $\unicode[STIX]{x1D70C}_{f}$ and dynamic viscosity $\unicode[STIX]{x1D707}$ (figure 1). A buoyant spherical particle of radius $a$ and density $\unicode[STIX]{x1D70C}_{s}$ is immersed in a fluid, and we assume that this sphere is small, and sufficiently far from the walls of the cylinder that hydrodynamic interactions between the particle and the walls of the cylinder can be neglected. Defining a radial coordinate system centred at the axis of the cylinder, with unit azimuthal and radial basis vectors $\boldsymbol{e}_{r}$ and $\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ , the velocity field in the absence of the sphere is solid-body rotation, $\boldsymbol{u}=r\unicode[STIX]{x1D6FA}\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ . The location of the centre of the sphere is denoted by $\boldsymbol{x}$ , with radial coordinates $(r,\unicode[STIX]{x1D703})$ and Cartesian coordinates $(x,y)$ and the acceleration arising from gravity by $\boldsymbol{g}=-g(\boldsymbol{e}_{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D703}+\boldsymbol{e}_{r}\sin \unicode[STIX]{x1D703})$ . The system has three independent dimensionless groups: the density ratio $\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x1D70C}_{f}$ (which takes the value $0.69$ in our experiments), and two Reynolds numbers, based on rotation rate and the settling velocity,

(2.1a,b ) $$\begin{eqnarray}Re=\frac{\unicode[STIX]{x1D70C}_{f}a^{2}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x1D707}}\quad \text{and}\quad Re_{p}=\frac{\unicode[STIX]{x1D70C}_{f}V_{T}a}{\unicode[STIX]{x1D707}},\end{eqnarray}$$

where

(2.2) $$\begin{eqnarray}V_{T}=\frac{2ga^{2}(\unicode[STIX]{x1D70C}_{f}-\unicode[STIX]{x1D70C}_{s})}{9\unicode[STIX]{x1D707}}\end{eqnarray}$$

is the rising speed of the (buoyant) sphere in quiescent fluid. The quantity $Re$ , as defined in (2.1), is often known as the Taylor number (e.g. Childress Reference Childress1964).

The equation of motion for the sphere is

(2.3) $$\begin{eqnarray}m_{p}\frac{\text{d}^{2}\boldsymbol{x}}{\text{d}t^{2}}=m_{p}\boldsymbol{g}+\oint _{dV}\unicode[STIX]{x1D70E}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}S,\end{eqnarray}$$

where $m_{p}=(4/3)\unicode[STIX]{x03C0}a^{3}\unicode[STIX]{x1D70C}_{s}$ is the mass of the spherical particle, $dV$ represents the particle surface, $\boldsymbol{n}$ is a unit outward normal and $\unicode[STIX]{x1D70E}$ is the Cauchy stress in the fluid. At infinitesimal Reynolds number, the contributions to this stress integral for a spherical particle in an unbounded domain can be described by the history, buoyancy, drag, centrifugal and added mass forces (Maxey & Riley Reference Maxey and Riley1983). Away from the Stokes flow limit, Magnaudet (Reference Magnaudet1997) reviews the forces and inertial corrections occurring at low to moderate Reynolds numbers, when the particle is also subject to a lift force. We now evaluate the contribution from each of these forces.

The buoyancy and Stokes drag forces are given by

(2.4) $$\begin{eqnarray}\boldsymbol{F}_{B}=-{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}a^{3}\unicode[STIX]{x1D70C}_{f}\boldsymbol{g}\end{eqnarray}$$

and

(2.5) $$\begin{eqnarray}\boldsymbol{F}_{D}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}ar\unicode[STIX]{x1D6FA}\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

respectively (see e.g. Batchelor Reference Batchelor1967; Maxey & Riley Reference Maxey and Riley1983). If all inertial and acceleration forces can be neglected (namely when $Re\rightarrow 0$ , $Re_{p}\rightarrow 0$ , $\unicode[STIX]{x1D6FA}^{2}R/g\rightarrow 0$ ), then the stress on the sphere results from these two forces alone. In this limit, the radial force balance implies that any equilibrium must lie at the same vertical position as the origin.

The fluid acceleration generated by the rotation of the cylinder results in a radial pressure gradient, and a consequent inward-pointing radial force on the particle, given by

(2.6) $$\begin{eqnarray}\boldsymbol{F}_{C}=-{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}a^{3}\unicode[STIX]{x1D6FA}^{2}r\unicode[STIX]{x1D70C}_{f}\boldsymbol{e}_{r}.\end{eqnarray}$$

When a spherical particle in an unbounded parallel shear flow is subjected to a force, it experiences an additional lift force in the perpendicular direction (Segré & Silberberg Reference Segré and Silberberg1962), which cannot be explained by viscous stresses alone (Bretherton Reference Bretherton1962). For a spherical particle in a shear flow (where a Reynolds number $Re_{s}=\unicode[STIX]{x1D70C}_{f}a^{2}\dot{\unicode[STIX]{x1D6FE}}/\unicode[STIX]{x1D707}$ is defined from the shear rate $\dot{\unicode[STIX]{x1D6FE}}$ ), Saffman (Reference Saffman1965) calculated the strength of this lift force in the low Reynolds number regime $Re_{p}\ll Re_{s}^{1/2}\ll 1$ , showing that it scales with $Re_{s}^{1/2}$ .

Saffman’s calculation of the lift force was subsequently extended by McLaughlin (Reference McLaughlin1991) to the regime in which both Reynolds numbers are much less than one, but where $Re_{p}$ is not necessarily small compared to $Re_{s}^{1/2}$ . Numerical calculation (Dandy & Dwyer Reference Dandy and Dwyer1990) indicates that the expression of Saffman (Reference Saffman1965), formally valid only in the asymptotic regime $Re_{p}\ll Re_{s}^{1/2}\ll 1$ , is nonetheless accurate if $Re_{p}<Re_{s}^{1/2}<1$ (Mei Reference Mei1992).

In the case relevant to our problem, where the background flow is in solid-body rotation rather than parallel shear, Gotoh (Reference Gotoh1990) calculated the leading-order effects of inertia using a similar analysis to that of Saffman (Reference Saffman1965). At small Reynolds number (when $Re_{p}\ll Re^{1/2}\ll 1$ ) and for a particle at rest, Gotoh (Reference Gotoh1990) shows that the leading-order contribution of inertia is at $O(Re^{1/2})$ , and takes the form

(2.7) $$\begin{eqnarray}\boldsymbol{F}_{L}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}ar\unicode[STIX]{x1D6FA}Re^{1/2}[k_{1}\boldsymbol{e}_{\unicode[STIX]{x1D703}}-k_{2}\boldsymbol{e}_{r}],\end{eqnarray}$$

where

(2.8a,b ) $$\begin{eqnarray}k_{1}=3\frac{\sqrt{2}(19+9\sqrt{3})}{280}\approx 0.524\quad \text{and}\quad k_{2}=3\frac{\sqrt{2}(19-9\sqrt{3})}{280}\approx 0.0517.\end{eqnarray}$$

This expression is valid only for particles that are stationary in the laboratory frame (that is, particles that may be rotating, but for which $\text{d}\boldsymbol{x}/\text{d}t=0$ ). The finite- $Re$ adjustment (2.7) comprises both a radial lift component analogous to that found by Saffman (Reference Saffman1965) and an azimuthal component, corresponding to a modification of the Stokes drag law at small finite Reynolds number. We note that the mechanisms of lift and drag studied by Gotoh (Reference Gotoh1990) differ significantly from those that occur at high Reynolds number (Rastello et al. Reference Rastello, Marié, Grosjean and Lance2009; Bluemink et al. Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2010). At low Reynolds number, the lift force on a sphere in a rotating flow is directed inwards, towards the centre of rotation (Gotoh Reference Gotoh1990). Surprisingly, this is the opposite direction to the lift force in a low Reynolds number simple shear flow of the same vorticity. The explanation for this difference, given by Van Nierop et al. (Reference Van Nierop, Luther, Bluemink, Magnaudet, Prosperetti and Lohse2007), is that a sphere in a rotating base flow experiences two lift forces: an outward-directed force similar to the one occurring in simple shear flows (Saffman Reference Saffman1965), and an inward-directed force, slightly larger in magnitude, resulting from curvature of the sphere wake.

Candelier (Reference Candelier2008) showed that the steady lift and drag corrections obtained by Gotoh (Reference Gotoh1990) in fact arise as a special case of the time-dependent history force on the particle, in the case where the particle is stationary. This history force is not restricted to steady states, but describes the forces on a particle moving arbitrarily when $Re_{p}\ll Re^{1/2}\ll 1$ ; for example, the lift and drag in the ‘centrifuging’ regime first calculated by Herron, Davis & Bretherton (Reference Herron, Davis and Bretherton1975) and the Basset–Boussinesq–Oseen force (Basset Reference Basset1888) are also captured by the history force of Candelier (Reference Candelier2008) in the appropriate regimes of particle motion. The expression for this history force is

(2.9) $$\begin{eqnarray}\boldsymbol{F}_{H}=-6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}a\sqrt{Re}\left(\int _{-\infty }^{t}K_{1}\left(\unicode[STIX]{x1D6FA}(t-\unicode[STIX]{x1D70F})\right)\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{v}_{s}}{\text{d}t}\,\text{d}\unicode[STIX]{x1D70F}+\int _{-\infty }^{t}K_{2}\left(\unicode[STIX]{x1D6FA}(t-\unicode[STIX]{x1D70F})\right)\boldsymbol{\cdot }\boldsymbol{v}_{s}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\right),\end{eqnarray}$$

where $\boldsymbol{v}_{s}=\text{d}\boldsymbol{x}/\text{d}t-\boldsymbol{u}(\boldsymbol{x})$ is the slip velocity,

(2.10a,b ) $$\begin{eqnarray}K_{1}(t)=\left(\begin{array}{@{}cc@{}}f_{1}(t) & -g_{1}(t)\\ g_{1}(t) & f_{1}(t)\end{array}\right)\cdot P(t),\quad K_{2}(t)=\unicode[STIX]{x1D6FA}\left(\begin{array}{@{}cc@{}}g_{1}(t)+f_{2}(t) & f_{1}(t)-g_{2}(t)\\ -f_{1}(t)+g_{2}(t) & g_{1}(t)+f_{2}(t)\end{array}\right)\cdot P(t),\end{eqnarray}$$

(2.11a-c ) $$\begin{eqnarray}P(t)=\left(\begin{array}{@{}cc@{}}\cos (t) & -\text{sin}(t)\\ \sin (t) & \cos (t)\end{array}\right),\quad f_{2}(t)=\frac{10}{7}\frac{\sin (2t)}{\sqrt{\unicode[STIX]{x03C0}t}},\quad g_{2}(t)=\frac{6}{5}\frac{\cos (2t)}{\sqrt{\unicode[STIX]{x03C0}t}},\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle f_{1}(t) & = & \displaystyle \frac{1}{112}\left(\frac{80\cos (2t)t^{3}+20\sin (2t)t^{2}+6\cos (2t)t-3\sin (2t)}{t^{2}\sqrt{\unicode[STIX]{x03C0}t}}\right),\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle g_{1}(t) & = & \displaystyle -\frac{3}{40}\left(\frac{8\sin (2t)t^{2}-2\cos (2t)t+\sin (2t)}{t^{3}\sqrt{\unicode[STIX]{x03C0}t}}\right).\end{eqnarray}$$

With the contributions to the surface integral in (2.3) calculated in (2.4), (2.5), (2.6) and (2.9), the equation of motion (2.3) then reads

(2.14) $$\begin{eqnarray}m_{p}\frac{\text{d}^{2}\boldsymbol{x}}{\text{d}t^{2}}=m_{p}\boldsymbol{g}+\boldsymbol{F}_{B}+\boldsymbol{F}_{C}+\boldsymbol{F}_{D}+\boldsymbol{F}_{H}.\end{eqnarray}$$

We integrate (2.14) in time numerically using a second-order implicit backward differentiation formula (BDF2). The integrals in (2.9) are evaluated using a second-order trapezoidal rule, modified to account for the $1/\sqrt{t}$ singularity in the integrand. Since the history forces (2.9) involve the complete history of motion of the sphere, we must specify this motion; here we assume that the sphere is at rest for $t<0$ , and starts to move at $t=0$ when integration of the governing equations begins. We use the result that the history force reduces to the lift force of Gotoh (Reference Gotoh1990) for a sphere at rest (i.e. $\boldsymbol{F}_{H}(t^{\prime })=\boldsymbol{F}_{L}(t^{\prime })$ if $\text{d}\boldsymbol{x}/\text{d}t=0$ for all $t\leqslant t^{\prime }$ ) to calculate this history integral exactly for $t<0$ , and thus avoid numerical error due to truncation of the integral domain.

We use this time-dependent theory to study the existence and stability of equilibrium points. For a sphere in stationary equilibrium (2.14) reads

(2.15) $$\begin{eqnarray}m_{p}\boldsymbol{g}+\boldsymbol{F}_{B}+\boldsymbol{F}_{C}+\boldsymbol{F}_{D}+\boldsymbol{F}_{L}=0\end{eqnarray}$$

(since $\boldsymbol{F}_{H}$ reduces to $\boldsymbol{F}_{L}$ for a stationary sphere), with azimuthal and radial components

(2.16) $$\begin{eqnarray}\displaystyle \cos \unicode[STIX]{x1D703} & = & \displaystyle -\frac{Re}{Re_{p}}(1+k_{1}Re^{1/2})\frac{r}{a}\quad \text{and}\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle \sin \unicode[STIX]{x1D703} & = & \displaystyle \frac{\unicode[STIX]{x1D6FA}^{2}r}{g}+k_{2}Re^{1/2}\frac{\unicode[STIX]{x1D6FA}r}{V_{T}}=\frac{Re^{3/2}}{Re_{p}}\left[\frac{1}{3}\sqrt{Re}+k_{2}\right]\frac{r}{a}.\end{eqnarray}$$

Solving for $r$ and $\unicode[STIX]{x1D703}$ , we find the equilibrium position of the sphere to be

(2.18) $$\begin{eqnarray}\displaystyle x & = & \displaystyle r\cos \unicode[STIX]{x1D703}=-a\frac{Re_{p}}{Re}\frac{1+k_{1}Re^{1/2}}{(1+k_{1}Re^{1/2})^{2}+Re\left(k_{2}+\frac{1}{3}Re^{1/2}\right)^{2}},\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle y & = & \displaystyle r\sin \unicode[STIX]{x1D703}=a\frac{Re_{p}}{Re^{1/2}}\frac{{\textstyle \frac{1}{3}}Re^{1/2}+k_{2}}{(1+k_{1}Re^{1/2})^{2}+Re(k_{2}+{\textstyle \frac{1}{3}}Re^{1/2})^{2}}.\end{eqnarray}$$

Since $y>0$ , the equilibrium position for a buoyant sphere lies above the origin. This arises from the requirement that the radial component of the buoyancy force $F_{B}$ is directed outwards, to balance the inward-directed lift and centrifugal forces. This differs from the prediction $y<0$ obtained from models where rotating background flow is approximated as a simple shear flow, where the Saffman (Reference Saffman1965) expression for the perpendicular component of lift is used (e.g. Coimbra & Kobayashi Reference Coimbra and Kobayashi2002; Ramirez et al. Reference Ramirez, Lim, Coimbra and Kobayashi2003). We note that the vertical displacement of the equilibrium position is much smaller than the horizontal displacement (by an order of $Re^{1/2}$ ), and so the equilibrium points lie close to the horizontal diameter of the cylinder on a radius, which for our experimental parameters is ${\lesssim}3^{\circ }$ above the central plane. The stability of these equilibrium points is discussed in § 4.

3 Experiment

3.1 Experimental set-up and procedure

Figure 2. Schematic diagram of the experiment. A glass cylinder was mounted horizontally between bearings and completely filled with glycerol. A light sphere was placed inside it and the cylinder was rotated around its axis using a motor, a gear box and a smooth belt. A shaft encoder was used to measure the frequency of rotation of the cylinder.

A schematic diagram of the apparatus is given in figure 2. The apparatus comprised a precision-bored rig glass cylindrical drum of length $225.000\pm 0.005~\text{mm}$ , inner diameter $120.000\pm 0.005~\text{mm}$ and with a wall thickness of $5.200\pm 0.005~\text{mm}$ . The cylinder was completely filled with glycerol of density $\unicode[STIX]{x1D70C}_{g}=1.261~\text{g}~\text{cm}^{-3}$ and viscosity $\unicode[STIX]{x1D708}=1100\pm 0.3~\text{mm}^{2}~\text{s}^{-1}$ (measured using an Ubbelohde suspended level viscometer). The experiments were performed in a temperature controlled room with a measured air temperature of $20\pm 0.5\,^{\circ }\text{C}$ . This gave rise to an estimated temperature variation of $\pm 0.2\,^{\circ }\text{C}$ in the glycerol.

The spheres used were polypropylene, of density $\unicode[STIX]{x1D70C}_{s}=0.87\pm 0.02~\text{g}~\text{cm}^{-3}$ and radius $a=3.15,4.76,6.32,7.05,7.90,9.50~\text{mm}\;(\pm 0.005~\text{mm})$ .

The cylinder was mounted on bearings on a machined steel platform with three adjustable legs, which were used to level the cylinder using an engineer’s spirit level. Careful levelling of the system was required in order to minimise buoyancy-induced drift of the sphere along the axis of the cylinder. The cylinder was rotated using a DC motor with feedback control via a 10:1 gear box, which was connected to the cylinder by a smooth belt. The speed of rotation was controlled by a commercial servo control and the rotation frequency was measured using an optical encoder, attached to the shaft of the motor, which produced 500 pulses per revolution. The pulse count was monitored using a universal counter and this was used to calibrate the rotation rate of the cylinder.

The angular velocity of the cylinder, $\unicode[STIX]{x1D6FA}=2\unicode[STIX]{x03C0}f$ , was measured by the encoder to an accuracy of 0.01 %. The maximum angular velocity used in this experiment was $\unicode[STIX]{x1D6FA}\approx 25~\text{rad}~\text{s}^{-1}$ giving a maximum Reynolds number $Re=a^{2}\unicode[STIX]{x1D6FA}/\unicode[STIX]{x1D708}\approx 2$ for the largest sphere. However, the majority of the experiments were performed with $0.05<Re<0.7$ .

The inside of the cylinder was polished and ground, as was a 1 cm band on the outside at the ends. The end caps were machined together to ensure that, when mounted, any misalignment of the ends was minimal. The excellent agreement between the experimental results and the theory for the fixed points suggests that any secondary flows were weak.

Two time scales were considered when taking into account how long it would take the fluid to achieve solid-body rotation after the rotational velocity were modified. As described by Greenspan & Howard (Reference Greenspan and Howard1963), the viscous diffusion time is $T_{d}=R^{2}/\unicode[STIX]{x1D707}\approx 3.2~\text{s}$ , and the spin-up time is $T_{s}=R/\sqrt{\unicode[STIX]{x1D707}\,\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6FA}}\approx 1.8~\text{s}$ , for a change in rotation rate $\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6FA}$ . After a change of rotation rate, we waited a time much longer than $T_{d}$ and $T_{s}$ (at least two minutes) before making any measurements, to ensure that any transients had decayed and the fluid inside the cylinder was in solid-body rotation.

When the cylinder was filled with glycerol, air also entered the fluid creating bubbles. The presence of an air bubble of comparable size to the sphere produced unwanted complications in the dynamics, analogous to those found when there is more than one spherical particle in the flow; see Mullin et al. (Reference Mullin, Li, Del Pino and Ashmore2005). Hence, before each set of experiments, bubbles were removed by first rotating the cylinder at high speed so that the bubbles merged on the axis of the cylinder. The larger bubble which formed was removed by injecting degassed fluid into one of two sealable holes drilled into the end plate of the cylinder.

The cylinder was front-illuminated using two light sources, and the back lid of the cylinder was painted white to improve the contrast between the background and the dark green polypropylene sphere. Images were taken using a fast digital camera (Sony XCD-X710) and the position of the sphere was determined by a dedicated image analysis MATLAB routine. The contrast between the sphere and the white background was used to identify the position of the centre of the sphere.

5 Discussion and conclusions

We have investigated the motion of a light sphere immersed in a rotating viscous flow at low Reynolds numbers. Our experimental results show a set of stable fixed points which are eccentric to the axis of the cylinder over a range of $Re$ . The balance of forces required for stability is explained using the buoyancy force arguments of Gotoh (Reference Gotoh1990), Magnaudet (Reference Magnaudet1997) and Candelier (Reference Candelier2008). We present results of a model for the equilibrium position of the sphere, based on these arguments, and the predictions of the equilibrium points are in excellent accord with the experiments.

We also found experimentally that these equilibrium positions become unstable to oscillatory motion below a critical value $Re_{c}$ and the sphere follows circular orbits. The centre of the orbit is located on the fixed point predicted by the theory, but the existence of a stable orbit is not predicted by the theory, which instead predicts a stable fixed point. As $Re$ decreases, the radius of the experimentally observed orbit scales initially as $Re^{-2}$ , which is slower than the growth following a typical Hopf bifurcation.

Two clear possibilities exist for the difference between the stability predictions of the theory and the experimental observations. Firstly, the effect of the outer walls of the drum are neglected in the theory, but become significant in the experiments as $Re$ is decreased and the predicted equilibrium position moves away from the axis of rotation towards the drum wall. At these lower values of $Re$ , the experimentally observed orbits become elliptical and decrease in size, until the sphere restabilises at a stable equilibrium point adjacent to the cylinder wall. This change in stability occurs via a standard Hopf bifurcation, as reported previously for heavy spheres (Mullin et al. Reference Mullin, Li, Del Pino and Ashmore2005). It is possible that the orbits we observe are caused by wall effects destabilising even the fixed points very close to the axis of rotation. To test this, experiments were performed in which a 100 mm long Plexiglas cylinder of radius 3 mm glued to the inner wall of the cylinder. No change to the orbits was observed, even with this large perturbation, suggesting that wall effects (and indeed any small geometrical imperfections in the experiment) are not the cause of the observed orbits.

Secondly, the theory is formally valid only in the asymptotic limit $Re_{p}\ll Re^{1/2}\ll 1$ . In our experiments, $Re^{1/2}<1$ is satisfied always, and $Re_{p}<Re^{1/2}$ is satisfied for all but three measurements of the spheres of 9.5 mm radius. We observe orbits only for smaller values of $Re$ , which suggests that while $Re^{1/2}\ll 1$ may be satisfied, finite- $Re_{p}$ effect not included in the model may act to destabilise the equilibrium points as $Re_{p}$ approaches $Re^{1/2}$ .

Notably, the observed dependence of orbit radius on $Re$ is not the standard $R\sim |Re-Re_{c}|^{1/2}$ expected of a Hopf bifurcation, and further work is required to explain the mechanism of instability and the origins of this behaviour. A full numerical simulation of the system may shed light on these issues, but it remains a challenging problem to resolve, in three dimensions, the subtle balance of stabilising and destabilising forces acting on the sphere at finite Reynolds number.